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# Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

Bram Lentjes Department of Mathematics, Hasselt University, Diepenbeek Campus,
3590 Diepenbeek, Belgium (bram.lentjes@uhasselt.be). Len Spek Department of
Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands
<EMAIL_ADDRESS>Maikel M. Bosschaert Department of Mathematics, Hasselt
University, Diepenbeek Campus, 3590 Diepenbeek, Belgium
(maikel.bosschaert@uhasselt.be). Yuri A. Kuznetsov Department of Mathematics,
Utrecht University, 3508 TA Utrecht, The Netherlands and Department of Applied
Mathematics, University of Twente, 7500 AE Enschede, The Netherlands
(i.a.kouznetsov@uu.nl).

###### Abstract

A recent work [28] by the authors on the existence of a periodic smooth
finite-dimensional center manifold near a nonhyperbolic cycle in delay
differential equations motivates derivation of periodic normal forms. In this
paper, we prove the existence of a special coordinate system on the center
manifold that will allow us to describe the local dynamics on the center
manifold in terms of these periodic normal forms. Furthermore, we characterize
the center eigenspace by proving the existence of time periodic smooth Jordan
chains for the original and the adjoint system.

delay differential equations, dual perturbation theory, sun-star calculus,
center manifold, Jordan chains, normal forms, generalized eigenfunctions,
nonhyperbolic cycles

34C20, 34K13, 34K17, 34K18, 34K19

##  1 Introduction

Bifurcation theory allows us to analyze the behavior of complicated high
dimensional nonlinear dynamical systems near bifurcations by reducing the
system to a low dimensional invariant manifold, called the center manifold.
Using normal form theory, the dynamics on the center manifold can be described
by a simple canonical equation called the normal form. These bifurcations and
normal forms can be categorized, and their properties can be understood in
terms of certain coefficients of the normal form, see [27] for more details.
Methods to compute these normal form coefficients have been implemented in
software like `MatCont` [9] and `DDE-BifTool` [16, 30] to study various
classes of dynamical systems.

For bifurcations of limit cycles in continuous-time dynamical systems, there
are three generic codimension one bifurcations: fold (or limit point), period-
doubling (or flip) and Neimark-Sacker (or torus) bifurcation. These
bifurcations are well understood for ordinary differentials equations (ODEs)
[22, 23, 26, 33], but for delay differential equations (DDEs) the theory is
still lacking. To understand these bifurcations, one should first prove the
existence of a center manifold near a nonhyperbolic cycle. The authors proved
in [28] that indeed such a center manifold near a nonhyperbolic cycle exists
and is sufficiently smooth. The next step is to study the local dynamics near
a nonhyperbolic cycle via normal forms.

The aim of this paper is to show for classical DDEs that the local dynamics
near a nonhyperbolic cycle can be studied via periodic normal forms. We
generalize the results from Iooss [22, 23] on periodic normal forms for
finite-dimensional ODEs towards infinite-dimensional DDEs. This task will be
accomplished by using the rigorous perturbation framework of dual semigroups
(sun-star calculus). In an upcoming paper, we present explicit computational
formulas for the critical normal form coefficients, along the lines of the
periodic normalization method [26, 33], for all codimension one bifurcations
of limit cycles in classical DDEs, completely avoiding Poincaré maps. Finally,
we plan to implement the obtained computational formulas into a software
package like `DDE-BifTool`.

###  1.1 Background

Consider a classical delay differential equation (DDE)

$\dot{x}(t)=F(x_{t}),\quad t\geq 0,$ (1)

where $x(t)\in\mathbb{R}^{n}$ for each $t\geq 0$, and

$x_{t}(\theta):=x(t+\theta),\quad\theta\in[-h,0],$

represents the _history_ at time $t$ of the unknown $x$, and $0<h<\infty$
denotes the upper bound of (finite) delays. The $\mathbb{R}^{n}$-valued smooth
operator $F$ is defined on the Banach space $X:=C([-h,0],\mathbb{R}^{n})$
consisting of $\mathbb{R}^{n}$-valued continuous functions defined on the
compact interval $[-h,0]$, endowed with the supremum norm. Furthermore, we
assume that (1) has a $T$-periodic solution
$\gamma:\mathbb{R}\to\mathbb{R}^{n}$ such that the periodic orbit (cycle)
$\Gamma:=\\{\gamma_{t}\in X\ :\ t\in\mathbb{R}\\}$ is nonhyperbolic.

