# Effective chaos for the Kirchhoff equation on tori

Pietro Baldi, Filippo Giuliani, Marcel Guardia, Emanuele Haus

###### Abstract

We consider the Kirchhoff equation on tori of any dimension and we construct
solutions whose Sobolev norms oscillates in a chaotic way on certain long time
scales. The chaoticity is encoded in the time between oscillations of the
norm, which can be chosen in any prescribed way. This phenomenon, that we name
as _effective chaos_ (it occurs over a long, but finite, time scale), is
consequence of the existence of symbolic dynamics for an effective system.
Since the first order resonant dynamics has been proved to be essentially
stable, we need to perform a second order analysis to find an effective model
displaying chaotic dynamics. More precisely, after some reductions, this model
behaves as two weakly coupled pendulums.

###### Contents

  1. 1 Introduction and main result
    1. 1.1 Main result
    2. 1.2 Main ideas of the proof
  2. 2 Effective dynamics for the Kirchhoff equation
    1. 2.1 A quasilinear partial normal form
    2. 2.2 The effective system
    3. 2.3 The truncated effective system
  3. 3 The truncated effective system with two triplets
    1. 3.1 Meaningfulness condition for the solutions
    2. 3.2 First integrals and a linear change of coordinates
    3. 3.3 The Hamiltonian structure
    4. 3.4 Normalization of coefficients by rescaling
    5. 3.5 Large Fourier frequency as a perturbation parameter
  4. 4 Chaos for two weakly coupled pendulums
    1. 4.1 Partially hyperbolic periodic orbit
    2. 4.2 Transverse intersection of invariant manifolds
    3. 4.3 Symbolic dynamics
  5. 5 Back to the truncated effective system
    1. 5.1 Positivity of the superactions
    2. 5.2 Initial data in the normal form ball
    3. 5.3 Construction of a compatible initial datum
    4. 5.4 Joining the two amplitude parameters
  6. 6 Approximation argument
    1. 6.1 Motion of the Sobolev norms
    2. 6.2 Back to the solutions of the Kirchhoff equation

##  1 Introduction and main result

We consider the Kirchhoff equation

$\partial_{tt}u-\Delta u\Big{(}1+\int_{{\mathbb{T}}^{d}}|\nabla
u|^{2}\,dx\Big{)}=0$ (1.1)

on the torus ${\mathbb{T}}^{d}$,
${\mathbb{T}}:={\mathbb{R}}/2\pi{\mathbb{Z}}$, in any dimension $d\geq 1$
(periodic boundary conditions), where the unknown $u=u(t,x)$,
$x\in{\mathbb{T}}^{d}$, is a real-valued function.

Equation (1.1) was first introduced by Kirchhoff [26] in 1876, to model
nonlinear transverse oscillations of strings and plates ($d=1,2$). It is a
quasilinear wave equation, with cubic, nonlocal nonlinearity and Hamiltonian
structure. Given its physical relevance, equation (1.1) has been largely
studied along the years; nonetheless, its study is still challenging, because
several basic questions remain open.

While it has long been known (Dickey [12], Arosio-Panizzi [1]) that the Cauchy
problem for (1.1) is locally wellposed with initial data
$(u(0),\partial_{t}u(0))$ in the Sobolev space
$H^{\frac{3}{2}}({\mathbb{T}}^{d},{\mathbb{R}})\times
H^{\frac{1}{2}}({\mathbb{T}}^{d},{\mathbb{R}})$, it is still an open problem
whether the solutions with initial data of any given Sobolev regularity are
global in time or not. In particular, it is not even known if $C^{\infty}$ (or
even Gevrey) initial data of small amplitude produce solutions that are global
in time. For initial data in analytic class, instead, global wellposedness is
known since the work of Bernstein [6] in 1940.

Moreover, below the regularity threshold $H^{\frac{3}{2}}\times
H^{\frac{1}{2}}$, neither local wellposedness nor illposedness have been
established. A partial, interesting result in this direction has been recently
obtained by Ghisi and Gobbino [13].

More general questions regard the lifespan of the solutions and their behavior
as time evolves, at least close to the equilibrium $u=0$. First of all, as a
consequence of the linear theory, for initial data of size $\varepsilon$ in
$H^{\frac{3}{2}}\times H^{\frac{1}{2}}$, the existence of the solution is
guaranteed at least for a time of the order $\varepsilon^{-2}$. Since (1.1) is
a quasilinear equation, it is not a priori obvious that one can obtain better
estimates. For instance, in the well-known example by Klainerman and Majda
[27] all nontrivial space-periodic solutions of size $\varepsilon$ blow up in
a time of order $\varepsilon^{-2}$. On the other hand, in the papers [3], [4],
[5], using techniques from the normal form theory, it is proved that for the
Kirchhoff equation the situation is more favorable. More precisely, in [3],
performing one step of quasilinear normal form, it is proved that the lifespan
of all solutions of small amplitude is at least of order $\varepsilon^{-4}$.
This is a consequence of the fact that the only resonant cubic terms that
cannot be erased in the first step of normal form give no contribution to the
energy estimates. In [4] the second step of quasilinear normal form is
computed, and it is proved that there are resonant terms of degree five that
cannot be erased and that give a nontrivial contribution to the time evolution
of Sobolev norms.

This is a starting point for describing interesting long-time dynamics for the
Kirchhoff equation. The qualitative behavior of solutions of the Kirchhoff
equation over long-time scales is poorly understood, even for small, compactly
Fourier supported initial data, which obey to finite dimensional systems.

Broadly speaking, for the dynamics of small data we can look for two different
types of regimes:

  * •

Stable regime: this is the case in which the long-time behavior of Fourier
modes resembles the dynamics of the linearized equation, namely the energy of
the modes remains almost constant over long time. We mention [5] where stable
motions of equation (1.1) are obtained for a suitable set of nonresonant
initial data, for which the effect of the resonant terms of degree five remain
small on a longer timescale of order $\varepsilon^{-6}$. We also mention [2]
and [11], where the existence of invariant tori is proved for a forced version
of (1.1).

  * •

Unstable regime: in this case the nonlinear terms lead to a new type of
dynamics, very different from the linear one. Of particular interest is
understanding how the nonlinear effects create exchanges of energy among
different modes.

Concerning the unstable regime, some remarkable results in literature regard
the “energy cascade” for nonlinear Schrödinger equations, where the energy
travels from low to high modes (or vice versa), in strong connection with the
weak turbulence theory. Such phenomenon, which can be measured in terms of an
arbitrarily large growth of Sobolev norms, was considered by Bourgain one of
the most important problems in Hamiltonian PDEs, see [8], and also [10, 19,
20, 21, 22].

In the unstable regime, other interesting dynamical behaviors are also based
on the mechanism of energy exchange among Fourier modes. Such exchanges can be
recurrent (i.e., periodic or quasi-periodic in time) or chaotic. Recurrent
energy exchanges are obtained, for instance, in [15, 16, 17, 23, 24]. To the
best of our knowledge, the only paper in literature in which chaotic exchanges
of energy are constructed for PDEs is [14]. In [14] the authors consider cubic
wave and beam equations and prove the existence of solutions essentially
Fourier supported on a finite number of resonant modes that exchange energy
among themselves in a chaotic way. The chaoticity reflects in the fact that it
is possible to provide energy exchanges among modes at a sequence of
prescribed times (randomness of exchanging time) or among modes belonging to a
prescribed resonant tuple (randomness of active and inactive modes).

Both in [14] and in the present paper the existence of chaotic solutions is
due to the presence of chaotic dynamics for the normal form of the equation,
up to a certain degree. More precisely, the normalized system leaves invariant
a finite dimensional subspace; then the chaotic behavior arises from the
existence of a Smale horseshoe, which gives rise to symbolic dynamics. The
orbits of the normalized system are globally defined in time, and the chaotic
behavior is displayed for an infinitely long time. However, this does not
imply the existence of chaotic solutions of the full PDE for an infinitely
long time. The chaotic behavior for the full PDE is obtained by proving the
vicinity of certain solutions of it to the chaotic orbits of the normalized
system, and this approximation only holds over a long, but finite, time
interval. We call this behavior _effective chaoticity_ , in the sense that the
dynamics behaves as chaotic in rather long time scales (in analogy to the
stability over long time scales, often called _effective stability_ in
Hamiltonian dynamics).

###  1.1 Main result

We denote by ${\mathbb{N}}_{0}$ the set $\\{0,1,2,\ldots\\}$ of nonnegative
integers. The next theorem, which shows the existence of solutions of the
Kirchhoff equation displaying chaotic-like, small amplitude, oscillations in
the Sobolev norms, is the main result of the paper.

######  Theorem 1.1.

There exist universal positive constants
$M,\tau,\varepsilon_{*},C,r_{0},b,K,K_{0}$ with the following property. Let
$d\geq 1$. For every sequence
$(m_{j})_{j\in{\mathbb{N}}_{0}}=(m_{0},m_{1},m_{2},\ldots)$ of integers such
that $m_{j}\geq M$ for all $j\in{\mathbb{N}}_{0}$, there exists a sequence

$0=s_{0}<\bar{s}_{0}<s_{1}<\bar{s}_{1}<s_{2}\leq\bar{s}_{2}<\ldots,\quad\
s_{j+1}=s_{j}+\tau(m_{j}+\theta_{j}),\quad\ 0\leq\theta_{j}<1,$

such that for every $\varepsilon\in(0,\varepsilon_{*}]$ there exists a
solution $u(t,x)$ of the Kirchhoff equation (1.1) on ${\mathbb{T}}^{d}$,
global in time, with finite Fourier support, whose norm

$\mathcal{N}(t):=\Big{(}\|u(t)\|_{H^{\frac{3}{2}}({\mathbb{T}}^{d})}^{2}+\|\partial_{t}u(t)\|_{H^{\frac{1}{2}}({\mathbb{T}}^{d})}^{2}\Big{)}^{\frac{1}{2}}$

satisfies

$\mathcal{N}(t)\leq C\varepsilon\quad\ \forall t\in{\mathbb{R}}$

and it oscillates around the central value
$A_{\varepsilon}:=\varepsilon+\varepsilon^{2}r_{0}$ with oscillations
described in terms of the amplitude $B_{\varepsilon}:=\varepsilon^{2}r_{0}$
and the error $\delta_{\varepsilon}:=\frac{1}{10}B_{\varepsilon}$ as

$\displaystyle-\delta_{\varepsilon}$
$\displaystyle\leq\mathcal{N}(t)-A_{\varepsilon}\leq
B_{\varepsilon}+\delta_{\varepsilon}\quad$ $\displaystyle\forall t\in
I_{j}=[t_{j},\bar{t}_{j}],\qquad$ $\displaystyle\max_{t\in
I_{j}}\mathcal{N}(t)-A_{\varepsilon}$ $\displaystyle\geq
B_{\varepsilon}-\delta_{\varepsilon},$ $\displaystyle-
B_{\varepsilon}-\delta_{\varepsilon}$
$\displaystyle\leq\mathcal{N}(t)-A_{\varepsilon}\leq\delta_{\varepsilon}\quad$
$\displaystyle\forall t\in E_{j}=[\bar{t}_{j},t_{j+1}],\qquad$
$\displaystyle\min_{t\in E_{j}}\mathcal{N}(t)-A_{\varepsilon}$
$\displaystyle\leq-B_{\varepsilon}+\delta_{\varepsilon},$

where

$t_{j}=\frac{s_{j}}{b\varepsilon^{3}},\quad\
\bar{t}_{j}=\frac{\bar{s}_{j}}{b\varepsilon^{3}},$

for all intervals $I_{j},E_{j}$ contained in the time interval
$[0,T_{\varepsilon}]$, where

$T_{\varepsilon}=K\varepsilon^{-3}\log(\varepsilon^{-1}).$

One has $I_{j},E_{j}\subset[0,T_{\varepsilon}]$ for all $j=0,\ldots,N$, where
the integer $N$ satisfies

$\sum_{j=0}^{N}m_{j}\leq K_{0}\log(\varepsilon^{-1}).$

In other words, Theorem 1.1 says that, around the equilibrium $u=0$, the
Kirchhoff equation possesses solutions whose norm $\mathcal{N}(t)$ exhibits
oscillations that follow any prescribed sequence of times on the time interval
$[0,T_{\varepsilon}]$, and the number $N$ of oscillations within that
interval, or more generally the sum of the time lengths of the oscillations,
is arbitrarily large for $\varepsilon$ small enough. These oscillations can
also be seen as a chaotic-like modulation of a stable motion, meaning that the
oscillating solutions are of size $\varepsilon$, they are
$\varepsilon^{2}$-close to effectively stable solutions (over long time
scales), but they exhibit chaotic-like exchanges of size $\varepsilon^{2}$
between the amplitude of different Fourier modes.

######  Remark 1.2.

The solution $u(t,x)$ in Theorem 1.1 is Fourier supported on the set
$\\{k\in{\mathbb{Z}}^{d}:|k|\in\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}\\}$,
where

$\alpha_{1}=m,\quad\ \alpha_{2}=m+p,\quad\ \alpha_{3}=2m+p,\quad\
\alpha_{4}=3m+2p,$

and $m,p$ are integers with $2\leq m<p$ and ratio $\sigma=m/p\leq\sigma_{*}$,
where $\sigma_{*}$ is a universal constant.

In fact, the ratio $\sigma$ is the perturbation parameter we use in the entire
construction. In principle, the constant $M$ in Theorem 1.1 depends on the
ratio $\sigma=m/p$ and it is of the order $M\sim\log(\sigma^{-1})$, see
(4.15). Theorem 1.1 is stated after fixing $m,p$ with $m=2$ and $p$ the
minimum integer such that $p>2$ and $2/p\leq\sigma_{*}$. ∎

######  Remark 1.3.

In Theorem 1.1 the Sobolev norm $\mathcal{N}(t)$ is used to describe the
transfer of energy between Fourier modes, because
$H^{\frac{3}{2}}({\mathbb{T}}^{d})\times H^{\frac{1}{2}}({\mathbb{T}}^{d})$ is
the space of the standard local wellposedness for the Kirchhoff equation.
Since the solution $u(t,x)$ in Theorem 1.1 has a fixed, finite Fourier support
for all times, all the Sobolev norms of $(u,\partial_{t}u)$ are equivalent,
and all are equally able to describe the chaotic transfer of energy among the
Fourier modes — all except the norm of the energy space
$H^{1}({\mathbb{T}}^{d})\times L^{2}({\mathbb{T}}^{d})$, which corresponds to
a conserved quantity of the approximating system that we use in the
construction; see Remark 3.2. ∎

######  Remark 1.4.

The factor $1/10$ in the definition of $\delta_{\varepsilon}$ in Theorem 1.1
comes from an arbitrary choice. We could replace 1/10 by any other positive
number; in that case, the constants $M,\varepsilon_{*},C,r_{0},b,K,K_{0}$ must
be chosen accordingly. ∎

######  Remark 1.5.

For simplicity, Theorem 1.1 and its proof are stated entirely in terms of
nonnegative times. However, with only minor changes, one proves that the
result holds over the time interval $[-T_{\varepsilon},T_{\varepsilon}]$. ∎

######  Remark 1.6.

Adapting the formulation of the symbolic dynamics for the approximating system
(see Proposition 4.7), one can prove an alternative version of Theorem 1.1,
where the prescribed random behavior of the norm $\mathcal{N}(t)$ is not only
given by the sequence of the time lengths of its oscillations, but also by any
sequence $(a_{0},a_{1},a_{2},\ldots)$ with $a_{j}\in\\{0,1\\}$ prescribing the
ordered sequence of “up” and “down” movements of $\mathcal{N}(t)$. In that
case, $\mathcal{N}(t)$ still makes oscillations of order $\varepsilon^{2}$
around a central value of order $\varepsilon$, varying in a range, say,
$[\ell_{\varepsilon},h_{\varepsilon}]$; the difference with respect to Theorem
1.1 is that, in the $j$-th time interval, $\mathcal{N}(t)$ get close to the
low value $\ell_{\varepsilon}$, and it remains in the slightly enlarged lower
half of the range, if $a_{j}=0$, while $\mathcal{N}(t)$ get close to the high
value $h_{\varepsilon}$, and it remains in the slightly enlarged upper half of
the range, if $a_{j}=1$.

In other words, around the equilibrium $u=0$, the Kirchhoff equation possesses
solutions whose norm $\mathcal{N}(t)$ exhibits oscillations that follow any
prescribed sequence of “up” and “down” on the time interval
$[0,T_{\varepsilon}]$. ∎

######  Remark 1.7.

The result in [3] shows that there are _no_ transfers of energy of size
$\varepsilon$ between Fourier spheres in a time interval of length
$\varepsilon^{-4}$. This could make one think that, on such a time scale,
between Fourier spheres there are no energy transfers at all. Theorem 1.1
shows that this is not true; in particular, it proves the existence of chaotic
transfers of energy of smaller size on a shorter time scale, i.e., transfers
of size $\varepsilon^{2}$ on a time scale
$\varepsilon^{-3}\log(\varepsilon^{-1})$. ∎

###  1.2 Main ideas of the proof

The main steps of the proof of Theorem 1.1 can be summarized as follows:

  1. 1.

Derive an effective resonant model for small solutions of (1.1). This reduced
system is obtained by using normal form arguments and introducing some
“macroscopic” variables describing the collective behavior of Fourier
frequencies with the same modulus.

  2. 2.

Show that, choosing carefully a finite set of Fourier frequencies, one can
make the effective system nearly integrable.

  3. 3.

Prove the existence of chaotic dynamics (a Smale horseshoe) for the effective
system.

  4. 4.

Show that certain solutions of the Kirchhoff equation (1.1) follow closely
those in the Smale horseshoe of the effective system for a sufficiently long
time interval.

The effective system is obtained with a normal form analysis. To this end, in
Section 2, we perform two steps of _quasilinear normal form_ (following [3,
4]), and introduce a set of special variables, found in [4], which allow to
reduce the dimension of the problem without losing information on the time
evolution of the Sobolev norm of the solution. Thus, the resulting reduced
model can be seen as a “macroscopic” effective system, where we do not
distinguish the evolution of the energy of each single Fourier mode.

The reduction to a finite dimensional effective system is done in Section 3.
We restrict the Fourier support to two coupled resonant triplets. The space of
functions supported on these modes is invariant for equation (1.1), thanks to
the particular form of its nonlinearity. Relying on symmetries of the problem,
we are able to further reduce the model to obtain a four dimensional system.

The next step is to construct chaotic motions for such a system. Since we rely
on perturbative techniques, we want the system to be nearly integrable; this
is obtained by choosing resonant triplets with Fourier modes as explained in
Remark 1.2. In particular, the system behaves as a pair of weakly coupled
pendulums.

Then, in Section 4, we apply the classical Poincaré-Melnikov theory [29] to
prove that the system has a hyperbolic periodic orbit with transverse
homoclinic orbits. By the classical Smale-Birkhoff Theorem, this implies the
existence of a Smale horseshoe, which is a hyperbolic invariant set with
symbolic dynamics. Note that the set is invariant and therefore one can
describe the dynamics of its orbits _for all times_.

Finally, it remains to translate the dynamics of the effective system to the
original equation (1.1). We prove that there exist solutions of the full PDE
that follow closely those of the effective system. Even if the approximation
argument is done through a Gronwall estimate (see Section 6), this is a rather
delicate procedure. Indeed, since we have performed several reductions,
rescalings, and two steps of normal form, we have to ensure that the solutions
of the Kirchhoff equation shadowing those of the effective system satisfy all
the required constraints over a sufficiently long time scale. This final part
of the proof is done in Sections 5 and 6.

The general strategy of the proof is similar to the one developed in [14] for
the cubic wave and beam equations. The proof of Theorem 1.1, however, is based
on a higher order normal form analysis, which is needed to consider systems
which are integrable at first order. This is the typical situation for PDEs on
one-dimensional spatial domains. Indeed, resonant Hamiltonian monomials of low
degree, which provide the dominant dynamics close to the origin, usually do
not change drastically the Fourier actions (and so, the Sobolev norms). Main
examples are given by the KdV, Klein-Gordon and Schrödinger equations and pure
gravity water waves equation in infinite depth, under Dirichlet or periodic
boundary conditions. This is somewhat the case also for equation (1.1), even
if the spatial domain is the torus ${\mathbb{T}}^{d}$ of any dimension $d\geq
1$, and even if the integrability property of the equation at the cubic order
only holds for the macroscopic variables. This makes the implementation of the
above strategy rather delicate. One of the issues in performing this kind of
analysis is that interesting instability phenomena only occur after a longer
time.

Another relevant difference with respect to the equations considered in [14]
is that equation (1.1) is quasilinear, namely the nonlinearity contains
derivatives of the same order as the linear part. This fact is not trivial,
because, even if one is able to construct normal form transformations for the
quasilinear equation (1.1) (as done in [3, 4]), here one has to be able to
provide a result of approximation between the effective model and the full PDE
for a long-time scale. This requires to consider an equation for the
difference of a special orbit of the effective system and a solution of (1.1).
This equation is quasilinear itself and presents a time-dependent linear part.
For such equation one has to provide a result of long-time stability.

Another difference with respect to [14] regards a quantitative aspect in the
energy exchange between Fourier frequencies: in [14] a large portion of the
energy transfers between Fourier frequencies having similar modulus; here, on
the contrary, a very small portion of the energy transfers between Fourier
frequencies of modulus $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ (see
Remark 1.2), where $\alpha_{2},\alpha_{3},\alpha_{4}$ are much larger than
$\alpha_{1}$.

### Acknowledgments

P.B. and E.H. are supported by the Italian Project PRIN 2020XB3EFL
_Hamiltonian and dispersive PDEs_. F.G. and E.H. have received funding from
INdAM-GNAMPA, Project CUP_E55F22 000270001. M.G. is supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 757802). M.G. is also supported by
the Catalan Institution for Research and Advanced Studies via an ICREA
Academia Prize 2019. This work is also supported by the Spanish State Research
Agency, through the Severo Ochoa and María de Maeztu Program for Centers and
Units of Excellence in R&D (CEX2020-001084-M).