Using the perturbation framework of dual semigroups, called sun-star calculus,
developed in [5, 6, 7, 8, 13], the existence of a periodic smooth finite-
dimensional center manifold $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$
near $\Gamma$ for (1) has been recently rigorously established in [28] by the
authors. We also refer to [1, 3, 4, 11, 14, 25] for results on the existence
of (parameter-dependent) center manifolds near nonhyperbolic equilibria and
cycles in classical, impulsive and abstract DDEs. Furthermore, we mention the
work by Hupkes and Verduyn Lunel on the existence and smoothness of center
manifolds near equilibria [20] and periodic orbits [21] for so-called
functional differential equations of mixed type (MFDEs).

To study the local dynamics of (1) on
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$, we will characterize the
center manifold by proving the existence of a special coordinate system on
this manifold. Afterwards, we show that any solution of (1) on this invariant
manifold can be locally parametrized in terms of these coordinates. The
existence of such a coordinate system for finite-dimensional ODEs was already
carried out by Iooss in [22, 23], and so we will generalize his results
towards the setting of classical DDEs, using the sun-star calculus framework.
It turns out that the linear part of our coordinate system will be closely
related to the coordinate system invented by Hale and Weedermann [19], which
was used to study perturbations of periodic orbits in classical DDEs. Iooss
indicated in [22] that his results would be easily extendable to the infinite-
dimensional setting. However, we will show in this paper some results that
were truly not expected by the authors. For example, an interplay between
history and periodicity for Jordan chains in thm:eigenfunctions was a
remarkable observation, since the history concept is not present in ODEs.
Furthermore, the proof on the existence of this coordinate system on the
center manifold happened to be far more involved, see for example
thm:normalformI and especially the role of the sun-star calculus machinery in
the proof. As a consequence of the results, the periodic normal forms for
bifurcations of limit cycles in classical DDEs are exactly the same as for
ODEs.

As already addressed, we will present in an upcoming article explicit
computational formulas for the critical normal form coefficients of all
codimension one bifurcations of limit cycles in DDEs. To obtain these
formulas, it is no surprise that we need to compute (generalized)
eigenfunctions and adjoint (generalized) eigenfunctions of a certain operator.
The first fundaments to obtain such formulas will be carried out in this paper
by proving the existence of time periodic smooth (adjoint) (generalized)
eigenfunctions. It turns out that the existence of the periodic smooth
(generalized) eigenfunctions are also a necessary for the construction of the
coordinate system on $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$.

###  1.2 Overview

The paper is organized as follows. In sec:periodic center manifolds we review
and extend the results from [28] on periodic smooth finite-dimensional center
manifolds near nonhyperbolic cycles in the setting of classical DDEs.

In sec:periodic spectral computations we mainly characterize the center
eigenspace and its associated adjoint. To do this, we prove that there exists
a periodic smooth basis of the center eigenspace by extending the results from
[23, Proposition III.1] towards classical DDEs, see thm:eigenfunctions. We
show an interesting interplay between the history and periodicity of the
(generalized) eigenfunctions, a phenomenon that is not present in the setting
of ODEs.

In sec:characterization we prove the existence of a special coordinate system
on $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ and generalize the normal
form theorems from finite-dimensional ODEs [22, 23] towards infinite-
dimensional DDEs, see thm:normalformI, thm:normalformII and thm:normalformIII
for the main results.

##  2 Periodic center manifolds for classical DDEs

In this section, we primarily summarize the results from [28] and secondly
recall and extend some results from (time-dependent) dual perturbation theory
for which the book [14] together with the article [5] are standard references.
All unreferenced claims relating to basic properties of time-dependent
perturbations of delays equations can be found in both references.