##  2 Effective dynamics for the Kirchhoff equation

In this section we recall how the “macroscopic” quantities
$S_{\lambda},B_{\lambda}$ in (2.12) are derived from the Kirchhoff equation
(1.1), starting with the normal form procedure.

###  2.1 A quasilinear partial normal form

Written as a first order evolution equation, (1.1) becomes

$\begin{cases}\partial_{t}u=v,\\\
\partial_{t}v=\big{(}1+\int_{{\mathbb{T}}^{d}}|\nabla u|^{2}dx\big{)}\Delta
u.\end{cases}$ (2.1)

It is proved in [3] and [4] (see also the shorter, unified description in the
Appendix A of [5]) that system (2.1) can be transformed, after two steps of a
quasilinear, partial normal form procedure, into another system, where the
cubic and the quintic terms are in normal form (up to harmless terms that do
not contribute to energy estimates). More precisely, it is proved that,
renaming $(\tilde{u},\tilde{v})$ the original, “physical” variables of system
(2.1), with the change of variable $(\tilde{u},\tilde{v})=\Phi(u,v)$, system
(2.1) becomes

$\partial_{t}(u,v)=W(u,v),$ (2.2)

where

$W(u,v)=(1+\mathcal{P}(u,v))\big{(}\mathcal{D}_{1}(u,v)+\mathcal{Z}_{3}(u,v)+\mathcal{Z}_{5}(u,v)\big{)}+W_{\geq
7}(u,v).$ (2.3)

The unknown $(u,v)$ for the transformed system (2.2) is a pair of complex
conjugate functions with zero average over ${\mathbb{T}}^{d}$. The term
$(1+\mathcal{P}(u,v))$ is a scalar multiplicative factor, close to $1$,
depending on $(u,v)$, and it is a function of time, independent of the space
variable $x$. Also, $\mathcal{D}_{1}$ is the linear operator
$\mathcal{D}_{1}(u,v):=(-i|D_{x}|u,i|D_{x}|v)$, where $|D_{x}|$ is the Fourier
multiplier $e^{ik\cdot x}\mapsto|k|e^{ik\cdot x}$, $k\in{\mathbb{Z}}^{d}$.
Next, $\mathcal{Z}_{3}$ is the cubic resonant operator
$\mathcal{Z}_{3}(u,v)=((\mathcal{Z}_{3})_{1}(u,v),(\mathcal{Z}_{3})_{2}(u,v))$
with components

$\displaystyle(\mathcal{Z}_{3})_{1}(u,v):=-\frac{i}{4}\sum_{\begin{subarray}{c}j,k\in{\mathbb{Z}}^{d}\setminus\\{0\\}\\\
|j|=|k|\end{subarray}}u_{j}u_{-j}|j|^{2}v_{k}e^{ik\cdot x},\quad\
(\mathcal{Z}_{3})_{2}(u,v):=\frac{i}{4}\sum_{\begin{subarray}{c}j,k\in{\mathbb{Z}}^{d}\setminus\\{0\\}\\\
|j|=|k|\end{subarray}}v_{j}v_{-j}|j|^{2}u_{k}e^{ik\cdot x},$

where $u_{j},v_{j}$ are the Fourier coefficients of $u,v$. The entire term
$(1+\mathcal{P}(u,v))(\mathcal{D}_{1}(u,v)+\mathcal{Z}_{3}(u,v))$ gives no
contribution to the energy estimates. The term $\mathcal{Z}_{5}$ is the
resonant quintic operator
$\mathcal{Z}_{5}(u,v)=((\mathcal{Z}_{5})_{1}(u,v),(\mathcal{Z}_{5})_{2}(u,v)),$
where

$\displaystyle(\mathcal{Z}_{5})_{1}(u,v)$
$\displaystyle=\frac{i}{32}\sum_{\begin{subarray}{c}j,\ell,k\in{\mathbb{Z}}^{d}\setminus\\{0\\}\\\
|j|=|\ell|\end{subarray}}u_{j}u_{-j}v_{\ell}v_{-\ell}u_{k}e^{ik\cdot
x}|j|^{2}|\ell|^{2}\Big{(}\frac{1}{|j|+|k|}-\frac{(1-\delta_{|\ell|}^{|k|})}{|\ell|-|k|}\Big{)}$
$\displaystyle\quad+\frac{3i}{32}\sum_{\begin{subarray}{c}j,\ell,k\in{\mathbb{Z}}^{d}\setminus\\{0\\}\\\
|k|=|j|+|\ell|\end{subarray}}u_{j}u_{-j}u_{\ell}u_{-\ell}v_{k}e^{ik\cdot
x}|j||\ell||k|$
$\displaystyle\quad+\frac{i}{16}\sum_{\begin{subarray}{c}j,\ell,k\in{\mathbb{Z}}^{d}\setminus\\{0\\}\\\
|j|=|k|\end{subarray}}u_{j}u_{-j}u_{\ell}v_{-\ell}v_{k}e^{ik\cdot
x}|j|^{2}|\ell|\Big{(}6+\frac{|\ell|}{|\ell|+|j|}+\frac{|\ell|(1-\delta_{|\ell|}^{|j|})}{|\ell|-|j|}\Big{)}$
$\displaystyle\quad+\frac{3i}{16}\sum_{\begin{subarray}{c}j,\ell,k\in{\mathbb{Z}}^{d}\setminus\\{0\\}\\\
|k|=|j|-|\ell|\end{subarray}}u_{j}u_{-j}v_{\ell}v_{-\ell}v_{k}e^{ik\cdot
x}|j||\ell||k|,$ (2.4)

and $(\mathcal{Z}_{5})_{2}(u,v)$ is obtained from $(\mathcal{Z}_{5})_{1}(u,v)$
by complex conjugation. For the coefficients in (2.4) we adopt the convention
that $\frac{0}{0}=0$, where $\delta$ is the usual Kronecker delta. The term
$W_{\geq 7}(u,v)$ in (2.3) contains only terms of homogeneity at least 7 in
$(u,v)$, and it is estimated in [4] and [5].

The map $\Phi$ that transforms (2.1) into (2.2) is obtained by composition,
and it is

$\Phi=\Phi_{1}\circ\Phi_{2}\circ\Phi_{3}\circ\Phi_{4}\circ\Phi_{5}.$ (2.5)

The maps $\Phi_{1}$ and $\Phi_{2}$ are simply the linear operators that
symmetrize and diagonalize the linear part of system (2.1) (i.e., the linear
wave equation), see (6.34). The map $\Phi_{3}$ is a nonlinear, preparatory
transformation, which is required because the problem is quasilinear. The map
$\Phi_{4}$ is the transformation of the first step of the normal form
procedure, and $\Phi_{5}$ is the one of the second step. The explicit
expressions of $\Phi_{3}$ and $\Phi_{4}$ are given in [3] and the one of
$\Phi_{5}$ in [4]. Unlike $\Phi_{1}$ and $\Phi_{2}$, the transformations
$\Phi_{3},\Phi_{4},\Phi_{5}$ are all close to the identity map. In the present
paper we do not use the explicit formula of $\Phi_{3},\Phi_{4},\Phi_{5}$, but
only the following properties of their composition.

For $s\in{\mathbb{R}}$, let $H^{s}_{0}({\mathbb{T}}^{d},{\mathbb{C}})$ be the
Sobolev space of zero average, complex-valued functions

$H^{s}_{0}({\mathbb{T}}^{d},{\mathbb{C}}):=\\{u:{\mathbb{T}}^{d}\to{\mathbb{C}}:u_{0}=0,\
\|u\|_{s}<\infty\\},\quad\
\|u\|_{s}^{2}:=\sum_{k\in{\mathbb{Z}}^{d}\setminus\\{0\\}}|u_{k}|^{2}|k|^{2s},$

where $u_{k}$, $k\in{\mathbb{Z}}^{d}$, are the Fourier coefficients of $u$,
and let

$H^{s}_{0}({\mathbb{T}}^{d},c.c.):=\\{(u,v):u,v\in
H^{s}_{0}({\mathbb{T}}^{d},{\mathbb{C}}),\
v=\overline{u}\\}=\\{(u,\overline{u}):u\in
H^{s}_{0}({\mathbb{T}}^{d},{\mathbb{C}})\\},$

where the notation “$c.c.$” reminds that they are pairs of complex conjugate
functions. Given $d\in{\mathbb{N}}$, let

$m_{1}:=1\quad\text{if }d=1;\qquad m_{1}:=2\quad\text{if }d\geq 2.$ (2.6)

######  Lemma 2.1 (From Lemma 2.9 of [5]).

There exist universal constants $\delta,C>0$ such that for all $(u,v)\in
H^{m_{1}}_{0}({\mathbb{T}}^{d},c.c.)$ in the ball $\|u\|_{m_{1}}\leq\delta$,
for all $k\in{\mathbb{Z}}^{d}$, the $k$-th Fourier coefficient
$f_{k}=\overline{g_{-k}}$ of
$(f,g):=(\Phi_{3}\circ\Phi_{4}\circ\Phi_{5})(u,v)$ satisfies

$|f_{k}-u_{k}|\leq C\|u\|_{m_{1}}^{2}(|u_{k}|+|u_{-k}|).$ (2.7)

From (2.7) it follows that $\|f-u\|_{s}\leq 2C\|u\|_{m_{1}}^{2}\|u\|_{s}$ for
all $s\in{\mathbb{R}}$, and that the map $\Phi_{3}\circ\Phi_{4}\circ\Phi_{5}$
is well defined in the ball $\|u\|_{m_{1}}\leq\delta$, with
$\|f\|_{s}\leq\|u\|_{s}(1+2C\|u\|_{m_{1}}^{2})$, $s\in{\mathbb{R}}$.

Inequality (2.7) also implies the invariance of the Fourier support: if
$u_{k}=u_{-k}=0$ for some $k\in{\mathbb{Z}}^{d}$, then also $f_{k}=f_{-k}=0$,
and vice versa (since $C\delta^{2}<1/2$).

######  Lemma 2.2 (From Lemma 2.3 in [5]).

There exist universal constants $\delta_{1},C_{1},C_{0}>0$ with the following
properties. Let $(u_{0},v_{0})\in H^{m_{1}}_{0}({\mathbb{T}}^{d},c.c.)$ and

$\|u_{0}\|_{m_{1}}\leq\delta_{1}.$ (2.8)

Then the Cauchy problem of system (2.2) with initial condition

$(u(0),v(0))=(u_{0},v_{0})$ (2.9)

has a unique solution $(u,v)\in
C([0,T_{\mathrm{NF}}],H^{m_{1}}_{0}({\mathbb{T}}^{d},c.c.))$ on the time
interval $[0,T_{\mathrm{NF}}]$. The solution satisfies

$\|u(t)\|_{m_{1}}\leq C_{1}\|u_{0}\|_{m_{1}}\leq\delta\quad\forall
t\in[0,T_{\mathrm{NF}}],\qquad T_{\mathrm{NF}}=C_{0}\|u_{0}\|_{m_{1}}^{-4},$
(2.10)

where $\delta$ is the constant in Lemma 2.1. As a consequence, for all
$t\in[0,T_{\mathrm{NF}}]$ the solution $(u(t),v(t))$ remains in the ball
$\|u(t)\|_{m_{1}}\leq\delta$ where $\Phi_{3}\circ\Phi_{4}\circ\Phi_{5}$ is
well defined, the function

$(\tilde{u}(t),\tilde{v}(t)):=\Phi(u(t),v(t))$ (2.11)

(where $\Phi$ is the map in (2.5)) solves the original system (2.1) on the
time interval $[0,T_{\mathrm{NF}}]$, and $\tilde{u}(t)$ solves the Kirchhoff
equation (1.1) on $[0,T_{\mathrm{NF}}]$.

As the notation suggests, $T_{\mathrm{NF}}$ is the existence time we obtain by
the normal form procedure. For more details on the map $\Phi$ and on the
transformed vector field $W(u,v)$ see [3], [4], [5].

###  2.2 The effective system

We recall the derivation of the _effective system_ (or _effective equation_)
from [4], [5]. Let

$\Gamma:=\\{|k|:k\in{\mathbb{Z}}^{d},\ k\neq
0\\}\subseteq\\{\sqrt{n}:n\in{\mathbb{N}}\\}\subset[1,\infty).$

For any pair $(u,v)\in L^{2}({\mathbb{T}}^{d},c.c.)$ of complex conjugate
functions, for any $\lambda\in\Gamma$ we define

$S_{\lambda}:=\sum_{\begin{subarray}{c}k\in{\mathbb{Z}}^{d}\\\
|k|=\lambda\end{subarray}}|u_{k}|^{2}=\sum_{\begin{subarray}{c}k\in{\mathbb{Z}}^{d}\\\
|k|=\lambda\end{subarray}}u_{k}v_{-k},\qquad
B_{\lambda}:=\sum_{\begin{subarray}{c}k\in{\mathbb{Z}}^{d}\\\
|k|=\lambda\end{subarray}}u_{k}u_{-k},$ (2.12)

and note that

$S_{\lambda}\geq 0,\qquad B_{\lambda}\in{\mathbb{C}},\qquad|B_{\lambda}|\leq
S_{\lambda}.$

The quantity $S_{\lambda}$ is called the “superaction” of $u$ on the sphere
$|k|=\lambda$. Its evolution on the time interval $[0,T_{\mathrm{NF}}]$
remains confined between two multiples of its initial value, as is observed in
the next lemma.

######  Lemma 2.3 (From Lemma 2.4 of [5]).

Let $(u_{0},v_{0})\in H^{m_{1}}_{0}({\mathbb{T}}^{d},c.c.)$, with $u_{0}$ in
the ball (2.8). Let $(u(t),v(t))$ be the solution of the Cauchy problem (2.2),
(2.9) on the time interval $[0,T_{\mathrm{NF}}]$, with $T_{\mathrm{NF}}$ in
(2.10), given by Lemma 2.2. For every $t\in[0,T_{\mathrm{NF}}]$, let
$S_{\lambda}(t)$ be the sum defined in (2.12). Then

$C_{1}S_{\lambda}(0)\leq S_{\lambda}(t)\leq C_{2}S_{\lambda}(0)$ (2.13)

for all $t\in[0,T_{\mathrm{NF}}]$, for all $\lambda\in\Gamma$, where
$C_{1},C_{2}>0$ are universal constants.

By (2.13), for every $\lambda\in\Gamma$, either $S_{\lambda}(t)>0$ for all
$t\in[0,T_{\mathrm{NF}}]$, or $S_{\lambda}(t)=0$ for all
$t\in[0,T_{\mathrm{NF}}]$. Hence, we decompose $\Gamma$ as the disjoint union
of

$\Gamma_{0}:=\\{\lambda\in\Gamma:S_{\lambda}=0\\},\qquad\Gamma_{1}:=\\{\lambda\in\Gamma:S_{\lambda}>0\\}.$
(2.14)

It is observed in [4], [5] that, if $(u(t),v(t))$ solves (2.2) on some time
interval, then, for every $\lambda\in\Gamma$, calculating the Fourier
coefficients of $W(u,v)$ in (2.3) and taking the sum over all indices
$k\in{\mathbb{Z}}^{d}$ on the sphere $|k|=\lambda$, the corresponding
quantities $S_{\lambda}(t),B_{\lambda}(t)$ in (2.12) satisfies the equations

$\displaystyle\partial_{t}S_{\lambda}$
$\displaystyle=\frac{3i}{32}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\alpha+\beta=\lambda\end{subarray}}(B_{\alpha}B_{\beta}\overline{B_{\lambda}}-\overline{B_{\alpha}}\overline{B_{\beta}}B_{\lambda})\alpha\beta\lambda+\frac{3i}{16}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\alpha-\beta=\lambda\end{subarray}}(B_{\alpha}\overline{B_{\beta}}\overline{B_{\lambda}}-\overline{B_{\alpha}}B_{\beta}B_{\lambda})\alpha\beta\lambda+R_{S_{\lambda}},$
$\displaystyle\partial_{t}B_{\lambda}$
$\displaystyle=-2i(1+\mathcal{P})\Big{(}\lambda+\frac{1}{4}\lambda^{2}S_{\lambda}\Big{)}B_{\lambda}+R_{B_{\lambda}}$
(2.15)

on the same time interval, where the terms $R_{S_{\lambda}},R_{B_{\lambda}}$
satisfies the following estimates.

######  Lemma 2.4 (Lemma 2.2 of [5]).

Let $(u,v)\in H^{m_{1}}_{0}({\mathbb{T}}^{d},c.c.)$ with
$\|u\|_{m_{1}}\leq\delta$, where $\delta$ is the constant given by Lemma 2.1
and appearing in (2.10). Then, for all $\lambda\in\Gamma$, the terms
$R_{S_{\lambda}},R_{B_{\lambda}}$ in (2.15) satisfy

$|R_{S_{\lambda}}|\leq C\|u\|_{m_{1}}^{6}S_{\lambda},\quad\
|R_{B_{\lambda}}|\leq C\|u\|_{m_{1}}^{4}S_{\lambda},$ (2.16)

where $C>0$ is a universal constant.

Define

$Z_{\alpha\beta\lambda}:=B_{\alpha}B_{\beta}\overline{B_{\lambda}},\quad\
\vartheta_{\alpha\beta\lambda}:=\mathrm{Im}(Z_{\alpha\beta\lambda})=\frac{B_{\alpha}B_{\beta}\overline{B_{\lambda}}-\overline{B_{\alpha}B_{\beta}}B_{\lambda}}{2i}.$
(2.17)

If $S_{\lambda},B_{\lambda}$ satisfy system (2.15), then
$S_{\lambda},Z_{\alpha\beta\lambda}$ satisfy

$\displaystyle\partial_{t}S_{\lambda}$
$\displaystyle=-\frac{3}{16}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\alpha+\beta=\lambda\end{subarray}}\vartheta_{\alpha\beta\lambda}\,\alpha\beta\lambda+\frac{3}{8}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\beta+\lambda=\alpha\end{subarray}}\vartheta_{\beta\lambda\alpha}\,\alpha\beta\lambda+R_{S_{\lambda}},$
(2.18) $\displaystyle\partial_{t}Z_{\alpha\beta\lambda}$
$\displaystyle=-2i(1+\mathcal{P})\Big{(}\alpha+\beta-\lambda+\frac{1}{4}(\alpha^{2}S_{\alpha}+\beta^{2}S_{\beta}-\lambda^{2}S_{\lambda})\Big{)}Z_{\alpha\beta\lambda}+\widetilde{R}_{Z_{\alpha\beta\lambda}},$

where

$\widetilde{R}_{Z_{\alpha\beta\lambda}}:=R_{B_{\alpha}}B_{\beta}\overline{B_{\lambda}}+B_{\alpha}R_{B_{\beta}}\overline{B_{\lambda}}+B_{\alpha}B_{\beta}\overline{R_{B_{\lambda}}}.$

For $\alpha+\beta=\lambda$, isolating the first nontrivial contribution from
terms of higher homogeneity orders, one has

$\partial_{t}Z_{\alpha\beta\lambda}=-\frac{i}{2}(\alpha^{2}S_{\alpha}+\beta^{2}S_{\beta}-\lambda^{2}S_{\lambda})Z_{\alpha\beta\lambda}+R_{Z_{\alpha\beta\lambda}}$
(2.19)

where

$R_{Z_{\alpha\beta\lambda}}:=-\frac{i}{2}\mathcal{P}_{2}(\alpha^{2}S_{\alpha}+\beta^{2}S_{\beta}-\lambda^{2}S_{\lambda})Z_{\alpha\beta\lambda}+\widetilde{R}_{Z_{\alpha\beta\lambda}}.$
(2.20)

The remainder $R_{S_{\lambda}}$ in (2.18) has been bounded in Lemma 2.4. The
remainder $R_{Z_{\alpha\beta\lambda}}$ in (2.20) is estimated in the next
lemma.

######  Lemma 2.5 (Lemma 2.5 of [5]).

Assume the hypotheses of Lemma 2.4. Then for all
$\alpha,\beta,\lambda\in\Gamma$ with $\alpha+\beta=\lambda$ the remainder
$R_{Z_{\alpha\beta\lambda}}$ defined in (2.20) satisfies

$|R_{Z_{\alpha\beta\lambda}}|\leq
C\|u\|_{m_{1}}^{4}S_{\alpha}S_{\beta}S_{\lambda},$ (2.21)

where $C>0$ is a universal constant.