In the setting of classical DDEs, we work with the real Banach space
$X:=C([-h,0],\mathbb{R}^{n})$ as the state space for some (maximal) finite
_delay_ $0<h<\infty$ equipped with the supremum norm $\|\cdot\|_{\infty}$.
Consider for an integer $k\geq 1$ a $C^{k+1}$-smooth operator
$F:X\to\mathbb{R}^{n}$ together with the initial value problem

$\begin{cases}\dot{x}(t)=F(x_{t}),\quad&t\geq 0,\\\
x_{0}=\varphi,\quad&\varphi\in X,\end{cases}$ (DDE)

where the _history of $x$ at time $t\geq 0$_, denoted by $x_{t}\in X$, is
defined as

$x_{t}(\theta):=x(t+\theta),\quad\forall\theta\in[-h,0].$ (2)

By a _solution_ of $\eqref{eq:DDEphi}$ we mean a continuous function
$x:[-h,t_{\varphi})\to\mathbb{R}^{n}$ for some _final time_
$0<t_{\varphi}\leq\infty$ that is continuously differentiable on
$[0,t_{\varphi})$ and satisfies (DDE). When $t_{\varphi}=\infty$, we call $x$
a _global solution_. We say that a function
$\gamma:\mathbb{R}\to\mathbb{R}^{n}$ is a _periodic solution_ of (DDE) if
$\gamma$ is a solution of (DDE) and there exists a minimal $T>0$, called the
_period of $\gamma$_, such that $\gamma_{T}=\gamma_{0}$. It follows from [18,
Corollary 10.3.1] that $\gamma\in C^{k+2}(\mathbb{R},\mathbb{R}^{n})$. We call
$\Gamma:=\\{\gamma_{t}\in X\ :\ t\in\mathbb{R}\\}$ a _periodic orbit_ or
_(limit) cycle_ in $X$.

We want to study (DDE) near the periodic solution $\gamma$, and it is
therefore convenient to translate $\gamma$ towards the origin. More
specifically, if $x$ is a solution of (DDE), then for $y$ defined as
$x=\gamma+y$, we have that $y$ satisfies the time-dependent DDE

$\dot{y}(t)=L(t)y_{t}+G(t,y_{t}),\\\ $ (3)

where the $C^{k}$-smooth bounded linear operator
$L(t):=DF(\gamma_{t})\in\mathcal{L}(X,\mathbb{R}^{n})$ denotes the Fréchet
derivative of $F$ evaluated at $\gamma_{t}$ and the $C^{k}$-smooth operator
$G(t,\cdot):=F(\gamma_{t}+\cdot)-F(\gamma_{t})-L(t)$ consists of solely
nonlinear terms. Regarding the linear part, it is traditional to apply a
vector-valued version of the Riesz representation theorem [14, Theorem 1.1] as

$L(t)\varphi=\int_{0}^{h}d_{2}\zeta(t,\theta)\varphi(-\theta)=:\langle\zeta(t,\cdot),\varphi\rangle,\quad\forall
t\in\mathbb{R},\ \varphi\in X.$ (4)

The _kernel_ $\zeta:\mathbb{R}\times[0,h]\to\mathbb{R}^{n\times n}$ is a
matrix-valued function, $\zeta(t,\cdot)$ is of bounded variation, right-
continuous on the open interval $(0,h)$, $T$-periodic in the first component
and normalized by the requirement that $\zeta(t,0)=0$ for all
$t\in\mathbb{R}$. The integral appearing in (4) is of Riemann-Stieltjes type
and the subscript in $d$ reflects on the fact that we integrate over the
second variable of $\zeta$.

The starting point of applying the sun-star calculus construction towards the
setting of classical DDEs is by studying the _trivial DDE_

$\begin{cases}\dot{x}(t)=0,\quad&t\geq 0,\\\ x_{0}=\varphi,\quad&\varphi\in
X,\end{cases}$ (5)

which has the unique global solution

$x(t)=\begin{cases}\varphi(t),\quad&-h\leq t\leq 0,\\\ \varphi(0),\quad&t\geq
0.\end{cases}$ (6)

We define the $\mathcal{C}_{0}$-semigroup $T_{0}$ on $X$, also called the
_shift semigroup_ , as

$(T_{0}(t)\varphi)(\theta):=\begin{cases}\varphi(t+\theta),\quad&-h\leq
t+\theta\leq 0,\\\ \varphi(0),\quad&t+\theta\geq 0,\end{cases}\quad\forall
t\geq 0,\ \varphi\in X,\ \theta\in[-h,0].$

and notice that $T_{0}$ generates the solution of (6) in the sense of
$T_{0}(t)\varphi=x_{t}$ for all $t\geq 0$. The shift semigroup has
(infinitesimal) generator $A_{0}:\mathcal{D}(A_{0})\to X$ defined by

$\mathcal{D}(A_{0})=\\{\varphi\in C^{1}([-h,0],\mathbb{R}^{n})\ :\
\dot{\varphi}(0)=0\\},\quad A_{0}\varphi=\dot{\varphi}.$

For this specific combination of $X$, $T_{0}$ and $A_{0}$, the abstract
duality structure can be constructed explicitly, see [14, Section II.5]. We
only summarize here the basic results that are needed for the upcoming
(sub)sections.