###  2.3 The truncated effective system

If we remove the remainders from equations (2.18) and (2.19), we obtain a
system that we call _truncated effective system_ , which is

$\displaystyle\partial_{t}S_{\lambda}$
$\displaystyle=-\frac{3}{16}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\alpha+\beta=\lambda\end{subarray}}\vartheta_{\alpha\beta\lambda}\alpha\beta\lambda+\frac{3}{8}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\beta+\lambda=\alpha\end{subarray}}\vartheta_{\beta\lambda\alpha}\beta\lambda\alpha,$
(2.22) $\displaystyle\partial_{t}Z_{\alpha\beta\lambda}$
$\displaystyle=-\frac{i}{2}\omega_{\alpha\beta\lambda}Z_{\alpha\beta\lambda},$
(2.23)

where

$\omega_{\alpha\beta\lambda}:=\alpha^{2}S_{\alpha}+\beta^{2}S_{\beta}-\lambda^{2}S_{\lambda}.$
(2.24)

For all $\alpha,\beta,\lambda\in\Gamma$ such that $\alpha+\beta=\lambda$, let

$r_{\alpha\beta\lambda}:=\mathrm{Re}\,(Z_{\alpha\beta\lambda}).$

Since $\omega_{\alpha\beta\lambda}$ is real, the real and imaginary part of
equation (2.23) is given by the system

$\partial_{t}r_{\alpha\beta\lambda}=\frac{1}{2}\omega_{\alpha\beta\lambda}\vartheta_{\alpha\beta\lambda},\qquad\partial_{t}\vartheta_{\alpha\beta\lambda}=-\frac{1}{2}\omega_{\alpha\beta\lambda}r_{\alpha\beta\lambda}.$
(2.25)

The solutions $(r_{\alpha\beta\lambda}(t),\vartheta_{\alpha\beta\lambda}(t))$
of (2.25) remain on a circle, because they satisfy

$\partial_{t}(|Z_{\alpha\beta\lambda}|^{2})=\partial_{t}\big{(}r_{\alpha\beta\lambda}^{2}+\vartheta_{\alpha\beta\lambda}^{2}\big{)}=0.$
(2.26)

Thus $|Z_{\alpha\beta\lambda}|$ is a prime integral of the truncated effective
system (2.22)-(2.23). Therefore, if $S_{\lambda},Z_{\alpha\beta\lambda}$ solve
(2.22)-(2.23), then the real and imaginary part of $Z_{\alpha\beta\lambda}$
satisfy

$r_{\alpha\beta\lambda}(t)=\rho_{\alpha\beta\lambda}\cos(\varphi_{\alpha\beta\lambda}(t)),\qquad\vartheta_{\alpha\beta\lambda}(t)=\rho_{\alpha\beta\lambda}\sin(\varphi_{\alpha\beta\lambda}(t)),$
(2.27)

where

$\rho_{\alpha\beta\lambda}=|Z_{\alpha\beta\lambda}|\geq 0$

is a constant, and $\varphi_{\alpha\beta\lambda}(t)$ is an angle. Moreover,
plugging (2.27) into (2.25) gives the equation for the evolution of the angle.
Hence the truncated effective system (2.22)-(2.23) becomes

$\displaystyle\partial_{t}S_{\lambda}$
$\displaystyle=-\frac{1}{2}\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\alpha+\beta=\lambda\end{subarray}}c_{\alpha\beta\lambda}\sin(\varphi_{\alpha\beta\lambda})+\sum_{\begin{subarray}{c}\alpha,\beta\in\Gamma\\\
\beta+\lambda=\alpha\end{subarray}}c_{\beta\lambda\alpha}\sin(\varphi_{\beta\lambda\alpha}),$
(2.28) $\displaystyle\partial_{t}\varphi_{\alpha\beta\lambda}$
$\displaystyle=-\frac{1}{2}(\alpha^{2}S_{\alpha}+\beta^{2}S_{\beta}-\lambda^{2}S_{\lambda}),$
(2.29)

where

$c_{\alpha\beta\lambda}:=\frac{3}{8}\rho_{\alpha\beta\lambda}\alpha\beta\lambda=\frac{3}{8}|Z_{\alpha\beta\lambda}|\alpha\beta\lambda=\frac{3}{8}|B_{\alpha}||B_{\beta}||B_{\lambda}|\alpha\beta\lambda$
(2.30)

is a constant. We note that (2.28) and (2.29) form a closed system for the
variables $S_{\lambda},\varphi_{\alpha\beta\lambda}$.

##  3 The truncated effective system with two triplets

We consider the case in which the Fourier support $\Gamma_{1}$ in (2.14) has
only $4$ distinct elements, forming two resonant triplets with two elements in
common, in the following way:

$\displaystyle\Gamma_{1}=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\},\qquad\alpha_{1}<\alpha_{2}<\alpha_{3}<\alpha_{4},$
$\displaystyle\alpha_{1}+\alpha_{2}=\alpha_{3},\qquad\alpha_{2}+\alpha_{3}=\alpha_{4},\qquad
2\alpha_{1}\neq\alpha_{2}.$ (3.1)

We can assume, without loss of generality, that the four elements of
$\Gamma_{1}$ are natural numbers. Examples of such sets are any four
consecutive elements of the Fibonacci sequence greater than $1$, like
$\\{2,3,5,8\\}$, or, more generally, any set of the form

$\alpha_{1}:=m,\quad\ \alpha_{2}:=m+p,\quad\ \alpha_{3}:=2m+p,\quad\
\alpha_{4}:=3m+2p,$ (3.2)

where $m,p$ are distinct positive integers with

$2\leq m<p.$

######  Lemma 3.1.

Assume (3.1), (3.2). If $\alpha,\beta,\lambda\in\Gamma_{1}$ satisfy
$\alpha+\beta=\lambda$, then the ordered triplet $(\alpha,\beta,\lambda)$ must
be

$(\alpha_{1},\alpha_{2},\alpha_{3})\quad\text{or}\quad(\alpha_{2},\alpha_{1},\alpha_{3})\quad\text{or}\quad(\alpha_{2},\alpha_{3},\alpha_{4})\quad\text{or}\quad(\alpha_{3},\alpha_{2},\alpha_{4}),$

and there are no other options.

###### Proof.

For example, one has

$\alpha_{3}=\alpha_{1}+\alpha_{2}<\alpha_{2}+\alpha_{2}<\alpha_{2}+\alpha_{3}=\alpha_{4},$

therefore $2\alpha_{2}\notin\Gamma_{1}$, and the triplet
$(\alpha_{2},\alpha_{2},2\alpha_{2})$ is not admissible; the other cases can
be checked similarly. ∎

To slightly shorten the notation, we denote

$S_{1}:=S_{\alpha_{1}},\quad\
\varphi_{123}:=\varphi_{\alpha_{1}\alpha_{2}\alpha_{3}},\quad\
c_{123}:=c_{\alpha_{1}\alpha_{2}\alpha_{3}},$

and so on. Hence, system (2.28)-(2.29) becomes

$\displaystyle\partial_{t}S_{1}$ $\displaystyle=c_{123}\sin(\varphi_{123}),$
(3.3) $\displaystyle\partial_{t}S_{2}$
$\displaystyle=c_{123}\sin(\varphi_{123})+c_{234}\sin(\varphi_{234}),$
$\displaystyle\partial_{t}S_{3}$
$\displaystyle=-c_{123}\sin(\varphi_{123})+c_{234}\sin(\varphi_{234}),$
$\displaystyle\partial_{t}S_{4}$ $\displaystyle=-c_{234}\sin(\varphi_{234}),$
$\displaystyle\partial_{t}\varphi_{123}$
$\displaystyle=-\frac{1}{2}(\alpha_{1}^{2}S_{1}+\alpha_{2}^{2}S_{2}-\alpha_{3}^{2}S_{3}),$
$\displaystyle\partial_{t}\varphi_{234}$
$\displaystyle=-\frac{1}{2}(\alpha_{2}^{2}S_{2}+\alpha_{3}^{2}S_{3}-\alpha_{4}^{2}S_{4}),$

which is a system of 6 equations in 6 unknowns.

###  3.1 Meaningfulness condition for the solutions

Our strategy is this: We want to find solutions of the truncated effective
system (3.3) with a prescribed, interesting dynamical behavior, and to show
that the solution of the effective system (2.18)-(2.19) is so close to the
solution of the truncated effective system (3.3) that the dynamical behaviors
of the two solutions are very similar, on a sufficiently long interval of
time. Later, in Section 6, we show that there exists a solution of the
original PDE (2.1) which is very close (up to the change of coordinates
$\Phi$) to the solution of the effective system.

Hence, we look for solutions of the truncated effective system (3.3) that
satisfy the natural meaningfulness condition required by system (2.18)-(2.19),
which is simply this: If a solution is defined on a time interval $[0,T]$,
then it must satisfy

$S_{n}(t)>0\quad\ \forall n=1,2,3,4,\quad\ \forall t\in[0,T].$ (3.4)

So, we reject any solution of (3.3) such that some of the $S_{n}$ becomes non-
positive at some time $t$ (recall definitions (2.12) and (2.14)).

In the following analysis, we first ignore the constrain (3.4); later, we will
select only solutions satisfying it. Analogously, we first consider the
coefficients $c_{123},c_{234}$ as any two given constants; later, we will go
back to the identities (2.30).

###  3.2 First integrals and a linear change of coordinates

Given any linear combination
$E:=\mu_{1}S_{1}+\mu_{2}S_{2}+\mu_{3}S_{3}+\mu_{4}S_{4}$ of
$S_{1},\ldots,S_{4}$ with constant real coefficients $\mu_{1},\ldots,\mu_{4}$,
we have

$\partial_{t}E=(\mu_{1}+\mu_{2}-\mu_{3})c_{123}\sin(\varphi_{123})+(\mu_{2}+\mu_{3}-\mu_{4})c_{234}\sin(\varphi_{234})$

along the solutions of system (3.3). Hence any $E$ with coefficients
$\mu_{1},\ldots,\mu_{4}$ satisfying

$\mu_{1}+\mu_{2}-\mu_{3}=0,\qquad\mu_{2}+\mu_{3}-\mu_{4}=0$

is a first integral. We choose the two functionally independent first
integrals

$E_{1}:=S_{1}+S_{3}+S_{4},\quad\ E_{2}:=S_{2}+S_{3}+2S_{4}.$ (3.5)

######  Remark 3.2.

One has

$\alpha_{1}E_{1}+\alpha_{2}E_{2}=\alpha_{1}(S_{1}+S_{3}+S_{4})+\alpha_{2}(S_{2}+S_{3}+2S_{4})=\sum_{n=1}^{4}\alpha_{n}S_{n}$
(3.6)

because $\alpha_{1}+\alpha_{2}=\alpha_{3}$ and
$\alpha_{1}+2\alpha_{2}=\alpha_{2}+(\alpha_{1}+\alpha_{2})=\alpha_{2}+\alpha_{3}=\alpha_{4}$.
Hence, when $S_{n}$ are given by (2.12), identity (3.6) implies that the
Sobolev norm $\|u\|_{\frac{1}{2}}^{2}=\sum_{n=1}^{4}\alpha_{n}S_{n}$ is also a
first integral of (3.3). ∎

At each time $t$, the values $S_{1}(t),S_{2}(t)$ can be obtained from
$E_{1},E_{2},S_{3}(t),S_{4}(t)$ by (3.5), i.e.,

$S_{1}(t)=E_{1}-S_{3}(t)-S_{4}(t),\quad\ S_{2}(t)=E_{2}-S_{3}(t)-2S_{4}(t).$
(3.7)

Hence system (3.3) can be reduced to a system of 4 equations in the 4 unknowns
$S_{3},S_{4},\varphi_{123},\varphi_{234}$, obtained by replacing $S_{1},S_{2}$
by (3.7) in the last two equations of (3.3). We get

$\displaystyle\partial_{t}S_{3}$
$\displaystyle=-c_{123}\sin(\varphi_{123})+c_{234}\sin(\varphi_{234}),$
$\displaystyle\partial_{t}S_{4}$ $\displaystyle=-c_{234}\sin(\varphi_{234}),$
$\displaystyle\partial_{t}\varphi_{123}$
$\displaystyle=-\frac{1}{2}(\alpha_{1}^{2}E_{1}+\alpha_{2}^{2}E_{2})+\frac{1}{2}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})S_{3}+\frac{1}{2}(\alpha_{1}^{2}+2\alpha_{2}^{2})S_{4},$
$\displaystyle\partial_{t}\varphi_{234}$
$\displaystyle=-\frac{1}{2}\alpha_{2}^{2}E_{2}-\frac{1}{2}(\alpha_{3}^{2}-\alpha_{2}^{2})S_{3}+\frac{1}{2}(2\alpha_{2}^{2}+\alpha_{4}^{2})S_{4}.$
(3.8)

We summarize the observations above in the following lemma.

######  Lemma 3.3.

Let $c_{123},c_{234}$ be any two constants. The following properties hold.

  * (i)

Let $(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$
be a solution of system (3.3) on some time interval $I$. Then $E_{1},E_{2}$
defined by (3.5) are constant in time and
$(S_{3}(t),S_{4}(t),\varphi_{123}(t)$, $\varphi_{234}(t))$ solves system (3.8)
on $I$.

  * (ii)

Let $E_{1},E_{2}$ be constants, and let
$(S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$ be a solution of
system (3.8) on some time interval $I$. Define the functions
$S_{1}(t),S_{2}(t)$ by the identities (3.7). Then
$(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$
solves system (3.3) on $I$.

We note that the sum of the first two equations in (3.8) does not contain the
angle $\varphi_{234}$. Hence, we consider a linear change of variable that
treats the sum $S_{3}+S_{4}$ as a new variable.

######  Lemma 3.4.

Let $c_{123},c_{234}$ be any two constants. The following properties hold.

  * (i)

Let $E_{1},E_{2}$ be constants, and let
$(S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$ be a solution of
system (3.8) on some time interval $I$. Define the functions
$x_{1}(t),x_{2}(t),y_{1}(t),y_{2}(t)$ by the change of coordinates

$\varphi_{123}=x_{1},\quad\ \varphi_{234}=x_{2},\quad\
S_{3}=y_{1}-y_{2},\quad\ S_{4}=y_{2}.$ (3.9)

Then $(x_{1}(t),x_{2}(t),y_{1}(t),y_{2}(t))$ solves

$\displaystyle\partial_{t}x_{1}$
$\displaystyle=-\frac{1}{2}b_{1}+\frac{1}{2}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})y_{1}-\frac{1}{2}(\alpha_{3}^{2}-\alpha_{2}^{2})y_{2},$
(3.10) $\displaystyle\partial_{t}x_{2}$
$\displaystyle=-\frac{1}{2}b_{2}-\frac{1}{2}(\alpha_{3}^{2}-\alpha_{2}^{2})y_{1}+\frac{1}{2}(\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2})y_{2}$
$\displaystyle\partial_{t}y_{1}$ $\displaystyle=-c_{123}\sin(x_{1}),$
$\displaystyle\partial_{t}y_{2}$ $\displaystyle=-c_{234}\sin(x_{2}),$

on $I$, where $b_{1},b_{2}$ are the constants

$b_{1}:=\alpha_{1}^{2}E_{1}+\alpha_{2}^{2}E_{2},\quad\
b_{2}:=\alpha_{2}^{2}E_{2}.$ (3.11)

  * (ii)

Let $b_{1},b_{2}$ be constants, and let
$(x_{1}(t),x_{2}(t),y_{1}(t),y_{2}(t))$ be a solution of system (3.10) on some
time interval $I$. Define the constants $E_{1},E_{2}$ as

$E_{1}=\frac{b_{1}-b_{2}}{\alpha_{1}^{2}},\qquad
E_{2}=\frac{b_{2}}{\alpha_{2}^{2}},$ (3.12)

and define the functions $S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t)$
by (3.9). Then $(S_{3}(t),S_{4}(t),\varphi_{123}(t)$, $\varphi_{234}(t))$
solves system (3.8) on $I$.

###  3.3 The Hamiltonian structure

System (3.10) is the 2-dimensional Hamiltonian system

$\dot{x}_{n}=\partial_{y_{n}}H(x,y),\quad\
\dot{y}_{n}=-\partial_{x_{n}}H(x,y),\quad\ n=1,2,$

with Hamiltonian

$H(x,y)=-c_{123}\cos(x_{1})-c_{234}\cos(x_{2})-\frac{1}{2}b\cdot
y+\frac{1}{4}Ay\cdot y$ (3.13)

where $b=(b_{1},b_{2})\in{\mathbb{R}}^{2}$ and $A$ is the matrix

$A:=\begin{pmatrix}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})\ &\
-(\alpha_{3}^{2}-\alpha_{2}^{2})\\\ -(\alpha_{3}^{2}-\alpha_{2}^{2})\ &\
(\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2})\end{pmatrix}.$ (3.14)

The matrix $A$ is symmetric, positive definite and invertible with

$A^{-1}=\frac{1}{\det
A}\begin{pmatrix}(\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2})\ &\
(\alpha_{3}^{2}-\alpha_{2}^{2})\\\ (\alpha_{3}^{2}-\alpha_{2}^{2})\ &\
(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})\end{pmatrix}$ (3.15)

and

$\det
A=\alpha_{1}^{2}\alpha_{4}^{2}+(\alpha_{1}^{2}+\alpha_{4}^{2})(\alpha_{2}^{2}+\alpha_{3}^{2})+4\alpha_{2}^{2}\alpha_{3}^{2}>0.$
(3.16)

The invertibility of $A$ is the so-called _twist condition_ for the
Hamiltonian $H$; thanks to it, we can eliminate the linear term $b\cdot y$
from the Hamiltonian by a translation of the $y$ variables

$x_{1}=\tilde{x}_{1},\quad\ x_{2}=\tilde{x}_{2},\quad\
y_{1}=q_{1}+\tilde{y}_{1},\quad\ y_{2}=q_{2}+\tilde{y}_{2},$ (3.17)

with $q=A^{-1}b$, namely

$q_{1}=\frac{\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}}{\det
A}\,b_{1}+\frac{\alpha_{3}^{2}-\alpha_{2}^{2}}{\det A}\,b_{2},\qquad
q_{2}=\frac{\alpha_{3}^{2}-\alpha_{2}^{2}}{\det
A}\,b_{1}+\frac{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}{\det A}\,b_{2}.$
(3.18)

This change of coordinates is symplectic and the new Hamiltonian is just
$\tilde{H}(\tilde{x},\tilde{y})=H(\tilde{x},q+\tilde{y})$, whose equations are
given by

$\displaystyle\partial_{t}\tilde{x}_{1}$
$\displaystyle=\tfrac{1}{2}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})\tilde{y}_{1}-\tfrac{1}{2}(\alpha_{3}^{2}-\alpha_{2}^{2})\tilde{y}_{2},$
(3.19) $\displaystyle\partial_{t}\tilde{y}_{1}$
$\displaystyle=-c_{123}\sin(\tilde{x}_{1}),$
$\displaystyle\partial_{t}\tilde{x}_{2}$
$\displaystyle=-\tfrac{1}{2}(\alpha_{3}^{2}-\alpha_{2}^{2})\tilde{y}_{1}+\tfrac{1}{2}(\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2})\tilde{y}_{2},$
$\displaystyle\partial_{t}\tilde{y}_{2}$
$\displaystyle=-c_{234}\sin(\tilde{x}_{2}).$

The equivalence of systems (3.10) and (3.19) is described in the following
lemma.

######  Lemma 3.5.

Let $c_{123},c_{234}$ be any two constants. The following properties hold.

  * (i)

Let $b_{1},b_{2}$ be constants, and let
$(x_{1}(t),x_{2}(t),y_{1}(t),y_{2}(t))$ be a solution of system (3.10) on some
time interval $I$. Define the constants $q_{1},q_{2}$ by (3.18), and define
the functions
$\tilde{x}_{1}(t),\tilde{x}_{2}(t),\tilde{y}_{1}(t),\tilde{y}_{2}(t)$ by
(3.17). Then
$(\tilde{x}_{1}(t),\tilde{x}_{2}(t),\tilde{y}_{1}(t),\tilde{y}_{2}(t))$ solves
(3.19).

  * (ii)

Let $(\tilde{x}_{1}(t),\tilde{x}_{2}(t),\tilde{y}_{1}(t),\tilde{y}_{2}(t))$ be
a solution of system (3.19) on some interval $I$. Let $q_{1},q_{2}$ be any two
real numbers. Define constants $b_{1},b_{2}$ by the identity $b=Aq$, i.e.,
define

$b_{1}=(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})q_{1}-(\alpha_{3}^{2}-\alpha_{2}^{2})q_{2},\qquad
b_{2}=-(\alpha_{3}^{2}-\alpha_{2}^{2})q_{1}+(\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2})q_{2},$
(3.20)

and define the functions $x_{1}(t),x_{2}(t),y_{1}(t),y_{2}(t)$ by (3.17). Then
$(x_{1}(t),x_{2}(t),y_{1}(t),y_{2}(t))$ solves (3.10) on $I$.

###  3.4 Normalization of coefficients by rescaling

Now we want to normalize the leading coefficients of system (3.19), using a
rescaling of the time variable and dilations of the $\tilde{y}$ variables. We
consider the change of variables

$\tilde{x}_{1}(t)=\xi_{1}(\mathtt{B}t),\quad\
\tilde{x}_{2}(t)=\xi_{2}(\mathtt{B}t),\quad\
\tilde{y}_{1}(t)=\mathtt{A}_{1}\eta_{1}(\mathtt{B}t),\quad\
\tilde{y}_{2}(t)=\mathtt{A}_{2}\eta_{2}(\mathtt{B}t),\quad$ (3.21)

where $\mathtt{A}_{1},\mathtt{A}_{2},\mathtt{B}$ are defined as follows. We
assume that

$c_{123}>0$ (3.22)

and we fix

$\mathtt{A}_{1}:=\Big{(}\frac{2c_{123}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}\Big{)}^{\frac{1}{2}},\qquad\mathtt{B}:=\Big{(}\frac{c_{123}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})}{2}\Big{)}^{\frac{1}{2}},$
(3.23)
$\mathtt{A}_{2}:=\frac{2\mathtt{B}}{\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}}.$
(3.24)

Note that $\mathtt{A}_{1}$ and $\mathtt{A}_{2}$ are related by

$\mathtt{A}_{1}=\mathtt{A}_{2}\gamma,\qquad\text{with}\qquad\gamma:=\frac{\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}.$
(3.25)

Thus, system (3.19) becomes

$\begin{split}\dot{\xi}_{1}&=\eta_{1}-\mu_{1}\eta_{2},\\\
\dot{\eta}_{1}&=-\sin(\xi_{1}),\\\ \dot{\xi}_{2}&=\eta_{2}-\mu_{2}\eta_{1},\\\
\dot{\eta}_{2}&=-\lambda\sin(\xi_{2}),\end{split}$ (3.26)

where

$\displaystyle\mu_{1}$
$\displaystyle:=\frac{\alpha_{3}^{2}-\alpha_{2}^{2}}{\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}}=\frac{(\alpha_{3}^{2}-\alpha_{2}^{2})\mathtt{A}_{2}}{2\mathtt{B}},$
(3.27) $\displaystyle\mu_{2}$
$\displaystyle:=\frac{\alpha_{3}^{2}-\alpha_{2}^{2}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}=\mu_{1}\gamma=\frac{(\alpha_{3}^{2}-\alpha_{2}^{2})\mathtt{A}_{1}}{2\mathtt{B}},$
(3.28) $\displaystyle\lambda$
$\displaystyle:=\frac{c_{234}(\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2})}{c_{123}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})}=\frac{c_{234}}{c_{123}}\,\gamma=\frac{c_{234}}{\mathtt{A}_{2}\mathtt{B}}.$
(3.29)

We observe that the system with $\mu_{1}=0$ is given by the sum of two
uncoupled Hamiltonians, while for $\mu_{1}\neq 0$ the Hamiltonian structure is
lost. The equivalence of systems (3.19) and (3.26) is described in the
following lemma.