Let us first introduce a convention. For
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$ let $\mathbb{K}^{n}$ be the linear
space of column vectors, while $\mathbb{K}^{n\star}$ denotes the linear space
of row vectors, all with components in $\mathbb{K}$. A representation theorem
by Riesz [29] enables us to identify
$X^{\star}=C([-h,0],\mathbb{R}^{n})^{\star}$ with the Banach space
$\operatorname{NBV}([0,h],\mathbb{R}^{n\star})$ consisting of functions
$\zeta:[0,h]\to\mathbb{R}^{n\star}$ that are normalized by $\zeta(0)=0$, are
continuous from the right on $(0,h)$ and have bounded variation. Because $X$
is not reflexive, the _dual semigroup_ $T_{0}^{\star}$ of $T_{0}$ is in
general only weak⋆ continuous on $X^{\star}$. This is also visible on the
generator level, as the adjoint $A_{0}^{\star}$ of $A_{0}$ is only the weak⋆
generator of $T_{0}^{\star}$ and takes the form

$\displaystyle\mathcal{D}(A_{0}^{\star})=\bigg{\\{}f\in\operatorname{NBV}([0,h],\mathbb{R}^{n\star})\
:\ f(\theta)=f(0^{+})+\int_{0}^{\theta}g(\sigma)d\sigma\mbox{ for }\theta>0,$
$\displaystyle g\in\operatorname{NBV}([0,h],\mathbb{R}^{n\star})\mbox{ and
}g(h)=0\bigg{\\}},$ $\displaystyle\quad A_{0}^{\star}f=g,$ (7)

where $f(0^{+}):=\lim_{t\downarrow 0}f(t)$ and the function $g$ is called the
_derivative_ of $f$. The maximal subspace of strong continuity

$X^{\odot}:=\\{x^{\star}\in X^{\star}:t\mapsto T_{0}^{\star}(t)x^{\star}\mbox{
is norm continuous on }[0,\infty)\\}$

is a norm closed $T_{0}^{\star}(t)$-invariant weak⋆ dense subspace of
$X^{\star}$ and we have the characterization

$X^{\odot}=\overline{\mathcal{D}(A_{0}^{\star})},$ (8)

where the bar denotes the norm closure in $X^{\star}$. Expression (8) enables
us to compute the _sun dual_ $X^{\odot}$ by taking the closure of
$\mathcal{D}(A_{0}^{\star})$ with respect to the norm defined on
$\operatorname{NBV}([0,h],\mathbb{R}^{n\star})$. As the space consisting of
functions that have bounded variation are norm dense in $L^{1}$, $X^{\odot}$
has the same description as $\mathcal{D}(A_{0}^{\star})$, but the derivative
is allowed to be in $L^{1}([0,h],\mathbb{R}^{n\star})$.

Let $\operatorname{AC}_{0}([0,h],\mathbb{R}^{n\star})$ denote the space of
$\mathbb{R}^{n\star}$-valued functions that are absolute continuous on
$(0,h]$, have zero value at zero and zero derivative at $h$. From the
description of the sun dual, it is clear that
$X^{\odot}=\operatorname{AC}_{0}([0,h],\mathbb{R}^{n\star})$. It turns out
that another characterization of the sun dual will be helpful. As an element
$f\in X^{\odot}$ is completely specified by $f(0^{+})\in\mathbb{R}^{n\star}$
and $g\in L^{1}([0,h],\mathbb{R}^{n\star})$, we obtain the isometric
isomorphism

$X^{\odot}\cong\mathbb{R}^{n\star}\times L^{1}([0,h],\mathbb{R}^{n\star}).$

A representation of the dual space $X^{\odot\star}$ of the sun dual
$X^{\odot}$, and its restriction to the maximal subspace of strong continuity
$X^{\odot\odot}$ are given by

$X^{\odot\star}\cong\mathbb{R}^{n}\times L^{\infty}([-h,0],\mathbb{R}^{n})\
\mbox{ and }\ X^{\odot\odot}\cong\mathbb{R}^{n}\times
C([-h,0],\mathbb{R}^{n}).$

The linear canonical embedding $j:X\to X^{\odot\star}$ has action
$j\varphi=(\varphi(0),\varphi)$ for all $\varphi\in X$, mapping $X$ onto
$X^{\odot\odot}$, meaning that $X$ is $\odot$-_reflexive_ with respect to the
shift semigroup $T_{0}$.

Next, we turn our attention to the time-dependent linear DDE

$\begin{cases}\dot{y}(t)=L(t)y_{t},\quad&t\geq s,\\\
y_{s}=\varphi,\quad&\varphi\in X,\end{cases}$ (T-LDDE)

in the setting of time-dependent perturbation theory. Recall that
$L(t)=DF(\gamma_{t})$ for all $t\in\mathbb{R}$ and $s\in\mathbb{R}$ denotes
the _starting time_. For $i=1,\dots,n$ we denote
$r_{i}^{\odot\star}:=(e_{i},0)$, where $e_{i}$ is the $i$th standard basic
vector of $\mathbb{R}^{n}$. It is conventional and convenient to introduce the
shorthand notation

$wr^{\odot\star}:=\sum_{i=1}^{n}w_{i}r_{i}^{\odot\star},\quad\forall
w=(w_{1},\dots,w_{n})\in\mathbb{R}^{n},$

and note that $wr^{\odot\star}=(w,0)\in X^{\odot\star}$. We define the finite
rank $T$-periodic _time-dependent bounded linear perturbation_ $B$ as

$B(t)\varphi:=[L(t)\varphi]r^{\odot\star},\quad\forall t\in\mathbb{R},\
\varphi\in X,$

and since $F\in C^{k+1}(X,\mathbb{R}^{n})$, $L(t)=DF(\gamma_{t})$,
$t\mapsto\gamma_{t}$ is $T$-periodic and of the class $C^{k}$, we have that
$B\in C^{k}(\mathbb{R},\mathcal{L}(X,X^{\odot\star}))$ is $T$-periodic and
Lipschitz continuous. It is shown in [14, Theorem 3.1] that there is a one-to-
one correspondence between solutions of (T-LDDE) and the time-dependent linear
abstract integral equation

$u(t)=T_{0}(t-s)\varphi+j^{-1}\int_{s}^{t}T_{0}^{\odot\star}(s-\tau)B(\tau)u(\tau)d\tau,\quad\varphi\in
X,$ (T-LAIE)

with $t\geq s$ where the integral has to be interpreted as a weak⋆ Riemann
integral [14, Chapter III] and takes values in $j(X)$ under the running
assumption of $\odot$-reflexivity, see [5, Lemma 2.2]. The unique solution of
(T-LAIE) on an interval $I_{\varphi}:=[s,t_{\varphi})\subset\mathbb{R}$ for
some $s<t_{\varphi}\leq\infty$ is generated by a strongly continuous forward
evolutionary system $U:=\\{U(t,s)\\}_{(t,s)\in\Omega_{\mathbb{R}}}$ on $X$ in
the sense that $u(t)=U(t,s)\varphi$ for all $t\in I_{\varphi}$, where
$\Omega_{J}$ is defined by $\\{(t,s)\in J\times J\ :\ t\geq s\\}$ for some
interval $J\subseteq\mathbb{R}$, see [14, Definition XII.2.1 and Theorem
XII.2.7].