######  Lemma 3.6.

The following statements are satisfied.

  * (i)

Let $c_{123},c_{234}$ be any two constants, with $c_{123}>0$, and let
$(\tilde{x}_{1}(t),\tilde{x}_{2}(t),\tilde{y}_{1}(t),\tilde{y}_{2}(t))$ be a
solution of system (3.19) on some time interval $[0,T]$. Let
$\mathtt{A}_{1},\mathtt{A}_{2},\mathtt{B}$ be the constants defined in (3.23),
(3.24). Define the functions $\xi_{1},\xi_{2},\eta_{1},\eta_{2}$ as

$\xi_{1}(t)=\tilde{x}_{1}\Big{(}\frac{t}{\mathtt{B}}\Big{)},\quad\
\xi_{2}(t)=\tilde{x}_{2}\Big{(}\frac{t}{\mathtt{B}}\Big{)},\quad\
\eta_{1}(t)=\frac{1}{\mathtt{A}_{1}}\tilde{y}_{1}\Big{(}\frac{t}{\mathtt{B}}\Big{)},\quad\
\eta_{2}(t)=\frac{1}{\mathtt{A}_{2}}\tilde{y}_{2}\Big{(}\frac{t}{\mathtt{B}}\Big{)}.\quad$
(3.30)

Define the constants $\mu_{1},\mu_{2},\lambda$ by (3.27), (3.28), (3.29). Then
$(\xi_{1}(t),\xi_{2}(t),\eta_{1}(t),\eta_{2}(t))$ solves (3.26) on the time
interval $[0,\mathtt{B}T]$.

  * (ii)

Let $\lambda$ be any constant. Let $\mu_{1},\mu_{2}$ be the constants defined
in (3.27), (3.28). Let $(\xi_{1}(t),\xi_{2}(t),\eta_{1}(t),\eta_{2}(t))$ be a
solution of system (3.26) on some time interval $[0,T]$. Let $\mathtt{A}_{1}$
be any positive constant. Define the constant $c_{123}$ as

$c_{123}=\frac{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}{2}\mathtt{A}_{1}^{2}$
(3.31)

and define the constant $c_{234}$ by means of (3.29), i.e.,
$c_{234}=c_{123}\lambda/\gamma$, where $\gamma$ is defined in (3.25). Define
the constant $\mathtt{B}$ by the second identity in (3.23), and the constant
$\mathtt{A}_{2}$ by (3.24). Define the functions
$\tilde{x}_{1},\tilde{x}_{2},\tilde{y}_{1},\tilde{y}_{2}$ by (3.21). Then
$(\tilde{x}_{1}(t),\tilde{x}_{2}(t),\tilde{y}_{1}(t),\tilde{y}_{2}(t))$ solve
(3.19) on the time interval $[0,\mathtt{B}^{-1}T]$.

###  3.5 Large Fourier frequency as a perturbation parameter

Recall the definition (3.2) of $\alpha_{1},\ldots,\alpha_{4}$ as functions of
the two integer parameters $m,p$. Then

$\displaystyle\alpha_{3}^{2}-\alpha_{2}^{2}$
$\displaystyle=(2m+p)^{2}-(m+p)^{2}=3m^{2}+2mp,$
$\displaystyle\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}$
$\displaystyle=m^{2}+(m+p)^{2}+(2m+p)^{2}=6m^{2}+6mp+2p^{2},$
$\displaystyle\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}$
$\displaystyle=(m+p)^{2}+(2m+p)^{2}+(3m+2p)^{2}=14m^{2}+18mp+6p^{2}.$

We note that the monomial $p^{2}$ cancels out in the difference
$\alpha_{3}^{2}-\alpha_{2}^{2}$, while it is present in the other two sums.
For this reason, taking $p$ large with respect to $m$ gives a small parameter,
which we will use in our perturbation analysis (the other small parameter of
the problem is the size of the solution, i.e., the size of the initial data of
the Kirchhoff equation). Denoting

$\sigma:=\frac{m}{p},$ (3.32)

one has

$\displaystyle\mu_{1}$
$\displaystyle=\frac{2\sigma+3\sigma^{2}}{6+18\sigma+14\sigma^{2}}=\sigma\Big{(}\frac{1}{3}+\tilde{\mu}_{1}(\sigma)\Big{)},\qquad\gamma=\frac{6+18\sigma+14\sigma^{2}}{2+6\sigma+6\sigma^{2}}=3+O(\sigma),$
$\displaystyle\mu_{2}$
$\displaystyle=\mu_{1}\gamma=\frac{2\sigma+3\sigma^{2}}{2+6\sigma+6\sigma^{2}}=\sigma\big{(}1+\tilde{\mu}_{2}(\sigma)\big{)},\qquad\tilde{\mu}_{1}(\sigma),\tilde{\mu}_{2}(\sigma)=O(\sigma)\quad\text{as
}\sigma\to 0,$ (3.33)

where $\tilde{\mu}_{1}(\sigma),\tilde{\mu}_{2}(\sigma)$ are defined by the
identities (3.33). We also note that $1<\gamma<3$ for all $\sigma>0$.

Thus, for $\sigma=m/p$ small, the “coupling” terms $\mu_{1}\eta_{2}$ and
$\mu_{2}\eta_{1}$ in system (3.26) can be considered as perturbations of the
“unperturbed” system of two uncoupled pendulums

$\begin{cases}\dot{\xi}_{1}=\eta_{1},\\\
\dot{\eta}_{1}=-\sin(\xi_{1}),\end{cases}\qquad\quad\begin{cases}\dot{\xi}_{2}=\eta_{2},\\\
\dot{\eta}_{2}=-\lambda\sin(\xi_{2}).\end{cases}$ (3.34)

We want to normalize also the coefficient $\lambda$ appearing in the last
equation of system (3.26). Later, we will see that this normalization
corresponds to a constraint on the initial data for equation (2.23); at this
stage, however, we simply observe that the parameter $\lambda$ in part $(ii)$
of Lemma 3.6 is not subject to any constraint. Thus, in the following analysis
we fix $\lambda=1$ and simply do not consider other values of that parameter.
For $\lambda=1$, system (3.26) becomes

$\begin{cases}\dot{\xi}_{1}=\eta_{1}-\mu_{1}\eta_{2},\\\
\dot{\eta}_{1}=-\sin(\xi_{1}),\\\ \dot{\xi}_{2}=\eta_{2}-\mu_{2}\eta_{1},\\\
\dot{\eta}_{2}=-\sin(\xi_{2}).\end{cases}$ (3.35)

System (3.35) has a conserved quantity (obtained expressing the old
Hamiltonian in terms of the new variables), which is

$\mu_{2}\Big{(}\frac{1}{2}\eta_{1}^{2}+1-\cos\xi_{1}\Big{)}+\mu_{1}\Big{(}\frac{1}{2}\eta_{2}^{2}+1-\cos\xi_{2}\Big{)}-\mu_{1}\mu_{2}\eta_{1}\eta_{2}.$
(3.36)

The only parameters in system (3.35) are the constants $\mu_{1},\mu_{2}$
defined in (3.27), (3.28), which depend only on the ratio $\sigma=m/p$, and
tend to zero as $\sigma\to 0$ (see (3.33)). The “unperturbed” part of system
(3.35) is (3.34) with $\lambda=1$, that is,

$\begin{cases}\dot{\xi}_{1}=\eta_{1},\\\
\dot{\eta}_{1}=-\sin(\xi_{1}),\end{cases}\qquad\quad\begin{cases}\dot{\xi}_{2}=\eta_{2},\\\
\dot{\eta}_{2}=-\sin(\xi_{2}).\end{cases}$ (3.37)

System (3.37) is fully normalized and it is the 2-dimensional Hamiltonian
system of two uncoupled normalized pendulums

$\dot{\xi}_{n}=\partial_{\eta_{n}}H,\quad\
\dot{\eta}_{n}=-\partial_{\xi_{n}}H,\quad\ n=1,2,$

with Hamiltonian

$H=H_{1}(\xi_{1},\eta_{1})+H_{2}(\xi_{2},\eta_{2}),\qquad
H_{n}(\xi_{n},\eta_{n})=\frac{1}{2}\eta_{n}^{2}+\big{(}1-\cos(\xi_{n})\big{)},\quad\
n=1,2.$ (3.38)

The constant term 1 in the formula of $H_{n}$ has been added just to give zero
energy to the elliptic equilibrium.

It is convenient to consider the system (3.35) as a perturbed double-pendulum
system where the perturbative parameter is given by $\sigma$, instead of
$\mu_{1},\mu_{2}$. Namely, using (3.33),

$\begin{aligned}
\dot{\xi}_{1}&=\eta_{1}-\sigma\Big{(}\frac{1}{3}+\tilde{\mu}_{1}(\sigma)\Big{)}\eta_{2},\\\
\dot{\eta}_{1}&=-\sin(\xi_{1}),\end{aligned}\qquad\quad\begin{aligned}
\dot{\xi}_{2}&=\eta_{2}-\sigma\big{(}1+\tilde{\mu}_{2}(\sigma)\big{)}\eta_{1},\\\
\dot{\eta}_{2}&=-\sin(\xi_{2}).\end{aligned}$ (3.39)

The conserved quantity in (3.36), divided by $\sigma$, is

$\begin{split}\mathcal{E}(\xi_{1},\eta_{1},\xi_{2},\eta_{2}):=&\Big{(}\frac{1}{3}+\tilde{\mu}_{1}(\sigma)\Big{)}H_{1}(\xi_{1},\eta_{1})+\big{(}1+\tilde{\mu}_{2}(\sigma)\big{)}H_{2}(\xi_{2},\eta_{2})\\\
&-\sigma\Big{(}\frac{1}{3}+\tilde{\mu}_{1}(\sigma)\Big{)}\big{(}1+\tilde{\mu}_{2}(\sigma)\big{)}\eta_{1}\eta_{2}.\end{split}$
(3.40)

##  4 Chaos for two weakly coupled pendulums

In this section we prove the following result about chaotic solutions of
system (3.39). We denote ${\mathbb{N}}_{0}:=\\{0,1,2,\ldots\\}$ the set of
nonnegative integers. Given an energy parameter $a\in(0,2)$, we denote by
$(\xi_{1}^{*}(t),\eta_{1}^{*}(t))$ the periodic solution of the pendulum
satisfying

$H_{1}(\xi_{1}^{*}(t),\eta_{1}^{*}(t))=a,\quad\ \xi_{1}^{*}(0)=0\quad\
\eta_{1}^{*}(0)>0.$ (4.1)

To emphasize its dependence on $a$ we will also denote
$\xi_{1}^{*}(t)=\xi_{1}^{*}(t;a)$ and $\eta_{1}^{*}(t)=\eta_{1}^{*}(t;a)$.

######  Proposition 4.1.

There exists a universal constant $a_{0}\in(0,2)$ (see Lemma 4.6) such that
the following holds. Let $a\in[a_{0},2)$ and let $T_{a}$ be the period of
$(\xi_{1}^{*}(t;a),\eta_{1}^{*}(t;a))$. There exist universal constants
$\sigma_{0}\in(0,1)$, $C_{1},C_{2}>0$ such that for every
$\sigma\in(0,\sigma_{0})$ there exists $M_{0}(\sigma)$ in the interval

$C_{1}\log(\sigma^{-1})\leq M_{0}(\sigma)\leq C_{2}\log(\sigma^{-1})$

such that the following properties hold. Let
$(m_{0},m_{1},m_{2},\ldots)=(m_{j})_{j\in{\mathbb{N}}_{0}}$ be any sequence of
integers with $m_{j}\geq M_{0}(\sigma)$ for all $j\in{\mathbb{N}}$. Then,
there exists a solution $(\xi_{1},\eta_{1},\xi_{2},\eta_{2})(t)$ of system
(3.39) such that the following holds.

  * $(i)$

The function $(\xi_{1},\eta_{1})$ satisfies

$\sup_{t\in{\mathbb{R}}}|\xi_{1}(t)-\xi_{1}^{*}(t)|\leq
C\sigma,\qquad\sup_{t\in{\mathbb{R}}}|\eta_{1}(t)-\eta_{1}^{*}(t)|\leq
C\sigma,$

for some universal constant $C>0$.

  * $(ii)$

There exists a sequence of times
$(t_{0},t_{1},t_{2},\ldots)=(t_{j})_{j\in{\mathbb{N}}_{0}}$ with

$t_{0}=0,\quad\ t_{j+1}=t_{j}+T_{a}(m_{j}+\theta_{j}),\quad\
0\leq\theta_{j}<1,\quad\ \forall j\in{\mathbb{N}}_{0}$ (4.2)

such that

${\eta_{2}(t_{j})}=1\quad\ \forall j\in{\mathbb{N}}_{0}.$

Moreover, there exists another sequence of times
$(\bar{t}_{j})_{j\in{\mathbb{N}}_{0}}$ satisfying $t_{j}<\bar{t}_{j}<t_{j+1}$
such that

$1<{\eta_{2}(t)}\leq 2+C\sigma\quad\forall
t\in(t_{j},\bar{t}_{j}),\qquad\quad-C\sigma\leq{\eta_{2}(t)}<1\quad\forall
t\in(\bar{t}_{j},t_{j+1})$ (4.3)

and

$\max_{t\in[t_{j},\bar{t}_{j}]}\eta_{2}(t)\geq
2-C\sigma,\qquad\min_{t\in[\bar{t}_{j},t_{j+1}]}\eta_{2}(t)\leq C\sigma$ (4.4)

for some universal constant $C>0$.

We remark that $\eta_{2}=1$ is the value around which $\eta_{2}(t)$ is
oscillating up and down with the randomly chosen sequence of times.

In order to prove Proposition 4.1 we shall find a partially hyperbolic
periodic orbit of the full system (3.39) and show that, for $\sigma>0$ small
enough, its stable and unstable invariant manifolds intersect transversally.
This will imply the existence of a Smale horseshoe and the existence of
symbolic chaotic dynamics.

######  Remark 4.2.

The pendulum energy $a$ in (4.1) is used as a free parameter only in this
section. Proposition 4.1 will be applied in Sections 5 and 6 only for
$a=a_{0}$. Since $a_{0}$ is a universal constant, for $a=a_{0}$ any quantity
depending only on $a$ becomes a universal constant. ∎

###  4.1 Partially hyperbolic periodic orbit

We start by searching for the partially hyperbolic periodic solution. The
unperturbed system ($\sigma=0$) has plenty of partially hyperbolic periodic
orbits, which are given for instance by the product of librations in the first
pendulum (in the plane $(\xi_{1},\eta_{1})$) and the saddle of the second
pendulum (in the plane $(\xi_{2},\eta_{2})$). We select one of these orbits
and we apply an implicit function theorem argument to prove the existence of a
nearby periodic orbit with the same period.

We consider the periodic orbit

$\mathcal{P}=\mathcal{P}(t;a):=(\xi_{1}^{*}(t;a),\eta_{1}^{*}(t;a),\pi,0)$
(4.5)

of the unperturbed system (3.37) (which is system (3.39) with $\sigma=0$),
where $(\xi_{1}^{*},\eta_{1}^{*})$ is defined in (4.1) and $a\in(0,2)$ will be
fixed at the end of subsection 4.2. Note that the elliptic equilibrium
$(\xi_{1},\eta_{1})=(0,0)$ of the first pendulum has energy $H_{1}(0,0)=0$,
and its saddle $(\xi_{1},\eta_{1})=(\pi,0)$, as well as its homoclinic orbits,
has energy $H_{1}(\pi,0)=2$. The solution $\mathcal{P}$ is supported on the
curve

${\mathbb{T}}_{0}:=\\{(\xi_{1},\eta_{1},\xi_{2},\eta_{2}):H_{1}(\xi_{1},\eta_{1})=a,\
\xi_{2}=\pi,\ \eta_{2}=0\\}.$ (4.6)

We denote by $T$ the period $T_{a}$ of the orbit $\mathcal{P}$ and by
$\omega:=2\pi/T$ its frequency. Of course ${\mathbb{T}}_{0},T,\omega$ depend
on the energy parameter $a$; in fact, all the quantities in the present
subsection and in the next one (included, in particular, the smallness radius
$\sigma_{1}$ given by Proposition 4.3) depend on $a$. Nonetheless, in general,
we do not indicate explicitly the dependence on $a$; we just underline that,
after fixing $a$, every quantity appearing in subsections 4.1 and 4.2 will be
determined, with no dependence on any other hidden parameter.

The unperturbed ($\sigma=0$) homoclinic manifold of $\mathcal{P}$ is

$W_{0}(\mathcal{P})=\\{(\xi_{1},\eta_{1},\xi_{2},\eta_{2}):H_{1}(\xi_{1},\eta_{1})=a,\
H_{2}(\xi_{2},\eta_{2})=2\\}.$ (4.7)

We consider its time-parametrization

$\Gamma_{0}^{\pm}:=\\{(\xi_{1}^{*}(\tau_{1}),\eta_{1}^{*}(\tau_{1}),q_{h}(\tau_{2}),p_{h}^{\pm}(\tau_{2})):\tau_{1},\tau_{2}\in{\mathbb{R}}\\},$
(4.8)

where

$(q_{h}(s),p_{h}^{\pm}(s))=\Big{(}2\arcsin(\tanh(s)),\,\pm\frac{2}{\cosh(s)}\Big{)}.$

Now we prove that the periodic orbit $\mathcal{P}$ persists when $0<\sigma\ll
1$. More precisely, we prove the following result.

######  Proposition 4.3.

Let $\mathcal{P}$ be the $T$-periodic orbit of (3.37) defined in (4.5). Then,
there exist constants ${\sigma_{1}}>0$, $C>0$ such that, for all
$0<\sigma<\sigma_{1}$, there exists a $T$-periodic solution
$\mathcal{P}_{\sigma}(t)$ of (3.39) which is $\sigma$-close to $\mathcal{P}$
in the $C^{1}$-topology, namely

$\|\mathcal{P}_{\sigma}-\mathcal{P}\|_{C^{1}({\mathbb{R}})}\leq C\sigma.$

Moreover $\mathcal{P}_{\sigma}$ possesses one stable and one unstable
hyperbolic direction.

Recall that system (3.39) has the energy (3.40) as first integral. Then, the
existence of a hyperbolic periodic orbit at each energy level is a consequence
of classical perturbation theory. However, Proposition 4.3 gives the existence
of a periodic orbit for a fixed period. This could be shown by proving that
the period is monotone with $\mathcal{E}$. Below, to make this paper
selfcontained, we give an alternative proof of Proposition 4.3 based on a
symmetry argument.

###### Proof of Proposition 4.3.

To prove the persistence of the periodic orbit $\mathcal{P}$ we use the fact
that the system (3.39) is reversible with respect to the involution

$\rho\colon({\mathbb{T}}\times{\mathbb{R}})^{2}\to({\mathbb{T}}\times{\mathbb{R}})^{2}\qquad\rho(\xi_{1},\eta_{1},\xi_{2},\eta_{2})=(-\xi_{1},\eta_{1},-\xi_{2},\eta_{2}).$

This means that, if we denote by $X$ the vector field of (3.39), then
$X\circ\rho=-\rho_{*}X$, where $\rho_{*}$ is the differential of $\rho$, which
acts on the tangent space ${\mathbb{R}}^{4}$. To apply an implicit function
theorem argument it is convenient to pass to action-angle coordinates on the
first pendulum (plane $(\xi_{1},\eta_{1})$). This will simplify the analysis
of the linearized problem in the tangential directions at the periodic orbit
$\mathcal{P}$. In the domain

$S:=\left\\{(x_{1},y_{1})\in{\mathbb{T}}\times{\mathbb{R}}:H_{1}(x_{1},y_{1})\in(0,2)\right\\},$

we consider the action-angle variables transformation

$\Phi\colon{\mathbb{T}}\times\mathcal{I}\to
S,\qquad(\xi_{1},\eta_{1})=\Phi(\theta,I)=(f(\theta,I),g(\theta,I))$

for some open interval $\mathcal{I}\subset{\mathbb{R}}$. If we fix
$(\xi_{1},\eta_{1})$ and call $2\kappa=H_{1}(\xi_{1},\eta_{1})$, we can
express $f$ and $g$ using elliptic functions in the following way:

$\begin{cases}f(\theta,I)=2\arcsin\left(\sqrt{\kappa}\,\mathrm{sn}\left(\dfrac{2\mathtt{K}(\kappa)}{\pi}\theta\,\Big{|}\,\kappa\right)\right)\\\\[11.38109pt]
g(\theta,I)=2\sqrt{\kappa}\,\,\mathrm{cn}\left(\dfrac{2\mathtt{K}(\kappa)}{\pi}\theta\,\Big{|}\,\kappa\right),\end{cases}$
(4.9)

where $\mathtt{K}(\kappa)$ is the complete elliptic integral of first kind and
$\mathrm{sn}$ and $\mathrm{cn}$ are the elliptic sine and the elliptic cosine
respectively (see e.g. [18]).