As we have defined $U(t,s)$ for all $(t,s)\in\Omega_{\mathbb{R}}$, we are
interested in the associated (sun) dual(s). It is clear that one can define
$U^{\star}(s,t):=U(t,s)^{\star}\in\mathcal{L}(X^{\star}):=\mathcal{L}(X^{\star},X^{\star})$
and that
$U^{\star}:=\\{U^{\star}(s,t)\\}_{(s,t)\in\Omega_{\mathbb{R}}^{\star}}$ forms
a backward evolutionary system on $X^{\star}$, where $\Omega_{J}^{\star}$ is
defined as $\\{(s,t)\in J^{2}:t\geq s\\}$ for some interval
$J\subseteq\mathbb{R}$. Furthermore, the Lipschitz continuity on $B$ ensures
that the restriction $U^{\odot}(s,t):=U^{\star}(s,t)|_{X^{\odot}}$ leaves
$X^{\odot}$ invariant and, by construction,
$U^{\odot}:=\\{U^{\odot}(s,t)\\}_{(s,t)\in\Omega_{\mathbb{R}}^{\star}}$ is a
strongly continuous backward evolutionary system, see [5, Theorem 5.3]. This
allows us to define $U^{\odot\star}(t,s):=(U^{\odot}(s,t))^{\star}$ and it is
clear that
$U^{\odot\star}:=\\{U^{\odot\star}(t,s)\\}_{(t,s)\in\Omega_{\mathbb{R}}}$ is a
forward evolutionary system on $X^{\odot\star}$ that extends $U$, which was
previously defined on $X$.

Let us now characterize the (generalized) generators
$A(\tau),A^{\star}(\tau),A^{\odot}(\tau)$ and $A^{\odot\star}(\tau)$ for all
$\tau\in\mathbb{R}$ together with their domains. The weak⋆ continuous
generator $A^{\odot\star}(\tau)$ takes the form

$\displaystyle\mathcal{D}(A^{\odot\star}(\tau))$
$\displaystyle=\\{(\alpha,\varphi)\in X^{\odot\star}\ :\
\varphi\in\operatorname{Lip}([-h,0],\mathbb{R}^{n})\mbox{ and
}\varphi(0)=\alpha\\},$ $\displaystyle A^{\odot\star}(\tau)j\varphi$
$\displaystyle=(L(\tau)\varphi,\dot{\varphi}),\quad\forall\tau\in\mathbb{R}.$
(9)

Because $A(\tau)$ is defined as the preimage under $j$ of the part of
$A^{\odot\star}(\tau)$ in $j(X)$, we get

$\mathcal{D}(A(\tau))=\\{\varphi\in C^{1}([-h,0],\mathbb{R}^{n})\ :\
\dot{\varphi}(0)=L(\tau)\varphi\\},\quad
A(\tau)\varphi=\dot{\varphi},\quad\forall\tau\in\mathbb{R}.$ (10)

The domain and action of the _dual generator_
$A^{\star}(\tau):=[A(\tau)]^{\star}$ of $A(\tau)$ is treated in the following
lemma. For the dual generator $A^{\star}(\tau)$ it holds

$\displaystyle\mathcal{D}(A^{\star}(\tau))$
$\displaystyle=\mathcal{D}(A_{0}^{\star}),\quad
A^{\star}(\tau)f=\dot{f}+f(0^{+})\zeta(\tau,\cdot),\quad\forall\tau\in\mathbb{R}.$

The equality between the domains follows from a sun-variant of [5, Lemma 4.2].
To show the action, we first have to determine
$B^{\star}(\tau):=[B(\tau)]^{\star}$ and notice that we traditionally restrict
this map on $X^{\odot}$, see [14, page 58]. It follows from the uniqueness of
the adjoint that $B^{\star}(\tau):X^{\odot}\to X^{\star}$ is given by
$B^{\star}(\tau)(c,g)=c\zeta(\tau,\cdot)$ since

$\langle
c\zeta(\tau,\cdot),\varphi\rangle=\langle(\langle\zeta(\tau,\cdot),\varphi\rangle,0),(c,g)\rangle=\langle
B(\tau)\varphi,(c,g)\rangle=\langle B^{\star}(\tau)(c,g),\varphi\rangle$

for all $\tau\in\mathbb{R},\varphi\in X$ and $(c,g)\in X^{\odot}$. Hence,

$A^{\star}(\tau)f=A_{0}^{\star}f+B^{\star}(\tau)f=\dot{f}+f(0^{+})\zeta(\tau,\cdot),$

which completes the proof. The _sun generator_ $A^{\odot}(\tau)$ is defined as
the part of $A^{\star}(\tau)$ in $X^{\odot}$ and it is convenient to write
this in $X^{\odot}$-notation as

$\mathcal{D}(A^{\odot}(\tau))=\\{(c,g)\in\mathcal{D}(A_{0}^{\star})\ :\
g+c\zeta(\tau,\cdot)\in X^{\odot}\\},\quad
A^{\odot}(\tau)(c,g)=g+c\zeta(\tau,\cdot).$

Since $\tau\mapsto\zeta(\tau,\cdot)$ is of the class $C^{k}$, it is clear from
the definitions of the generators that $\tau\mapsto
A(\tau),A^{\star}(\tau),A^{\odot}(\tau)$ and $A^{\odot\star}(\tau)$ are of the
class $C^{k}$.