We now drop the sub-index from $\xi_{2},\eta_{2}$. Let us denote by

$\Psi=(\Phi,\mathrm{Id})\colon{\mathbb{T}}\times\mathcal{I}\times{\mathbb{T}}\times{\mathbb{R}}\to
S\times{\mathbb{T}}\times{\mathbb{R}},\quad\
\Psi(\theta,I,\xi,\eta)=(\Phi(\theta,I),\xi,\eta).$

The involution $\rho$ expressed in these new coordinates is just given by
$\nu=\Psi^{-1}\circ\rho\circ\Psi$. Since the elliptic sine is odd and the
elliptic cosine is even (with respect to its first variable), it is
straightforward to see that system (3.39) in the new coordinates is reversible
with respect to the involution

$\nu(\theta,I,\xi,\eta)=(-\theta,I,-\xi,\eta).$

We denote by $I\mapsto\Omega(I)$ the action-to-frequency map of the
unperturbed pendulum.

######  Remark 4.4.

Note that $\Omega^{\prime}(I)\neq 0$ for all $I\in\mathcal{I}$. ∎

The unperturbed periodic orbit $\mathcal{P}$ now reads as

$\theta(t)=\Omega(I_{0})t,\quad I(t)=I_{0},\quad\xi(t)=\pi,\quad\eta(t)=0$

for some $I_{0}\in\mathcal{I}$. We remark that $\Omega(I_{0})=\omega$, where
$\omega=2\pi/T$ was defined below (4.6). We consider the scaled time
$t\rightsquigarrow\omega t$, and the system (3.39) becomes

$\begin{cases}\omega\dot{\theta}&=\Omega(I)+\sigma\mathcal{R}_{1}(\theta,I,\eta),\\\
\omega\dot{I}&=\sigma\mathcal{R}_{2}(\theta,I,\eta),\\\
\omega\dot{\xi}&=\eta+\sigma\mathcal{R}_{3}(\theta,I),\\\
\omega\dot{\eta}&=-\sin(\xi)\end{cases}$ (4.10)

where the functions $\mathcal{R}_{i}$, $i=1,2,3$, are determined by the
relation

$[D\Psi(\theta,I,\xi,\eta)]^{-1}\begin{pmatrix}-(\frac{1}{3}+\tilde{\mu}_{1}(\sigma))\eta\\\
0\\\ -(1+\tilde{\mu}_{2}(\sigma))g(\theta,I)\\\
0\end{pmatrix}=\begin{pmatrix}\mathcal{R}_{1}(\theta,I,\eta)\\\
\mathcal{R}_{2}(\theta,I,\eta)\\\ \mathcal{R}_{3}(\theta,I)\\\
0\end{pmatrix}.$

We look for $2\pi$-periodic, smooth, reversible solutions of (4.10), namely
$u(t)=(\theta(t),I(t),\xi(t),\eta(t))$ such that $\nu u(-t)=u(t)$. In other
words, we look for solutions in the invariant set

$C^{1}_{\mathrm{odd,\,even}}({\mathbb{T}}):=\\{(\theta(t),I(t),\xi(t),\eta(t))\in
C^{1}({\mathbb{T}};{\mathbb{T}}\times\mathcal{I}\times{\mathbb{T}}\times{\mathbb{R}}):\theta(t),\xi(t)\,\,\mathrm{odd},\,\,I(t),\eta(t)\,\,\mathrm{even}\\}.$

Similarly, we define the space

$C^{0}_{\mathrm{even,\,odd}}({\mathbb{T}}):=\\{(g_{1}(t),g_{2}(t),g_{3}(t),g_{4}(t))\in
C^{0}({\mathbb{T}};{\mathbb{R}}^{4}):g_{1}(t),g_{3}(t)\,\,\mathrm{even},\,\,g_{2}(t),g_{4}(t)\,\,\mathrm{odd}\\}.$

######  Remark 4.5.

We note that $2\pi$-solutions of (4.10) correspond to $T$-periodic solutions
of (3.39). ∎

Let us define

$\mathcal{F}(\sigma;\cdot)\colon C^{1}_{\mathrm{odd,\,even}}({\mathbb{T}})\to
C^{0}_{\mathrm{even,\,odd}}({\mathbb{T}}),\qquad\mathcal{F}(\sigma;\theta,I,\xi,\eta)=\begin{pmatrix}\omega\dot{\theta}-\Omega(I)-\sigma\mathcal{R}_{1}(\theta,I,\eta)\\\
\omega\dot{I}-\sigma\mathcal{R}_{2}(\theta,I,\eta)\\\
\omega\dot{\xi}-\eta-\sigma\mathcal{R}_{3}(\theta,I)\\\
\omega\dot{\eta}+\sin(\xi)\end{pmatrix}.$

Since the maps $\Omega$, $\Psi$ and the vector field $X$ are analytic, we have
that $\mathcal{F}(\sigma;\cdot)$ is at least $C^{1}$. Then

$\mathcal{F}(0;\omega t,I_{0},\pi,0)=0.$

We now study the linearized system at the unperturbed solution. We fix
$(g_{1},g_{2},g_{3},g_{4})\in C^{0}_{\mathrm{even,\,odd}}({\mathbb{T}})$ and
we look for solutions of the linear system

$\begin{cases}\omega\partial_{t}{\theta}-\Omega^{\prime}(I_{0})I=g_{1}\\\
\omega\partial_{t}{I}=g_{2}\\\ \omega\partial_{t}{\xi}-\eta=g_{3}\\\
\omega\partial_{t}{\eta}-\xi=g_{4}.\end{cases}$ (4.11)

We observe that the above system is decoupled, hence we can study separately
the equations for $(\theta,I)$ and the ones for $(\xi,\eta)$. Concerning the
former, we first solve the equation for the actions. Since $g_{2}$ is odd we
have $\langle g_{2}\rangle=0$ and

$I-\langle I\rangle=(\omega\partial_{t})^{-1}g_{2},$

where we denote by $\langle\cdot\rangle$ the time average over ${\mathbb{T}}$
and we denote by $(\omega\partial_{t})^{-1}g_{2}$ the primitive of $g_{2}$
with zero average. Hence $I$ is determined up to its average. Substituting in
the equation for the angle we obtain

$\omega\partial_{t}\theta=\Omega^{\prime}(I_{0})(\omega\partial_{t})^{-1}g_{2}+\Omega^{\prime}(I_{0})\langle
I\rangle+g_{1}.$ (4.12)

Equation (4.12) can be solved only if the r.h.s. has zero average. Therefore
we fix

$\langle I\rangle=-\frac{\langle g_{1}\rangle}{\Omega^{\prime}(I_{0})},\quad\
\theta=(\omega\partial_{t})^{-1}\big{[}\Omega^{\prime}(I_{0})(\omega\partial_{t})^{-1}g_{2}+\Omega^{\prime}(I_{0})\langle
I\rangle+g_{1}\big{]}$

(note that $\Omega^{\prime}(I_{0})\neq 0$, see Remark 4.4). Concerning the
equations for the $(\xi,\eta)$ variables we have the following: by setting
$v:=\xi+\eta$, $w:=\xi-\eta$, $h:=g_{3}+g_{4}$ and $\tilde{h}:=g_{3}-g_{4}$ we
have that

$\omega\partial_{t}{v}-v=h,\qquad\omega\partial_{t}{w}+w=\tilde{h}.$ (4.13)

By Fourier series, one has that

$v(t)=\sum_{k\in{\mathbb{Z}}}\frac{h_{k}}{\mathrm{i}\omega
k-1}e^{\mathrm{i}kt},\qquad
w(t)=\sum_{k\in{\mathbb{Z}}}\frac{\tilde{h}_{k}}{\mathrm{i}\omega
k+1}e^{\mathrm{i}kt}$

are the unique solutions of (4.13). Since $\xi=(v+w)/2$ and $\eta=(v-w)/2$, we
recover the $\xi,\eta$ components of the solution of (4.11). From the explicit
expression of the solutions we have that for $g_{i}\equiv 0$, $i=1,\dots,4$,
the only solution of (4.11) is zero. Moreover the solutions are of class
$C^{1}$. This implies that $d\mathcal{F}(0;\omega t,I_{0},\pi,0)$ is
invertible and by the implicit function theorem there exists $\sigma_{1}>0$
and a $C^{1}$ function $\mathtt{g}\colon(-\sigma_{1},\sigma_{1})\to
C^{1}_{\mathrm{odd,\,even}}({\mathbb{T}})$ such that

$\mathcal{F}(\sigma;\mathtt{g}(\sigma))=0\quad\forall\sigma\in(-\sigma_{1},\sigma_{1}),\qquad\mathtt{g}(0;t)=(\omega
t,I_{0},\pi,0).$

We call $\mathcal{P}_{\sigma}$ the periodic orbit $\mathtt{g}(\sigma)$ written
in the original coordinates $(\xi_{1},\eta_{1},\xi_{2},\eta_{2})$. We also
notice that by (4.13) this orbit is hyperbolic in the $(\xi_{2},\eta_{2})$
directions. This concludes the proof. ∎

By classical theory of persistence of invariant manifolds,
$\mathcal{P}_{\sigma}$ has stable and unstable invariant manifolds
$W_{\sigma}^{s,u}(\mathcal{P}_{\sigma})$ that depend differentiably on
$\sigma$. Moreover these manifolds can be locally parametrized as $C^{1}$
graphs over the unperturbed invariant manifold (4.7).

###  4.2 Transverse intersection of invariant manifolds

In this section we prove that the stable and unstable invariant manifolds
$W_{\sigma}^{s,u}(\mathcal{P}_{\sigma})$ intersect transversally at some
point. Since the invariant manifolds have dimension $2$ and we look for
intersections within a $3$-dimensional energy level (see (3.40)), it is
sufficient to construct a $1$-dimensional section $\Lambda$ and measure the
distance between the manifolds on the projection of $\Lambda$.

We recall the time parameterization of the unperturbed separatrix
$\Gamma^{\pm}_{0}$ given in (4.8). By symmetry we can consider just a single
branch of the unperturbed homoclinic manifold, say
$\Gamma_{0}=\Gamma_{0}^{+}$. Let us consider a point
$z_{0}=z_{0}(\tau_{1},\tau_{2})\in\Gamma_{0}\subset\\{H_{1}=a,\,H_{2}=2\\}$
and define the section

$\Lambda:=\\{z_{0}+\lambda\nabla H_{1}(z):\lambda\in{\mathbb{R}}\\},$

where we use the notation
$H_{1}(\xi_{1},\eta_{1},\xi_{2},\eta_{2}):=H_{1}(\xi_{1},\eta_{1})$. The line
$\Lambda$ passes through $z_{0}$ and it is normal to $\Gamma_{0}$.

By the continuous dependence of the invariant manifolds on the parameters, for
$\sigma$ small enough, $\Lambda$ intersects transversally also
$W^{s,u}_{\sigma}(\mathcal{P}_{\sigma})$ at two points
$z_{\sigma}^{s,u}=z_{\sigma}^{s,u}(\tau_{1},\tau_{2})$. We use the unperturbed
energy of the first pendulum $H_{1}$ to measure the distance between
$z^{s}_{\sigma}$ and $z^{u}_{\sigma}$. Note that the gradient of $H_{1}$ never
vanishes on $\Gamma_{0}$, hence it is a good measure of a displacement in the
normal directions of $\Gamma_{0}$. We define the distance

$H_{1}(z_{\sigma}^{s})-H_{1}(z_{\sigma}^{u})=\sigma\,M+O(\sigma^{2}),$ (4.14)

where the first order of this distance is given by

$M=M(\tau_{1},\tau_{2}):=\frac{d}{d\sigma}_{|_{\sigma=0}}(H_{1}(z_{\sigma}^{s})-H_{1}(z_{\sigma}^{u})).$

By classical arguments (see for instance [31]) we have that the first order is
given by the Melnikov integral

$M(\tau_{1},\tau_{2})=\int_{-\infty}^{+\infty}DH_{1}(\Phi_{H}^{t}(\Gamma_{0}(\tau_{1},\tau_{2})))[Y(\Phi_{H}^{t}(\Gamma_{0}(\tau_{1},\tau_{2})))]\,dt,$

where $Y$ is the first order in $\sigma$ of the perturbation of the system
(3.39), namely

$Y(\xi_{1},\eta_{1},\xi_{2},\eta_{2}):=(-\eta_{2}/3,0,-\eta_{1},0),$

and $\Phi_{H}^{t}$ is the Hamiltonian flow of $H$ in (3.38). We observe that
(recall (4.8))

$\Phi_{H}^{t}(\Gamma_{0}(\tau_{1},\tau_{2}))=(\xi_{1}^{*}(\tau_{1}+t),\eta_{1}^{*}(\tau_{1}+t),q_{h}(\tau_{2}+t),p_{h}^{+}(\tau_{2}+t)).$

By the autonomous nature of system (3.39), the Melnikov integral depends just
on one parameter. We define $\tau:=\tau_{2}-\tau_{1}$ and we consider the
reduced Melnikov integral

$\mathcal{M}(\tau):=-\frac{1}{3}\int_{-\infty}^{+\infty}p_{h}^{+}(\tau+s)\,\sin(\xi_{1}^{*}(s))\,ds.$

Note that, for all $\tau_{1}\in{\mathbb{R}}$, one has
$\mathcal{M}(\tau)=M(\tau_{1},\tau_{1}+\tau)=M(0,\tau)$. We now prove the
following.

######  Lemma 4.6.

There exists a universal constant $a_{0}\in(0,2)$ such that, for all
$a\in[a_{0},2)$, the reduced Melnikov integral $\mathcal{M}(\tau)$ has a non-
degenerate zero at $\tau=0$.

###### Proof.

We observe that $\mathcal{M}(0)=0$ because $\xi_{1}^{*}(t)$ is an odd
function, while $p_{h}^{+}(t)$ is even. The derivative of the reduced Melnikov
integral at $\tau=0$ is

$\frac{d}{d\tau}\mathcal{M}(0)=\frac{1}{3}\int_{-\infty}^{+\infty}\sin(q_{h}(s))\sin(\xi_{1}^{*}(s))\,ds=\frac{2}{3}\int_{0}^{+\infty}\sin(q_{h}(s))\sin(\xi_{1}^{*}(s))\,ds.$

We recall that $(\xi_{1}^{*},\eta_{1}^{*})$ in (4.5) depends on the parameter
$a$, and we explicitly indicate the dependence on $a$ of the integral we want
to study, denoting

$J(a):=\int_{0}^{+\infty}\sin(q_{h}(s))\,\sin(\xi_{1}^{*}(s;a))\,ds.$

We have to prove that $J(a)$ is non zero for some value of $a\in(0,2)$. By the
classical theorem of continuous dependence on initial data for ODEs, one has
the following pointwise convergence:

$\text{for every }s\in[0,\infty),\quad\lim_{a\to 2}\xi_{1}^{*}(s;a)=q_{h}(s)$

(even more, the convergence is uniform on compact intervals). Hence, for every
$s\in[0,\infty)$, $f_{a}(s)$ converges to $f_{2}(s)$ as $a\to 2$, where

$f_{a}(s):=\sin(q_{h}(s))\sin(\xi_{1}^{*}(s;a)),\quad\
f_{2}(s):=\sin^{2}(q_{h}(s)).$

Moreover, since $q_{h}(s)\in[0,\pi)$ for all $s\in[0,\infty)$, one has

$|f_{a}(s)|\leq|\sin(q_{h}(s))|=\sin(q_{h}(s))=\frac{2\sinh(s)}{\cosh^{2}(s)}=:g(s)\quad\forall
s\in[0,\infty),$

and $g\in L^{1}(0,\infty)$. Hence, by the dominated convergence theorem,

$J_{*}:=\lim_{a\to
2}J(a)=\int_{0}^{\infty}f_{2}(s)\,ds=\int_{0}^{\infty}g^{2}(s)\,ds.$

The limit $J_{*}$ is finite by the exponential decay of $g^{2}$, and it is
positive because $g^{2}$ is positive. Hence there exists $a_{0}\in(0,2)$ such
that $|J(a)-J_{*}|\leq J_{*}/2$ for all $a\in[a_{0},2)$, and the lemma is
proved. ∎

By the above lemma and the Implicit Function Theorem, for $\sigma>0$ small
enough, there exists at least one zero of the distance (4.14), with transverse
intersection.

###  4.3 Symbolic dynamics

We introduce the section

$\Pi:=\\{(\xi_{1},\eta_{1},\xi_{2},\eta_{2}):\xi_{2}=0,\ \eta_{2}>0,\
\mathcal{E}(\xi_{1},\eta_{1},\xi_{2},\eta_{2})=\mathcal{E}(\mathcal{P}_{\sigma})\\},$

where $\mathcal{E}(\mathcal{P}_{\sigma})$ is the value of the prime integral
$\mathcal{E}$ in (3.40) at the solution $\mathcal{P}_{\sigma}$. This section
is transverse to the unperturbed flow $(\sigma=0)$ at a point of
$W_{0}(\mathcal{P})$ and so, for $\sigma>0$ small enough, also to the
perturbed one. Moreover, by Lemma 4.6, it contains points of
$W^{s}_{\sigma}(\mathcal{P}_{\sigma})\pitchfork
W^{u}_{\sigma}(\mathcal{P}_{\sigma})$ (where $\pitchfork$ means transverse
intersection).

Denote by $\Phi^{t}$ the flow of system (3.39). Fixed a point $z\in\Pi$, we
define $\mathcal{T}(z)>0$ as the first (forward) return time to $\Pi$. For
those points $z$ that do not hit back the section $\Pi$ (for instance, the
points of $W_{\sigma}^{s}(\mathcal{P}_{\sigma})$), we set
$\mathcal{T}(z)=+\infty$. We define the open set $\mathcal{U}\subset\Pi$ as

$\mathcal{U}:=\\{z\in\Pi:\mathcal{T}(z)<+\infty\\}$

and the associated Poincaré map $\mathtt{P}\colon\mathcal{U}\subset\Pi\to\Pi$
by $\mathtt{P}(z)=\Phi^{t}(z)|_{t=\mathcal{T}(z)}$.

######  Proposition 4.7.

(Smale Horseshoe) There exist universal constants $\sigma_{0},C_{1},C_{2}>0$
such that for all $\sigma\in(0,\sigma_{0})$ there exists a positive integer
$M_{0}(\sigma)$ in the interval

$C_{1}\log(\sigma^{-1})\leq M_{0}(\sigma)\leq C_{2}\log(\sigma^{-1})$ (4.15)

such that the Poincaré map $\mathtt{P}$ possesses an invariant set
$Y\subset\mathcal{U}$ whose dynamics is conjugated to the infinite symbols
shift. Namely, there exists a homeomorphism $h\colon\mathcal{A}\to Y$, where

$\mathcal{A}:=\\{\omega=\\{\omega_{k}\\}_{k\in{\mathbb{Z}}}:\omega_{k}\in\mathbb{N},\
\ \omega_{k}\geq M_{0}(\sigma)\ \forall k\in{\mathbb{Z}}\\},$

such that $\mathtt{P}_{|_{Y}}=h\circ\mathfrak{d}\circ h^{-1}$ where
$\mathfrak{d}\colon\mathcal{A}\to\mathcal{A}$ is the shift

$(\mathfrak{d}\omega)_{k}=\omega_{k+1},\quad\ k\in{\mathbb{Z}}.$

Moreover $h^{-1}$ can be defined as follows. Associated to $z\in Y$ one can
define the sequence of hitting times

$\displaystyle t_{0}=0,\quad\
t_{k}=t_{k-1}+\mathcal{T}(\mathtt{P}^{k-1}z)\quad\mathrm{for}\,\,\,k\geq
1,\quad\
t_{k}=t_{k+1}-\mathcal{T}(\mathtt{P}^{k}z)\quad\mathrm{for}\,\,\,k\leq-1,$

and $h^{-1}(z):=\omega=(\omega_{k})_{k\in{\mathbb{Z}}}$, with

$\omega_{k}=\left\lfloor\frac{t_{k}-t_{k-1}}{T}\right\rfloor,$

where $T=T_{a}$ is the period of the periodic orbit $\mathcal{P}_{\sigma}$,
and $\lfloor\cdot\rfloor$ denotes the integer part.

###### Proof.

The proof follows the same lines as the construction of symbolic dynamics done
by Moser in Chapter 3 of [30]. ∎

Proposition 4.7 concludes the proof of Proposition 4.1. Note that the return
times to the section are large since orbits get close to the hyperbolic
periodic orbit.

##  5 Back to the truncated effective system

In Section 4 we have proved the existence of chaotic solutions for system
(3.39). These solutions are global in time. In the next lemma we obtain the
corresponding solutions of the truncated effective system (3.3).

######  Lemma 5.1.

Let $a=a_{0}$ in Proposition 4.1, and let $\sigma_{0}$ be the corresponding
universal constant given by Proposition 4.1. Let $m,p$ be two integers, $2\leq
m<p$, with ratio $\sigma:=m/p$ in the interval $(0,\sigma_{0})$. Define
$\alpha_{1},\ldots\alpha_{4}$ by (3.2),
$\tilde{\mu}_{1}(\sigma),\tilde{\mu}_{2}(\sigma)$ by (3.33), and $\gamma$ by
the second identity in (3.25). Assume the hypotheses of Proposition 4.1, and
consider the solution $(\xi_{1}(t),\eta_{1}(t),\xi_{2}(t),\eta_{2}(t))$ of
system (3.39) obtained in Proposition 4.1.

Consider any three real numbers $\mathtt{A}_{1},q_{1},q_{2}$, with
$\mathtt{A}_{1}>0$, and define the following constants: define $c_{123}$ by
(3.31), define $c_{234}=c_{123}/\gamma$, define $\mathtt{B}$ by the second
identity in (3.23), define $\mathtt{A}_{2}$ by (3.24), define $b_{1},b_{2}$ by
(3.20), define $E_{1},E_{2}$ by (3.12). Define the functions

$\displaystyle S_{1}(t)$
$\displaystyle:=E_{1}-q_{1}-\mathtt{A}_{1}\eta_{1}(\mathtt{B}t),$ (5.1)
$\displaystyle S_{2}(t)$
$\displaystyle:=E_{2}-q_{1}-q_{2}-\mathtt{A}_{1}\eta_{1}(\mathtt{B}t)-\mathtt{A}_{2}\eta_{2}(\mathtt{B}t),$
$\displaystyle S_{3}(t)$
$\displaystyle:=q_{1}-q_{2}+\mathtt{A}_{1}\eta_{1}(\mathtt{B}t)-\mathtt{A}_{2}\eta_{2}(\mathtt{B}t),$
$\displaystyle S_{4}(t)$
$\displaystyle:=q_{2}+\mathtt{A}_{2}\eta_{2}(\mathtt{B}t),$
$\displaystyle\varphi_{123}(t)$ $\displaystyle:=\xi_{1}(\mathtt{B}t),$
$\displaystyle\varphi_{234}(t)$ $\displaystyle:=\xi_{2}(\mathtt{B}t).$

Then
$(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t),c_{123},c_{234})$
satisfies (3.3) for all $t\in{\mathbb{R}}$.

###### Proof.

We apply Lemma 3.6-$(ii)$ with $\lambda=1$ to go back from system (3.39) to
system (3.19), then Lemma 3.5-$(ii)$ to go back from system (3.19) to system
(3.10), then Lemma 3.4-$(ii)$ to go back from system (3.10) to system (3.8),
and finally Lemma 3.3-$(ii)$ to go back from system (3.8) to system (3.3). ∎

###  5.1 Positivity of the superactions

Now we come to the question whether the solutions of system (3.3) obtained in
Lemma 5.1 satisfy the inequalities (3.4). As a first step, we study the
constant terms

$E_{1}-q_{1},\qquad E_{2}-q_{1}-q_{2},\qquad q_{1}-q_{2},\qquad q_{2}$ (5.2)

appearing in the definition of $S_{1},\ldots,S_{4}$ in Lemma 5.1. We compute
the formula of $E_{1},E_{2}$ as functions of $q_{1},q_{2}$: from (3.12) and
(3.20) we get

$E_{1}=\frac{\alpha_{1}^{2}+2\alpha_{3}^{2}}{\alpha_{1}^{2}}q_{1}-\frac{2\alpha_{3}^{2}+\alpha_{4}^{2}}{\alpha_{1}^{2}}q_{2},\qquad
E_{2}=-\frac{\alpha_{3}^{2}-\alpha_{2}^{2}}{\alpha_{2}^{2}}q_{1}+\frac{\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}}{\alpha_{2}^{2}}q_{2}.$
(5.3)

Next, we observe in the following lemma that $q_{1},q_{2}$ can be chosen such
that the constant terms (5.2) are all positive.

######  Lemma 5.2.

Let $q_{1},q_{2}$ be positive real numbers with ratio $q_{1}/q_{2}$ in the
interval $(1+\frac{r}{2},1+r)$, $r=\alpha_{4}^{2}/\alpha_{3}^{2}$, and let
$E_{1},E_{2}$ be defined by (5.3). Then the constants (5.2) are all positive.
In particular, if

$q_{1}=\Big{(}1+\frac{2}{3}r\Big{)}q_{2},$ (5.4)

then

$E_{1}-q_{1}=\frac{1}{3}\frac{\alpha_{4}^{2}}{\alpha_{1}^{2}}q_{2},\qquad
E_{2}-q_{1}-q_{2}=\frac{1}{3}\frac{\alpha_{4}^{2}}{\alpha_{2}^{2}}q_{2},\qquad
q_{1}-q_{2}=\frac{2}{3}\frac{\alpha_{4}^{2}}{\alpha_{3}^{2}}q_{2}.$ (5.5)

###### Proof.

By (5.3),

$\displaystyle E_{1}-q_{1}$
$\displaystyle=\Big{(}\frac{2\alpha_{3}^{2}}{\alpha_{1}^{2}}\,\frac{q_{1}}{q_{2}}-\frac{2\alpha_{3}^{2}+\alpha_{4}^{2}}{\alpha_{1}^{2}}\Big{)}q_{2},\qquad
E_{2}-q_{1}-q_{2}=\Big{(}-\frac{\alpha_{3}^{2}}{\alpha_{2}^{2}}\,\frac{q_{1}}{q_{2}}+\frac{\alpha_{3}^{2}+\alpha_{4}^{2}}{\alpha_{2}^{2}}\Big{)}q_{2},$
$\displaystyle q_{1}-q_{2}$
$\displaystyle=\Big{(}\frac{q_{1}}{q_{2}}-1\Big{)}q_{2}.$

We write the ratio $q_{1}/q_{2}$ as $1+\vartheta r$, where
$\vartheta\in{\mathbb{R}}$ is a free parameter and
$r=\alpha_{4}^{2}/\alpha_{3}^{2}$. Then

$E_{1}-q_{1}=\frac{\alpha_{4}^{2}q_{2}}{\alpha_{1}^{2}}(2\vartheta-1),\qquad
E_{2}-q_{1}-q_{2}=\frac{\alpha_{4}^{2}q_{2}}{\alpha_{2}^{2}}(1-\vartheta),\qquad
q_{1}-q_{2}=\frac{\alpha_{4}^{2}q_{2}}{\alpha_{3}^{2}}\vartheta.$

The minimum

$\min\\{2\vartheta-1,1-\vartheta,\vartheta,1\\}=\min\\{2\vartheta-1,1-\vartheta\\}$

is positive for $\vartheta\in(\frac{1}{2},1)$, and it reaches its maximum
value at $\vartheta=2/3$. ∎

Note that, by (3.2), the ratio $r=\alpha_{4}^{2}/\alpha_{3}^{2}$ tends to $4$
as $\sigma=m/p\to 0$. By (3.25), (3.33), the constant $\mathtt{A}_{2}$ in
Lemma 5.1 satisfies

$\mathtt{A}_{2}=\frac{\mathtt{A}_{1}}{\gamma}=\frac{1+3\sigma+3\sigma^{2}}{3+9\sigma+7\sigma^{2}}\mathtt{A}_{1}\leq\mathtt{A}_{1}.$
(5.6)

The solutions $\eta_{1},\eta_{2}$, constructed in Proposition 4.1 and
appearing in Lemma 5.1, satisfy

$\sup_{t\in{\mathbb{R}}}|\eta_{1}(t)|\leq
3,\qquad\sup_{t\in{\mathbb{R}}}|\eta_{2}(t)|\leq 3$ (5.7)

(more accurate estimates about $\eta_{1},\eta_{2}$ have been obtained in
Proposition 4.1). Thus, we prove the following bound for $S_{n}$ from below.

######  Lemma 5.3.

Let $q_{2},\mathtt{A}_{1}$ be any two positive real numbers, and define
$q_{1}$ by (5.4). If

$\mathtt{A}_{1}\leq\frac{q_{2}}{9},$ (5.8)

then the functions $S_{n}(t)$ defined in Lemma 5.1 satisfy

$S_{n}(t)\geq\frac{q_{2}}{2}>0\qquad\forall
t\in{\mathbb{R}}\quad\text{and}\quad\forall n=1,2,3,4.$ (5.9)

###### Proof.

By (5.5) and (5.6), the functions $S_{n}(t)$ defined in Lemma 5.1 satisfy for
all $t\in{\mathbb{R}}$

$\displaystyle S_{1}(t)$ $\displaystyle\geq
E_{1}-q_{1}-3\mathtt{A}_{1}=\frac{1}{3}\frac{\alpha_{4}^{2}}{\alpha_{1}^{2}}q_{2}-3\mathtt{A}_{1},$
$\displaystyle S_{2}(t)$ $\displaystyle\geq
E_{2}-q_{1}-q_{2}-3\mathtt{A}_{1}-3\mathtt{A}_{2}\geq\frac{1}{3}\frac{\alpha_{4}^{2}}{\alpha_{2}^{2}}q_{2}-6\mathtt{A}_{1},$
$\displaystyle S_{3}(t)$ $\displaystyle\geq
q_{1}-q_{2}-3\mathtt{A}_{1}-3\mathtt{A}_{2}\geq\frac{2}{3}\frac{\alpha_{4}^{2}}{\alpha_{3}^{2}}q_{2}-6\mathtt{A}_{1},$
$\displaystyle S_{4}(t)$ $\displaystyle\geq q_{2}-3\mathtt{A}_{2}\geq
q_{2}-3\mathtt{A}_{1}.$

By (3.2), one has

$\frac{\alpha_{4}^{2}}{\alpha_{1}^{2}}>4,\qquad\frac{\alpha_{4}^{2}}{\alpha_{2}^{2}}>4,\qquad\frac{2\alpha_{4}^{2}}{\alpha_{3}^{2}}>4,$

which, together with (5.8), implies (5.9). ∎

By Lemma 5.3, the condition (5.8) implies the meaningfulness condition (3.4)
for the solutions of system (3.3). By Lemma 5.3, now we have two free
parameters, which are $q_{2}$ and $\mathtt{A}_{1}$, related by the inequality
$0<\mathtt{A}_{1}\leq q_{2}/9$, while $q_{1}$ is now constrained by formula
(5.4).

###  5.2 Initial data in the normal form ball

The special solutions of system (3.3) defined in Lemma 5.1 will be compared,
by a Gronwall argument, with those of the full (i.e., non-truncated) effective
system (2.18), (2.19) starting at the same initial data at time $t=0$. The
initial data we are interested in correspond to functions $u_{0}(x)$ in the
ball (2.8), because every $u_{0}$ in that ball produces a solution of the
Cauchy problem (2.2), (2.9) that remains, for a sufficiently long interval of
time, in the domain where the normal form transformation is well-defined, as
is explained quantitatively in Lemma 2.2. Recall that $\delta_{1}>0$ in (2.8)
is a universal constant, and $m_{1}$ is defined in (2.6). The following lemma
deals with the ball (2.8) written in terms of $S_{n}$.

######  Lemma 5.4.

Consider the solutions of system (3.3) given by Lemma 5.1, and assume (5.4),
(5.8). Then

$\sum_{n=1}^{4}\alpha_{n}^{2s}S_{n}(t)\leq 8\alpha_{4}^{2s}q_{2}\quad\forall
t\in{\mathbb{R}},\quad s\in[1,\infty).$ (5.10)

###### Proof.

For all $t\in{\mathbb{R}}$, if the smallness condition (5.8) is satisfied,
then the solutions of system (3.3) obtained in Lemma 5.1 satisfy

$\displaystyle\sum_{n=1}^{4}\alpha_{n}^{2s}S_{n}$
$\displaystyle\leq\alpha_{1}^{2s}(E_{1}-q_{1}+3\mathtt{A}_{1})+\alpha_{2}^{2s}(E_{2}-q_{1}-q_{2}+6\mathtt{A}_{1})+\alpha_{3}^{2s}(q_{1}-q_{2}+6\mathtt{A}_{1})+\alpha_{4}^{2s}(q_{2}+3\mathtt{A}_{1})$
$\displaystyle\leq\alpha_{1}^{2s}\frac{2\alpha_{4}^{2}q_{2}}{\alpha_{1}^{2}}+\alpha_{2}^{2s}\frac{2\alpha_{4}^{2}q_{2}}{\alpha_{2}^{2}}+\alpha_{3}^{2s}\frac{2\alpha_{4}^{2}q_{2}}{\alpha_{3}^{2}}+\alpha_{4}^{2s}2q_{2}\leq
8\alpha_{4}^{2s}q_{2},$

where we have used the bounds in (5.7) for $\eta_{1},\eta_{2}$, the identities
(5.5) for the constants terms (5.2), the bound (5.8) for $\mathtt{A}_{1}$, the
bound (5.6) for $\mathtt{A}_{2}$, and the fact that $s\geq 1$,
$\alpha_{1}<\dots<\alpha_{4}$. ∎

A consequence of this lemma is the following. If one chooses $q_{2}$ such that

$8\alpha_{4}^{2m_{1}}q_{2}\leq\delta_{1}^{2},$ (5.11)

where $m_{1},\delta_{1}$ are the universal constants of the ball (2.8), then

$\sum_{n=1}^{4}\alpha_{n}^{2m_{1}}S_{n}(t)\leq\delta^{2}\qquad\forall
t\in{\mathbb{R}}.$ (5.12)

###  5.3 Construction of a compatible initial datum

Given a trigonometric polynomial $u\in C({\mathbb{T}}^{d},{\mathbb{C}})$,
Fourier supported on the set
$\\{k\in{\mathbb{Z}}^{d}:|k|\in\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}\\}$,
with Fourier coefficients $u_{k}$, we use the superscript $u$ to denote

$S_{n}^{(u)}:=\sum_{|k|=\alpha_{n}}|u_{k}|^{2},\qquad
B_{n}^{(u)}:=\sum_{|k|=\alpha_{n}}u_{k}u_{-k},\qquad n=1,2,3,4,$ (5.13)
$Z_{123}^{(u)}:=B_{1}^{(u)}B_{2}^{(u)}\overline{B_{3}^{(u)}},\qquad\rho_{123}^{(u)}:=|Z_{123}^{(u)}|,\qquad
c_{123}^{(u)}:=\frac{3}{8}\rho_{123}^{(u)}\alpha_{1}\alpha_{2}\alpha_{3},$
(5.14)

analogous definitions for $Z_{234}^{(u)}$, $\rho_{234}^{(u)}$,
$c_{234}^{(u)}$, and, if $\rho_{123}^{(u)},\rho_{234}^{(u)}$ are positive, we
define the angles $\varphi_{123}^{(u)}$,
$\varphi_{234}^{(u)}\in{\mathbb{T}}={\mathbb{R}}/2\pi{\mathbb{Z}}$ by the
identities

$Z_{123}^{(u)}=\rho_{123}^{(u)}\exp(i\varphi_{123}^{(u)}),\qquad
Z_{234}^{(u)}=\rho_{234}^{(u)}\exp(i\varphi_{234}^{(u)}).$ (5.15)

We consider the following question:

_Given a solution
$(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$ of
system (3.3) obtained in Lemma 5.1, and taken, in particular, its value at
time $t=0$, does there exist a function $u_{0}(x)$ in the ball (2.8), Fourier
supported on the spheres of radius
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$, such that_

$\begin{cases}S_{n}^{(u_{0})}=S_{n}(0),\quad n=1,2,3,4,\\\
\varphi_{123}^{(u_{0})}=\varphi_{123}(0),\quad\varphi_{234}^{(u_{0})}=\varphi_{234}(0),\\\
c_{123}^{(u_{0})}=c_{123},\quad c_{234}^{(u_{0})}=c_{234}\quad?\end{cases}$
(5.16)

The equations for the angles must be interpreted as identities of elements of
${\mathbb{T}}={\mathbb{R}}/2\pi{\mathbb{Z}}$. The affirmative answer to this
question is given in Lemma 5.8 below, whose proof uses the next three simple
preparatory lemmas.

######  Lemma 5.5.

Let $(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$
be a solution of system (3.3) obtained in Lemma 5.1. Assume that
$q_{1},q_{2},\mathtt{A}_{1}$ satisfy (5.4), (5.8). If, in addition,
$\mathtt{A}_{1},q_{2}$ satisfy

$\mathtt{A}_{1}\leq\Big{(}\frac{3}{32}\,\frac{\alpha_{1}^{3}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}\Big{)}^{\frac{1}{2}}q_{2}^{\frac{3}{2}},$
(5.17)

then there exist real numbers $r_{n},\psi_{n}$, $n=1,2,3,4$, with

$0<r_{n}\leq S_{n}(0),$ (5.18)

such that

$\displaystyle\psi_{1}+\psi_{2}-\psi_{3}$
$\displaystyle=\varphi_{123}(0),\quad$
$\displaystyle\quad\frac{3}{8}(r_{1}\alpha_{1})(r_{2}\alpha_{2})(r_{3}\alpha_{3})$
$\displaystyle=c_{123},$ (5.19) $\displaystyle\psi_{2}+\psi_{3}-\psi_{4}$
$\displaystyle=\varphi_{234}(0),\quad$
$\displaystyle\quad\frac{3}{8}(r_{2}\alpha_{2})(r_{3}\alpha_{3})(r_{4}\alpha_{4})$
$\displaystyle=c_{234}.$ (5.20)

###### Proof.

Regarding $(\psi_{1},\psi_{2},\psi_{3},\psi_{4})$, there are infinitely many
solutions, because they are 4 unknowns that have to satisfy just 2 linear
constraints. For example, we can fix
$(\psi_{1},\psi_{2},\psi_{3},\psi_{4})=(\varphi_{123}(0),0,0,-\varphi_{234}(0))$.
Regarding $r_{n}$, we first recall that, from Lemma 5.1, the constants
$c_{123}$, $c_{234}$ are

$c_{123}=\frac{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}{2}\mathtt{A}_{1}^{2},\qquad
c_{234}=\frac{c_{123}}{\gamma},$

where $\gamma$ is defined in (3.25) and satisfies $1<\gamma<3$. Therefore
$c_{123},c_{234}$ are positive, because $\mathtt{A}_{1}$ is positive. We have
to choose $r_{n}$ such that

$\displaystyle(\alpha_{1}r_{1})(\alpha_{2}r_{2})(\alpha_{3}r_{3})$
$\displaystyle=\frac{8}{3}c_{123}=\frac{4}{3}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})\mathtt{A}_{1}^{2},$
$\displaystyle(\alpha_{2}r_{2})(\alpha_{3}r_{3})(\alpha_{4}r_{4})$
$\displaystyle=\frac{8}{3}\frac{c_{123}}{\gamma}=\frac{4}{3}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})\mathtt{A}_{1}^{2}\frac{1}{\gamma}.$

Hence we fix

$r_{1}=\frac{r_{0}}{\alpha_{1}},\quad r_{2}=\frac{r_{0}}{\alpha_{2}},\quad
r_{3}=\frac{r_{0}}{\alpha_{3}},\quad
r_{4}=\frac{r_{0}}{\gamma\alpha_{4}},\qquad
r_{0}:=\Big{(}\frac{4}{3}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})\mathtt{A}_{1}^{2}\Big{)}^{\frac{1}{3}}.$

Thus, $r_{n}$ are all positive and satisfy the required identities. It only
remains to check that $r_{n}\leq S_{n}(0)$. By (5.9), we know that
$q_{2}/2\leq S_{n}(0)$ for all $n=1,2,3,4$. Since
$\alpha_{1}<\alpha_{2}<\alpha_{3}<\alpha_{4}$ and $\gamma>1$, we have
$r_{4}<r_{3}<r_{2}<r_{1}$. Hence it is sufficient to check that $r_{1}\leq
q_{2}/2$, and, by the definition of $r_{1}=r_{0}/\alpha_{1}$, this holds if
$\mathtt{A}_{1},q_{2}$ satisfy (5.17). ∎

######  Lemma 5.6.

Let $s,r,\psi$ be real numbers such that $0<r\leq s$. Then there exists
$z_{1},z_{2}\in{\mathbb{C}}\setminus\\{0\\}$ such that
$|z_{1}|^{2}+|z_{2}|^{2}=s$ and $2z_{1}z_{2}=re^{i\psi}$.

######  Lemma 5.7.

Let $s_{n},r_{n},\psi_{n}$, $n=1,2,3,4$, be real numbers such that
$0<r_{n}\leq s_{n}$. Then, in any dimension $d\geq 1$, there exists a
trigonometric polynomial $u_{0}\in C({\mathbb{T}}^{d},{\mathbb{C}})$, Fourier
supported on the set
$\\{k\in{\mathbb{Z}}^{d}:|k|\in\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}\\}$,
such that, recalling the notation (5.13),

$S_{n}^{(u_{0})}=s_{n},\quad B_{n}^{(u_{0})}=r_{n}e^{i\psi_{n}},\quad
n=1,2,3,4.$ (5.21)

###### Proof.

For each $n=1,2,3,4$, fix an integer vector $k_{n}\in{\mathbb{Z}}^{d}$ with
$|k_{n}|=\alpha_{n}$, and apply Lemma 5.6 to determine two nonzero complex
numbers $z_{1,n},z_{2,n}$ such that

$|z_{1,n}|^{2}+|z_{2,n}|^{2}=s_{n},\qquad 2z_{1,n}z_{2,n}=r_{n}e^{i\psi_{n}}.$

We define $u_{0}$ as the trigonometric polynomial having $z_{1,n},z_{2,n}$ as
Fourier coefficients for the frequencies $k_{n},-k_{n}$, and having no other
frequencies in its support, i.e.

$u_{0}(x)=\sum_{n=1}^{4}(z_{1,n}e^{ik_{n}\cdot x}+z_{2,n}e^{-ik_{n}\cdot x}).$

Then

$S_{n}^{(u_{0})}=|z_{1,n}|^{2}+|z_{2,n}|^{2}=s_{n},\qquad
B_{n}^{(u_{0})}=2z_{1,n}z_{2,n}=r_{n}e^{i\psi_{n}}.\qed$

Lemma 5.7 deals with trigonometric polynomials supported on just one pair
$(k_{n},-k_{n})$ of points on the sphere $\\{k:|k|=\alpha_{n}\\}$; this is the
minimal situation, valid in any dimension $d\geq 1$. Of course, in dimension
$d\geq 2$ the Fourier support can contain more than one pair of opposite
frequencies on the same sphere, and therefore the construction of $u_{0}$ with
prescribed $S_{n}^{(u_{0})},B_{n}^{(u_{0})}$ has even more free parameters at
disposal.

The following lemma gives the answer to question (5.16).

######  Lemma 5.8.

Let $(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$
be a solution of system (3.3) obtained in Lemma 5.1. Also assume that
$q_{1},q_{2},\mathtt{A}_{1}$ satisfy (5.4), (5.8), (5.17). Then there exists a
trigonometric polynomial $u_{0}\in C({\mathbb{T}}^{d},{\mathbb{C}})$, Fourier
supported on the set
$\\{k\in{\mathbb{Z}}^{d}:|k|\in\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}\\}$,
satisfying all the identities in (5.16). If, in addition, $q_{2}$ satisfies
(5.11), then $u_{0}$ belongs to the ball (2.8).

###### Proof.

By Lemma 5.5, there exist constants $r_{n},\psi_{n}$ satisfying (5.18),
(5.19), (5.20). Define $s_{n}:=S_{n}(0)$, so that (5.18) becomes $0<r_{n}\leq
s_{n}$. By Lemma 5.7, there exists a trigonometric polynomial $u_{0}$, with
the desired Fourier support, satisfying (5.21). By the first identity in
(5.21) we directly have

$S_{n}^{(u_{0})}=s_{n}=S_{n}(0),\qquad n=1,2,3,4.$

By the second identity in (5.21), the first definition in (5.14), and the
first identity in (5.19), we obtain

$Z_{123}^{(u_{0})}=r_{1}r_{2}r_{3}\exp(i(\psi_{1}+\psi_{2}-\psi_{3}))=r_{1}r_{2}r_{3}\exp(i\varphi_{123}(0)).$
(5.22)

By (5.22), by the second and third definition in (5.14), and by the second
identity in (5.19), we get

$\rho_{123}^{(u_{0})}=r_{1}r_{2}r_{2},\qquad
c_{123}^{(u_{0})}=\frac{3}{8}r_{1}r_{2}r_{3}\alpha_{1}\alpha_{2}\alpha_{3}=c_{123}.$

Moreover, since $r_{1}r_{2}r_{3}>0$, (5.22) is a polar representation of
$Z_{123}^{(u_{0})}$, and hence $\varphi_{123}^{(u_{0})}=\varphi_{123}(0)$ as
elements of ${\mathbb{T}}={\mathbb{R}}/2\pi{\mathbb{Z}}$. Similar proof
applies for $c_{234}$, $\varphi_{234}(0)$.

Finally, if $q_{2}$ satisfies (5.11), then, by Lemma 5.4, bound (5.12) holds;
in particular, this bound at time $t=0$ implies that $u_{0}$ belongs to the
ball (2.8). ∎

###  5.4 Joining the two amplitude parameters

In Lemma 5.1 the solutions of system (3.3) obtained from Proposition 4.1 are
described by the three independent parameters $q_{1},q_{2},\mathtt{A}_{1}$.
Then, to get the positivity of the functions $S_{n}$, it is enough to use that
the ratio $q_{1}/q_{2}$ is bounded from below and from above by (5.4). After
that, only two independent parameters remain, which are $q_{2}$ and
$\mathtt{A}_{1}$. Then, to obtain the lower bound (5.9), we need (5.8), which
is a bound of the form $\mathtt{A}_{1}\leq Cq_{2}$ for some universal constant
$C$. Also, $q_{2}$ itself must satisfy the smallness condition (5.11), which
is an inequality of the form $q_{2}\leq K$, for some constant $K$ depending on
$m,p$. Next, $\mathtt{A}_{1},q_{2}$ must also satisfy the condition (5.17),
which is an inequality of the form $\mathtt{A}_{1}\leq Kq_{2}^{3/2}$, for some
constant $K$ depending on $m,p$.

We would like to obtain values of $\mathtt{A}_{1}$ as large as possible,
because $\mathtt{A}_{1}$ and its multiple $\mathtt{A}_{2}$ are the amplitudes
of the chaotic movements we want to construct. We fix $\mathtt{A}_{1}$ as the
largest value compatible with (5.8) and (5.17). Thus, we define

$\varepsilon:=q_{2}^{\frac{1}{2}},\qquad\mathtt{A}_{1}:=\Big{(}\frac{3}{32}\frac{\alpha_{1}^{3}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}\Big{)}^{\frac{1}{2}}\varepsilon^{3},$
(5.23)

so that (5.17) is satisfied. Note that (5.8) becomes

$\Big{(}\frac{3}{32}\frac{\alpha_{1}^{3}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}\Big{)}^{\frac{1}{2}}\varepsilon^{3}\leq\frac{1}{9}\varepsilon^{2}.$
(5.24)

We define

$\varepsilon_{0}:=\min\Big{\\{}1,\,\frac{1}{9}\Big{(}\frac{32}{3}\frac{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}{\alpha_{1}^{3}}\Big{)}^{\frac{1}{2}}\,,\,\frac{\delta_{1}}{\sqrt{8}\alpha_{4}^{m_{1}}}\Big{\\}},$
(5.25)

where $\delta_{1}$ is the universal constant in (5.11) and in (2.8), so that
both (5.24) and (5.11) are satisfied for all
$0<\varepsilon\leq\varepsilon_{0}$. The constant $\varepsilon_{0}$ depends
only on $m,p$. For $\sigma=m/p$ small enough, the minimum in (5.25) is the
third element of the set, as we note in the following lemma.

######  Lemma 5.9.

There exists a universal constant $\sigma_{1}\in(0,1)$ such that, if
$\sigma=m/p\leq\sigma_{1}$, then $\varepsilon_{0}$ defined in (5.25) is
$\varepsilon_{0}=\delta_{1}/(\sqrt{8}\alpha_{4}^{m_{1}})$.

###### Proof.

It is enough to recall that
$\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}>3\alpha_{1}^{2}$,
$\alpha_{4}^{m_{1}}\geq\alpha_{4}$, and
$\alpha_{4}>\alpha_{1}>\sqrt{\alpha_{1}}$. ∎

######  Lemma 5.10.

Let
$a,m,p,\sigma,\alpha_{1},\ldots\alpha_{4},\tilde{\mu}_{1}(\sigma),\tilde{\mu}_{2}(\sigma),\gamma,(\xi_{1}(t),\eta_{1}(t),\xi_{2}(t),\eta_{2}(t))$
be like in Lemma 5.1. Consider any $\varepsilon\in(0,\varepsilon_{0})$, where
$\varepsilon_{0}$ is the constant, depending only on $m,p$, defined in (5.25),
and define $q_{2}:=\varepsilon^{2}$, $\mathtt{A}_{1}$ given by the second
identity in (5.23), $q_{1}$ by (5.4), $E_{1},E_{2}$ by (5.3), and $c_{123}$,
$c_{234}$, $\mathtt{B}$, $\mathtt{A}_{2}$, $S_{1}(t)$, $S_{2}(t)$, $S_{3}(t)$,
$S_{4}(t)$, $\varphi_{123}(t)$, $\varphi_{234}(t)$ like in Lemma 5.1. Also
define $\rho_{123}=\frac{8}{3}c_{123}/(\alpha_{1}\alpha_{2}\alpha_{3})$,
$\rho_{234}=\frac{8}{3}c_{234}/(\alpha_{2}\alpha_{3}\alpha_{4})$.

Then $(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$
is a solution of system (3.3) for all $t\in{\mathbb{R}}$. Moreover, the
identity (5.4) is satisfied, and therefore (5.5) holds; the inequality (5.8)
is satisfied, and therefore the lower bound (5.9) holds; the inequality (5.11)
is satisfied, and therefore the bound (5.12) holds; the inequality (5.17) is
satisfied, and therefore the thesis of Lemma 5.8 holds.

###### Proof.

The proof follows from (5.23), (5.25) and the results of the previous
subsections. ∎

Recalling that the function $\eta_{2}$ given by Proposition 4.1 satisfies
(4.4), we also have the following lemma, where
$\varepsilon^{2},\varepsilon^{3}$ are isolated from the other parts of the
coefficients. The reason to consider the quantity $\mathcal{N}_{1}$ in (5.29)
is that it corresponds to the square of the Sobolev norm
$H^{1}({\mathbb{T}}^{d})$ of the solution of the Kirchhoff equation.

To simplify the exposition of the lemma, we rewrite (5.1) as

$\displaystyle S_{1}(t)$
$\displaystyle=\varepsilon^{2}s_{1}-\varepsilon^{3}a_{1}\eta_{1}(b\varepsilon^{3}t),$
(5.26) $\displaystyle S_{2}(t)$
$\displaystyle=\varepsilon^{2}s_{2}-\varepsilon^{3}a_{1}\eta_{1}(b\varepsilon^{3}t)-\varepsilon^{3}a_{2}\eta_{2}(b\varepsilon^{3}t),$
$\displaystyle S_{3}(t)$
$\displaystyle=\varepsilon^{2}s_{3}+\varepsilon^{3}a_{1}\eta_{1}(b\varepsilon^{3}t)-\varepsilon^{3}a_{2}\eta_{2}(b\varepsilon^{3}t),$
$\displaystyle S_{4}(t)$
$\displaystyle=\varepsilon^{2}s_{4}+\varepsilon^{3}a_{2}\eta_{2}(b\varepsilon^{3}t),$

where

$\begin{gathered}s_{1}:=\frac{\alpha_{4}^{2}}{3\alpha_{1}^{2}},\quad\quad
s_{2}:=\frac{\alpha_{4}^{2}}{3\alpha_{2}^{2}},\quad\quad
s_{3}:=\frac{2\alpha_{4}^{2}}{3\alpha_{3}^{2}},\quad\quad s_{4}:=1,\\\
a_{1}:=\Big{(}\frac{3}{32}\frac{\alpha_{1}^{3}}{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}\Big{)}^{\frac{1}{2}},\quad\quad
a_{2}:=\frac{1}{\gamma}a_{1},\quad\quad
b:=\frac{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}{2}a_{1},\end{gathered}$
(5.27)

so that

$\mathtt{A}_{1}=a_{1}\varepsilon^{3},\quad\
\mathtt{A}_{2}=a_{2}\varepsilon^{3},\quad\ \mathtt{B}=b\varepsilon^{3}.$
(5.28)

Note that the constants $s_{1},s_{2},s_{3},s_{4},a_{1},a_{2},b$ depend only on
$m,p$. We also define the associated function

$\displaystyle\mathcal{N}_{1}(t)$
$\displaystyle:=\sum_{n=1}^{4}\alpha_{n}^{2}S_{n}(t)=\varepsilon^{2}\frac{7\alpha_{4}^{2}}{3}+\varepsilon^{3}(2\alpha_{1}\alpha_{2})a_{1}\eta_{1}(b\varepsilon^{3}t)+\varepsilon^{3}(2\alpha_{2}\alpha_{3})a_{2}\eta_{2}(b\varepsilon^{3}t).$
(5.29)

######  Lemma 5.11.

There exists a universal constant $\sigma_{2}\in(0,1)$ with the following
property. Let $t_{j},\bar{t}_{j}$ be given by Proposition 4.1, and define

$t_{j}^{*}:=\frac{t_{j}}{B}=\frac{t_{j}}{b\varepsilon^{3}},\qquad\bar{t}_{j}^{*}:=\frac{\bar{t}_{j}}{B}=\frac{\bar{t}_{j}}{b\varepsilon^{3}},\qquad
I_{j}^{*}:=[t_{j}^{*},\bar{t}_{j}^{*}],\qquad
E_{j}^{*}:=[\bar{t}_{j}^{*},t_{j+1}^{*}].$ (5.30)

Assume the hypotheses of Lemma 5.10. If, in addition, the ratio $\sigma=m/p$
satisfies $\sigma\leq\sigma_{2}$, then $\mathcal{N}_{1}(t)$ defined in (5.29)
satisfies

$\displaystyle\varepsilon^{2}c_{1}+\frac{99}{100}\varepsilon^{3}r_{1}\leq\mathcal{N}_{1}(t)\leq\varepsilon^{2}c_{1}+\frac{201}{100}\varepsilon^{3}r_{1}\quad\forall
t\in I_{j}^{*},$
$\displaystyle\varepsilon^{2}c_{1}-\frac{1}{100}\varepsilon^{3}r_{1}\leq\mathcal{N}_{1}(t)\leq\varepsilon^{2}c_{1}+\frac{101}{100}\varepsilon^{3}r_{1}\quad\forall
t\in E_{j}^{*},$ $\displaystyle\max_{t\in
I_{j}^{*}}\mathcal{N}_{1}(t)\geq\varepsilon^{2}c_{1}+\frac{199}{100}\varepsilon^{3}r_{1},\quad\quad\min_{t\in
E_{j}^{*}}\mathcal{N}_{1}(t)\leq\varepsilon^{2}c_{1}+\frac{1}{100}\varepsilon^{3}r_{1},$
(5.31)

for all $j\geq 0$, where $c_{1}:=\frac{7}{3}\alpha_{4}^{2}$,
$r_{1}:=(2\alpha_{2}\alpha_{3})a_{2}$ depend only on $m,p$.

###### Proof.

To prove the first inequality in the last line of (5.31), consider the
interval $I_{j}^{*}$, and let $t_{j}^{\prime}$ be a point in that interval
where the function $\eta_{2}(b\varepsilon^{3}t)$ achieves its maximum value.
Since by (4.4) and (5.7)

$\eta_{2}(b\varepsilon^{3}t_{j}^{\prime})\geq
2-C\sigma\quad\text{and}\quad|\eta_{1}(t)|\leq 3\quad\forall
t\in{\mathbb{R}},$

from (5.29) and the identity $a_{1}=\gamma a_{2}$ (see (5.27)) we get

$\displaystyle\max_{t\in[t_{j}^{*},\bar{t}_{j}^{*}]}\mathcal{N}_{1}(t)\geq\mathcal{N}_{1}(t_{j}^{\prime})$
$\displaystyle\geq\varepsilon^{2}\frac{7\alpha_{4}^{2}}{3}+\varepsilon^{3}(2\alpha_{2}\alpha_{3})a_{2}(2-C\sigma)-\varepsilon^{3}(2\alpha_{1}\alpha_{2})3a_{1}$
$\displaystyle=\varepsilon^{2}\frac{7\alpha_{4}^{2}}{3}+\varepsilon^{3}(2\alpha_{2}\alpha_{3})a_{2}(2-C\sigma)\Big{(}1-\frac{3\gamma\alpha_{1}}{\alpha_{3}(2-C\sigma)}\Big{)}.$

The difference in the last parenthesis tends to 1 as $\sigma\to 0$, because
$\gamma\to 3$ and the ratio $\alpha_{1}/\alpha_{3}=m/(2m+p)$ tends to $0$ as
$\sigma=m/p\to 0$. The other inequalities in (5.31) are proved similarly,
using (4.3), (4.4), and the fact that

$\mathcal{N}_{1}(t)=\varepsilon^{2}\frac{7\alpha_{4}^{2}}{3}+\varepsilon^{3}(2\alpha_{2}\alpha_{3})a_{2}\Big{(}\eta_{2}(b\varepsilon^{3}t)+\frac{\alpha_{1}\gamma}{\alpha_{3}}\eta_{1}(b\varepsilon^{3}t)\Big{)}=\varepsilon^{2}c_{1}+\varepsilon^{3}r_{1}\big{(}\eta_{2}(b\varepsilon^{3}t)+O(\sigma)\big{)}.\
$ (5.32)

∎

######  Remark 5.12.

By (5.32), the oscillations of $\mathcal{N}_{1}$ at the main order in $\sigma$
are fully described by the ones of $\eta_{2}$, i.e., of $S_{4}$. ∎

Notation. From now on, we will sometimes be much less accurate than before in
keeping track of the explicit dependence of the various constants on
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$; we will denote generically by
$K$ (or sometimes $K^{\prime}$, or $K^{\prime\prime}$) any constant, possibly
different from line to line, that depends only on the integers $m,p$.

With the new notation, the constants defined in Lemma 5.10 become

$c_{123}=K\varepsilon^{6},\quad\ c_{234}=K\varepsilon^{6},\quad\
\rho_{123}=K\varepsilon^{6},\quad\ \rho_{234}=K\varepsilon^{6},$ (5.33)

where the four constants $K$ denote four (possibly different) values.

##  6 Approximation argument

Consider a solution
$(S_{1}(t),S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$ of
system (3.3) obtained in Lemma 5.10, and let $u_{0}$ be the corresponding
trigonometric polynomial in the $H^{m_{1}}$ ball (2.8) constructed in Lemma
5.8. Let $v_{0}:=\overline{u_{0}}$ be the complex conjugate of $u_{0}$, and
let $(u(t),v(t))$ be the solution of the Cauchy problem for the transformed
Kirchhoff equation (2.2) with initial condition (2.9). Since $u_{0}$ is a
trigonometric polynomial, the solution $(u,v)$ is global in time. Moreover,
since $u_{0}$ belongs to the ball (2.8), the solution $(u,v)$ satisfies (2.10)
on the time interval $[0,T_{\mathrm{NF}}]$, see Lemma 2.2.

The estimate in Lemma 5.4 computed at time $t=0$ gives the inequality

$\|u_{0}\|_{s}^{2}\leq 8\alpha_{4}^{2s}q_{2}\quad\forall s\geq 1,$ (6.1)

and therefore, recalling (5.23) and the notation at the end of the previous
section, $T_{\mathrm{NF}}$ in (2.10) can be also estimated in terms of
$\varepsilon$ as

$T_{\mathrm{NF}}\geq\frac{C_{0}}{(8\alpha_{4}^{2m_{1}}q_{2})^{2}}=K\varepsilon^{-4}.$
(6.2)

For each time $t\in[0,T_{\mathrm{NF}}]$, to slightly simplify the notations
(5.13), (5.14), (5.15), we denote

$S_{n}^{u}(t):=S_{n}^{(u(t))}$ (6.3)

the superactions of the function $u(t,\cdot)$, and we also introduce the
analogous notation for all the other quantities in (5.13), (5.14), (5.15).

By Lemma 2.3, one has

$C_{1}S_{n}^{u}(0)\leq S_{n}^{u}(t)\leq C_{2}S_{n}^{u}(0)\quad\forall
t\in[0,T_{\mathrm{NF}}],\quad n=1,2,3,4.$ (6.4)

By construction, one has $S_{n}^{u}(0)=S_{n}(0)$. Hence, by (6.4), (5.10),
(5.23),

$\displaystyle\|u(t)\|_{s}^{2}$
$\displaystyle=\sum_{n=1}^{4}\alpha_{n}^{2s}S_{n}^{u}(t)$ $\displaystyle\leq
C_{2}\sum_{n=1}^{4}\alpha_{n}^{2s}S_{n}^{u}(0)=C_{2}\sum_{n=1}^{4}\alpha_{n}^{2s}S_{n}(0)=C_{2}\|u_{0}\|_{s}^{2}\leq
8C_{2}\alpha_{4}^{2s}q_{2}=8C_{2}\alpha_{4}^{2s}\varepsilon^{2}$ (6.5)

for all $t\in[0,T_{\mathrm{NF}}]$, all $s\geq 1$. For $s\in\\{1,m_{1}\\}$, one
has $\alpha_{4}^{2s}\leq\alpha_{4}^{2m_{1}}$, and therefore we can simply
write

$\|u(t)\|_{s}\leq K\varepsilon\qquad\forall t\in[0,T_{\mathrm{NF}}],\quad
s\in\\{1,m_{1}\\}.$ (6.6)

Since $(u(t),v(t))$ solves (2.2) on $[0,T_{\mathrm{NF}}]$, the functions
$S_{n}^{u}(t)$, $B_{n}^{u}(t)$ solve the effective equations (2.15) on the
same time interval, and $Z_{123}^{u}(t)$, $Z_{234}^{u}(t)$ solve (2.19) on the
same time interval. Since $\rho_{123}^{u}(t),\rho_{234}^{u}(t)$ are continuous
and they are positive at time $t=0$, there exists $T_{\mathrm{polar}}>0$ such
that $\rho_{123}^{u}(t),\rho_{234}^{u}(t)$ are positive for all
$t\in[0,T_{\mathrm{polar}}]$.

######  Remark 6.1.

Note that, in general, $T_{\mathrm{polar}}$ could be smaller than
$T_{\mathrm{NF}}$. ∎

On the time interval $[0,T_{\mathrm{polar}}]$, the angles
$\varphi_{123}^{u}(t),\varphi_{234}^{u}(t)$ are well-defined by the identities

$Z_{123}^{u}(t)=\rho_{123}^{u}(t)\exp(i\varphi_{123}^{u}(t)),\quad
Z_{234}^{u}(t)=\rho_{234}^{u}(t)\exp(i\varphi_{234}^{u}(t)),\quad\forall
t\in[0,T_{\mathrm{polar}}].\quad$ (6.7)

Since $Z_{123}^{u}(t)$, $Z_{234}^{u}(t)$ solve (2.19), the functions
$\rho_{123}^{u}(t)$, $\rho_{234}^{u}(t)$ solve the equations

$\partial_{t}\rho_{123}^{u}=R_{\rho_{123}^{u}},\qquad\partial_{t}\rho_{234}^{u}=R_{\rho_{234}^{u}}$
(6.8)

on $[0,T_{\mathrm{polar}}]$, and the functions $\varphi_{123}^{u}(t)$,
$\varphi_{234}^{u}(t)$ solve the equations

$\displaystyle\partial_{t}\varphi_{123}^{u}$
$\displaystyle=-\frac{1}{2}(\alpha_{1}^{2}S_{1}^{u}+\alpha_{2}^{2}S_{2}^{u}-\alpha_{3}^{2}S_{3}^{u})+R_{\varphi_{123}^{u}},$
$\displaystyle\partial_{t}\varphi_{234}^{u}$
$\displaystyle=-\frac{1}{2}(\alpha_{2}^{2}S_{2}^{u}+\alpha_{3}^{2}S_{3}^{u}-\alpha_{4}^{2}S_{4}^{u})+R_{\varphi_{234}^{u}}$
(6.9)

on $[0,T_{\mathrm{polar}}]$, where

$R_{\rho_{123}^{u}}:=\mathrm{Re}\,\Big{(}\exp(-i\varphi_{123}^{u})R_{Z_{123}^{u}}\Big{)},\qquad
R_{\varphi_{123}^{u}}:=\frac{1}{\rho_{123}^{u}}\mathrm{Im}\,\Big{(}\exp(-i\varphi_{123}^{u})R_{Z_{123}^{u}}\Big{)},$
(6.10)

and analogous definition for $R_{\rho_{234}^{u}}$, $R_{\varphi_{234}^{u}}$
(just replace $123$ with $234$ everywhere in (6.10)). The remainders of the
type $R_{Z}$ are defined in (2.20) and estimated in (2.21). Moreover,
$S_{n}^{u}$, $n=1,2,3,4$, solve

$\displaystyle\partial_{t}S_{1}^{u}$
$\displaystyle=\frac{3}{8}\rho_{123}^{u}\alpha_{1}\alpha_{2}\alpha_{3}\sin(\varphi_{123}^{u})+R_{S_{1}^{u}},$
$\displaystyle\partial_{t}S_{2}^{u}$
$\displaystyle=\frac{3}{8}\rho_{123}^{u}\alpha_{1}\alpha_{2}\alpha_{3}\sin(\varphi_{123}^{u})+\frac{3}{8}\rho_{234}^{u}\alpha_{2}\alpha_{3}\alpha_{4}\sin(\varphi_{234}^{u})+R_{S_{2}^{u}},$
$\displaystyle\partial_{t}S_{3}^{u}$
$\displaystyle=-\frac{3}{8}\rho_{123}^{u}\alpha_{1}\alpha_{2}\alpha_{3}\sin(\varphi_{123}^{u})+\frac{3}{8}\rho_{234}^{u}\alpha_{2}\alpha_{3}\alpha_{4}\sin(\varphi_{234}^{u})+R_{S_{3}^{u}},$
$\displaystyle\partial_{t}S_{4}^{u}$
$\displaystyle=-\frac{3}{8}\rho_{234}^{u}\alpha_{2}\alpha_{3}\alpha_{4}\sin(\varphi_{234}^{u})+R_{S_{4}^{u}}$
(6.11)

on $[0,T_{\mathrm{polar}}]$, where the remainders of the type $R_{S}$ appear
in (2.15) and are estimated in (2.16). In the following lemma we prove a
formula for $T_{\mathrm{polar}}$.

######  Lemma 6.2.

Assume the hypotheses of Lemma 5.10. There exists a universal constant
$\sigma_{3}\in(0,1)$ such that, if $\sigma=m/p$, in addition to the hypotheses
of Lemma 5.10, also satisfies $\sigma\leq\sigma_{3}$, then $Z_{123}^{u}(t)\neq
0$, $Z_{234}^{u}(t)\neq 0$ for all $t\in[0,T_{\mathrm{polar}}]$, with

$T_{\mathrm{polar}}=\frac{C\alpha_{1}A_{1}^{2}}{\alpha_{4}^{4m_{1}+2}q_{2}^{5}}=K\varepsilon^{-4}\leq
T_{\mathrm{NF}},$ (6.12)

where $C>0$ is a universal constant and $K>0$ is a constant depending only on
$m,p$. As a consequence, on $[0,T_{\mathrm{polar}}]$,
$\rho_{123}^{u},\rho_{234}^{u}$ are positive and $\varphi_{123}^{u}$,
$\varphi_{234}^{u}$ are well-defined . Moreover,

$|\rho_{123}^{u}(t)-\rho_{123}|\leq\frac{1}{2}\rho_{123},\qquad|\rho_{234}^{u}(t)-\rho_{234}|\leq\frac{1}{2}\rho_{234}.$
(6.13)

###### Proof.

In this proof $C$ (and $C^{\prime}$, $C^{\prime\prime}$) denote universal
constants, possibly different from line to line. Suppose that $\rho_{123}^{u}$
is positive on some time interval $[0,T]\subset[0,T_{\mathrm{NF}}]$. Then, by
(6.8) and (6.10),

$|\partial_{t}\rho_{123}^{u}|=\big{|}\mathrm{Re}\,\big{(}\exp(-i\varphi_{123}^{u})R_{Z_{123}^{u}}\big{)}\big{|}\leq\big{|}\exp(-i\varphi_{123}^{u})R_{Z_{123}^{u}}\big{|}=|R_{Z_{123}^{u}}|.$

By (2.21), for all $s\in{\mathbb{R}}$,

$|R_{Z_{123}^{u}}|\leq
C\|u\|_{m_{1}}^{4}S_{1}^{u}S_{2}^{u}S_{3}^{u}=\frac{C\|u\|_{m_{1}}^{4}}{(\alpha_{1}\alpha_{2}\alpha_{3})^{2s}}(\alpha_{1}^{2s}S_{1}^{u})(\alpha_{2}^{2s}S_{2}^{u})(\alpha_{3}^{2s}S_{3}^{u})\leq\frac{C\|u\|_{m_{1}}^{4}\|u\|_{s}^{6}}{(\alpha_{1}\alpha_{2}\alpha_{3})^{2s}}$

and, by (6.5), for $s\geq 1$ and $t\in[0,T]$,

$\|u(t)\|_{m_{1}}^{4}\|u(t)\|_{s}^{6}\leq C\alpha_{4}^{4m_{1}+6s}q_{2}^{5}.$

Hence, since $\alpha_{4}\leq 2\alpha_{3}$ and $\alpha_{4}\leq 3\alpha_{2}$,
for $s=1$, we obtain

$|\partial_{t}\rho_{123}^{u}|\leq|R_{Z_{123}^{u}}|\leq\frac{C\alpha_{4}^{4m_{1}+6}q_{2}^{5}}{(\alpha_{1}\alpha_{2}\alpha_{3})^{2}}\leq\frac{C^{\prime}\alpha_{4}^{4m_{1}+2}q_{2}^{5}}{\alpha_{1}^{2}}\quad\forall
t\in[0,T].$ (6.14)

At time $t=0$ we have, by construction, and by the definition (3.31) of
$c_{123}$,

$\rho_{123}^{u}(0)=\rho_{123}=\frac{8c_{123}}{3\alpha_{1}\alpha_{2}\alpha_{3}}=\frac{4(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})}{3\alpha_{1}\alpha_{2}\alpha_{3}}\mathtt{A}_{1}^{2},$
(6.15)

which is positive because $\mathtt{A}_{1}>0$. Hence

$|\rho_{123}^{u}(t)-\rho_{123}|\leq\int_{0}^{t}|\partial_{t}\rho_{123}^{u}(s)|\,ds\leq\frac{C^{\prime}\alpha_{4}^{4m_{1}+2}q_{2}^{5}}{\alpha_{1}^{2}}t\quad\forall
t\in[0,T],$

and the last quantity is strictly less than $\rho_{123}$ for all $t\in[0,T]$
if

$T<\frac{C\alpha_{1}}{\alpha_{4}^{4m_{1}+2}}\frac{\mathtt{A}_{1}^{2}}{q_{2}^{5}}=:T_{123}^{*}.$
(6.16)

This implies that $\rho_{123}^{u}>0$ on $[0,T_{123}^{*})$. Moreover, by (5.17)
and (6.2),

$T_{123}^{*}=\frac{C\alpha_{1}}{\alpha_{4}^{4m_{1}+2}}\frac{\mathtt{A}_{1}^{2}}{q_{2}^{5}}\leq
C^{\prime}\frac{\alpha_{1}^{4}}{\alpha_{4}^{4}}\frac{1}{\alpha_{4}^{2m_{1}}q_{2}^{2}}\leq
C^{\prime\prime}\sigma^{4}T_{\mathrm{NF}},$

where $\sigma=m/p$. Then, for $C^{\prime\prime}\sigma^{4}\leq 1$, one has
$T_{123}^{*}\leq T_{\mathrm{NF}}$.

Proceeding similarly for $\rho_{234}^{u}$, we obtain the estimate

$|\partial_{t}\rho_{234}^{u}|\leq|R_{Z_{234}^{u}}|\leq\frac{C\alpha_{4}^{4m_{1}+6}q_{2}^{5}}{(\alpha_{2}\alpha_{3}\alpha_{4})^{2}}\leq
C^{\prime}\alpha_{4}^{4m_{1}}q_{2}^{5},$ (6.17)

and we deduce that $\rho_{234}^{u}>0$ on $[0,T_{234}^{*})$, with

$T_{234}^{*}=\frac{C\mathtt{A}_{1}^{2}}{\alpha_{4}^{4m_{1}+1}q_{2}^{5}}\leq
C^{\prime}\sigma^{3}T_{\mathrm{NF}}.$

Also, $T_{123}^{*}\leq T_{234}^{*}C\sigma\leq T_{234}^{*}$ for $C\sigma\leq
1$.

Finally, the estimates already proved also give that
$|\rho_{123}^{u}(t)-\rho_{123}|\leq\frac{1}{2}\rho_{123}$ for all
$t\in[0,T_{123}^{*}/2]$. Thus, we fix $T_{\mathrm{polar}}=T_{123}^{*}/2$. ∎

We recall that the superscript $u$ indicates quantities related to the
solution $(u,v)$ of the Cauchy problem (2.2), (2.9), while the absence of that
superscript corresponds to the solution of the truncated effective system
(3.3). By construction, we have

$S^{u}(0)=S(0),\qquad\rho^{u}(0)=\rho(0),\qquad\varphi^{u}(0)=\varphi(0).$

Moreover, $\rho$ is constant in time, and there are no remainders
$R_{S},R_{\rho},R_{\varphi}$ without the superscript $u$ because (3.3) is the
truncated effective system.

The next lemma gives estimates for the difference between $S^{u}(t)$ and
$S(t)$ in a certain time interval.

######  Lemma 6.3.

Assume the hypotheses of Lemma 5.10, and consider the associated solution
$(S_{1}(t),$ $S_{2}(t),S_{3}(t),S_{4}(t),\varphi_{123}(t),\varphi_{234}(t))$
of system (3.3) given by Lemma 5.10. Let $u_{0}$ be the corresponding
trigonometric polynomial in the ball (2.8) constructed in Lemma 5.8. Let
$v_{0}:=\overline{u_{0}}$ be the complex conjugate of $u_{0}$, and let
$(u(t),v(t))$ be the solution of the Cauchy problem for the transformed
Kirchhoff equation (2.2) with initial condition (2.9). Assume that the ratio
$\sigma=m/p$ satisfies $\sigma\leq\sigma_{3}$, where $\sigma_{3}$ is defined
in Lemma 6.2.

There exists $\varepsilon_{1}\in(0,\varepsilon_{0})$, depending only on $m,p$,
where $\varepsilon_{0}$ is the constant in (5.25), such that, for
$0<\varepsilon\leq\varepsilon_{1}$, one has

$\varepsilon^{-3}|S_{n}^{u}(t)-S_{n}(t)|\leq\varepsilon^{\frac{3}{4}}\quad\
\forall t\in[0,T_{\varepsilon}],\quad n=1,2,3,4,$ (6.18)

where

$T_{\varepsilon}=K\varepsilon^{-3}\log(\varepsilon^{-1})$ (6.19)

for some positive constant $K$ depending only on $m,p$.

###### Proof.

From the difference of equations (6.11) for $S_{n}^{u}$ and equations (3.3)
for $S_{n}$, we obtain that for all $t\in[0,T_{\mathrm{polar}}]$, $n=1,2,3,4$,

$\displaystyle|\partial_{t}(S_{n}^{u}-S_{n})|$
$\displaystyle\leq\alpha_{1}\alpha_{2}\alpha_{3}\Big{(}\rho_{123}|\varphi_{123}^{u}-\varphi_{123}|+|\rho_{123}^{u}-\rho_{123}|\Big{)}$
$\displaystyle\quad\
+\alpha_{2}\alpha_{3}\alpha_{4}\Big{(}\rho_{234}|\varphi_{234}^{u}-\varphi_{234}|+|\rho_{234}^{u}-\rho_{234}|\Big{)}+|R_{S_{n}^{u}}|.$

By (2.16) one has $|R_{S_{n}^{u}}|\leq C\|u\|_{m_{1}}^{6}S_{n}^{u}$. By (6.6),

$\|u\|_{m_{1}}\leq
K\varepsilon,\qquad|S_{n}^{u}|\leq\alpha_{n}|S_{n}^{u}|\leq\|u\|_{1}^{2}\leq
K\varepsilon^{2},$

and therefore $|R_{S_{n}^{u}}|\leq K\varepsilon^{8}$ on
$[0,T_{\mathrm{polar}}]$. Hence, for $t\in[0,T_{\mathrm{polar}}]$ and
$n=1,2,3,4$,

$\displaystyle|\partial_{t}(S_{n}^{u}-S_{n})|$ $\displaystyle\leq
K\Big{(}\rho_{123}|\varphi_{123}^{u}-\varphi_{123}|+|\rho_{123}^{u}-\rho_{123}|\Big{)}$
$\displaystyle\quad\
+K\Big{(}\rho_{234}|\varphi_{234}^{u}-\varphi_{234}|+|\rho_{234}^{u}-\rho_{234}|\Big{)}+K\varepsilon^{8}.$
(6.20)

Since $\rho_{123},\rho_{234}$ are constants, by the equations (6.8) for
$\rho_{123}^{u}$, $\rho_{234}^{u}$ one has

$\partial_{t}(\rho_{123}^{u}-\rho_{123})=\partial_{t}\rho_{123}^{u}=R_{\rho_{123}^{u}},\qquad\partial_{t}(\rho_{234}^{u}-\rho_{234})=\partial_{t}\rho_{234}^{u}=R_{\rho_{234}^{u}}.$

The time derivatives $\partial_{t}\rho_{123}^{u}$,
$\partial_{t}\rho_{234}^{u}$ have been already estimated in (6.14) and (6.17);
hence, integrating in time, we get

$|\rho_{123}^{u}(t)-\rho_{123}|\leq K\varepsilon^{10}t,\quad\
|\rho_{234}^{u}(t)-\rho_{234}|\leq K\varepsilon^{10}t\quad\ \forall
t\in[0,T_{\mathrm{polar}}].$

Also, by (6.15) and (5.28), $\rho_{123}\leq K\varepsilon^{6}$, $\rho_{234}\leq
K\varepsilon^{6}$. We plug these estimates into (6.20) and we get

$\displaystyle|\partial_{t}(S_{n}^{u}-S_{n})|$ $\displaystyle\leq
K\varepsilon^{6}\big{(}|\varphi_{123}^{u}-\varphi_{123}|+|\varphi_{234}^{u}-\varphi_{234}|\big{)}+K\varepsilon^{10}t+K\varepsilon^{8}$
(6.21)

for all $t\in[0,T_{\mathrm{polar}}]$, $n=1,2,3,4$.

Subtracting the equations (6.9) for $\varphi_{123}^{u}$, $\varphi_{234}^{u}$
and those for $\varphi_{123},\varphi_{234}$ in (3.3), we have

$\displaystyle|\partial_{t}(\varphi_{123}^{u}-\varphi_{123})|$
$\displaystyle\leq\alpha_{1}^{2}|S_{1}^{u}-S_{1}|+\alpha_{2}^{2}|S_{2}^{u}-S_{2}|+\alpha_{3}^{2}|S_{3}^{u}-S_{3}|+|R_{\varphi_{123}^{u}}|,$
$\displaystyle|\partial_{t}(\varphi_{234}^{u}-\varphi_{234})|$
$\displaystyle\leq\alpha_{2}^{2}|S_{2}^{u}-S_{2}|+\alpha_{3}^{2}|S_{3}^{u}-S_{3}|+\alpha_{4}^{2}|S_{4}^{u}-S_{4}|+|R_{\varphi_{234}^{u}}|.$

By (6.13), $\rho_{123}^{u}\geq\frac{1}{2}\rho_{123}$ on
$[0,T_{\mathrm{polar}}]$. Hence, by (6.10),
$|R_{\varphi_{123}^{u}}|\leq(2/\rho_{123})|R_{Z_{123}^{u}}|$, and
$R_{Z_{123}^{u}}$ has been estimated in (6.14). Also, $\rho_{123}$ is given in
(6.15). Therefore

$|R_{\varphi_{123}^{u}}|\leq\frac{2}{\rho_{123}}|R_{Z_{123}^{u}}|\leq\frac{Kq_{2}^{5}}{\mathtt{A}_{1}^{2}}\,=K\varepsilon^{4}$

and, similarly, $|R_{\varphi_{234}^{u}}|\leq K\varepsilon^{4}$. Thus

$|\partial_{t}(\varphi_{123}^{u}-\varphi_{123})|+|\partial_{t}(\varphi_{234}^{u}-\varphi_{234})|\leq
K\sum_{n=1}^{4}|S_{n}^{u}-S_{n}|+K\varepsilon^{4}.$ (6.22)

We apply Gronwall’s inequality to (6.21) and (6.22). It is convenient to
introduce a factor $\varepsilon^{\beta}$, with $\beta$ to be determined,
because the factors in (6.21) and (6.22) contain different powers of
$\varepsilon$. We define the vector

$\displaystyle\psi$
$\displaystyle:=\big{(}\varepsilon^{\beta}(S_{1}^{u}-S_{1}),\varepsilon^{\beta}(S_{2}^{u}-S_{2}),\varepsilon^{\beta}(S_{3}^{u}-S_{3}),\varepsilon^{\beta}(S_{4}^{u}-S_{4}),(\varphi_{123}^{u}-\varphi_{123}),(\varphi_{234}^{u}-\varphi_{234})\big{)}$
$\displaystyle=:(\psi_{1},\psi_{2},\psi_{3},\psi_{4},\psi_{5},\psi_{6}),$
(6.23)

which is a function of $t\in[0,T_{\mathrm{polar}}]$ taking values in
${\mathbb{R}}^{6}$. From (6.21) we obtain

$\displaystyle|\partial_{t}\psi_{n}|=\varepsilon^{\beta}|\partial_{t}(S_{n}^{u}-S_{n})|$
$\displaystyle\leq\varepsilon^{\beta}[K\varepsilon^{6}(|\psi_{5}|+|\psi_{6}|)+K\varepsilon^{10}t+K\varepsilon^{8}]$
$\displaystyle\leq
K\varepsilon^{6+\beta}|\psi|+K\varepsilon^{10+\beta}t+K\varepsilon^{8+\beta},\quad\
n=1,2,3,4,$

and from (6.22) we get

$\displaystyle|\partial_{t}\psi_{5}|+|\partial_{t}\psi_{6}|$
$\displaystyle\leq
K\varepsilon^{-\beta}(|\psi_{1}|+|\psi_{2}|+|\psi_{3}|+|\psi_{4}|)+K\varepsilon^{4}\leq
K\varepsilon^{-\beta}|\psi|+K\varepsilon^{4},$

where $|\ |$ is the usual Euclidean norm of ${\mathbb{R}}^{6}$. Therefore

$|\partial_{t}\psi|\leq
K(\varepsilon^{6+\beta}+\varepsilon^{-\beta})|\psi|+K\varepsilon^{10+\beta}t+K\varepsilon^{8+\beta}+K\varepsilon^{4}.$
(6.24)

We fix the value of $\beta$ that minimizes the sum
$\varepsilon^{6+\beta}+\varepsilon^{-\beta}$, that is, $\beta=-3$. Hence
(6.24) becomes

$|\partial_{t}\psi|\leq
K\varepsilon^{3}|\psi|+K\varepsilon^{7}t+K\varepsilon^{4}.$ (6.25)

Moreover, by the choice of the initial conditions $\psi(0)=0$. Then, by
Gronwall’s inequality,

$|\psi(t)|\leq\exp(K_{0}\varepsilon^{3}t)K_{1}(\varepsilon^{7}t^{2}+\varepsilon^{4}t)\quad\
\forall t\in[0,T_{\mathrm{polar}}],$ (6.26)

for some positive constants $K_{0},K_{1}$ depending only on $m,p$.

Now we consider the time
$T_{\varepsilon,h}:=K_{0}^{-1}\varepsilon^{-3}\log(\varepsilon^{-h})$, where
$h$ is any positive real constant and $K_{0}>0$ is the constant appearing in
(6.26); we note that $T_{\varepsilon,h}$ depends on $h,\varepsilon,m,p$. By
(6.12), one has $T_{\varepsilon,h}\leq T_{\mathrm{polar}}$ for
$0<\varepsilon\leq C$, for some positive constant $C$ depending only on
$h,m,p$. By (6.26), for all $t\in[0,T_{\varepsilon,h}]$ one has

$\displaystyle|\psi(t)|$
$\displaystyle\leq\exp(K_{0}\varepsilon^{3}t)K_{1}(\varepsilon^{7}t^{2}+\varepsilon^{4}t)\leq\varepsilon^{-h}K_{1}\Big{(}\varepsilon^{7}K_{0}^{-2}\varepsilon^{-6}\log^{2}(\varepsilon^{-h})+\varepsilon^{4}K_{0}^{-1}\varepsilon^{-3}\log(\varepsilon^{-h})\Big{)}$
$\displaystyle\leq\varepsilon^{1-h}K_{1}\Big{(}K_{0}^{-2}\log^{2}(\varepsilon^{-h})+K_{0}^{-1}\log(\varepsilon^{-h})\Big{)}.$

We also note that

$\varepsilon^{1-h}K_{1}\Big{(}K_{0}^{-2}\log^{2}(\varepsilon^{-h})+K_{0}^{-1}\log(\varepsilon^{-h})\Big{)}\leq\varepsilon^{1-2h}\quad\
\text{for }0<\varepsilon\leq C^{\prime},$ (6.27)

for some positive constant $C^{\prime}$ depending on $h,m,p$. We can fix, for
example, $h=1/8$. ∎

###  6.1 Motion of the Sobolev norms

We use the $H^{1}({\mathbb{T}}^{d})$ Sobolev norm $\|u(t)\|_{1}$ to describe
the transfer of energy, namely the exchanges in amplitude of the superactions
$S_{n}^{u}(t)$, $n=1,2,3,4$, as time evolves. Except the $H^{\frac{1}{2}}$
norm, which is constant in time for the solutions of the approximating system
(3.39) given by Lemma 5.1 (see Remark 3.6), any other $H^{s}$ norms could be
used to capture the transfer of energy among the superactions. We decide to
use the $H^{1}$ norm because $H^{1}$ here corresponds to the space
$H^{\frac{3}{2}}\times H^{\frac{1}{2}}$ for the “physical variables”, which is
the space of the standard local wellposedness for the Kirchhoff equation.

######  Lemma 6.4.

Assume all the hypotheses of Lemma 6.3, and also let $\sigma\leq\sigma_{2}$,
where $\sigma_{2}$ is given by Lemma 5.11. Let