Our next aim is to study how the time-dependent nonlinear delay differential
equation

$\begin{cases}\dot{y}(t)=L(t)y_{t}+G(t,y_{t}),\quad&t\geq s,\\\
y_{s}=\varphi,\quad&\varphi\in X,\end{cases}$ (T-DDE)

fits naturally in the setting of dual perturbation theory. Recall that
$G(t,\cdot)=F(\gamma_{t}+\cdot)-F(\gamma_{t})-L(t)$ and so $G$ is
$C^{k}$-smooth and $T$-periodic in the first component. The $T$-periodic
$C^{k}$-smooth _time-dependent nonlinear perturbation_ $R$ is given by

$R(t,\varphi):=G(t,\varphi)r^{\odot\star},\quad\forall(t,\varphi)\in\mathbb{R}\times
X,$

and it is shown in [28, Theorem 36] that there is a one-to-one correspondence
between solutions of (T-DDE) and the time-dependent nonlinear abstract
integral equation

$u(t)=U(t,s)\varphi+j^{-1}\int_{s}^{t}U^{\odot\star}(t,\tau)R(\tau,u(\tau))d\tau,\quad\varphi\in
X,$ (T-AIE)

for $t\geq s$ where the integral has to be interpreted as a weak⋆ Riemann
integral and takes values in $j(X)$ under the running assumption of
$\odot$-reflexivity, see [28, Lemma 2]. The $C^{k}$-smoothness of the
nonlinearity $R$ ensures that for any $\varphi\in X$ there exists a unique
(maximal) solution $u_{\varphi}$ of (T-AIE) on some (maximal) interval
$I_{\varphi}=[s,t_{\varphi})\subset\mathbb{R}$ with $s<t_{\varphi}\leq\infty$,
see [28, Proposition 2].

Let us now turn our attention towards the construction of the center manifold
around the cycle $\Gamma$. The spectrum $\sigma(U(s+T,s))$ of the _monodromy
operator_ (at time $s$) $U(s+T,s)\in\mathcal{L}(X)$ is a countable set in
$\mathbb{C}$, independent of the starting time $s$, consisting of $0$ and
isolated eigenvalues (called _Floquet multipliers_) that can possibly
accumulate to $0$. The number $\sigma\in\mathbb{C}$ satisfying
$\lambda=e^{\sigma T}$ is called the _Floquet exponent_ , and it is known that
$1$ is always a Floquet multiplier (called the _trivial Floquet multiplier_)
with associated eigenfunction $\dot{\gamma}_{s}$. The _(generalized)
eigenspace_ (at time $s$) associated to the Floquet multiplier $\lambda$ is
defined as $E_{\lambda}(s):=\mathcal{N}((\lambda I-U(s+T,s))^{k_{\lambda}})$,
where $k_{\lambda}$ is the order of a pole of $(\lambda I-U(s+T,s))^{-1}$.
Hence, $E_{\lambda}(s)$ is finite-dimensional, and its dimension, called the
_algebraic multiplicity_ (of $\lambda$) will be denoted by $m_{\lambda}$. The
_geometric multiplicity_ (of $\lambda$) reflects the dimension of the
eigenspace. As a consequence, the set of Floquet multipliers on the unit
circle $\Lambda_{0}:=\\{\lambda\in\sigma(U(s+T,s))\ :\ |\lambda|=1\\}$ is
finite, say $1\leq n_{0}+1<\infty$ counted with algebraic multiplicity. Then,
we define the $(n_{0}+1)$-dimensional _center eigenspace_ (at time $s$) as

$X_{0}(s):=\bigoplus_{\lambda\in\Lambda_{0}}E_{\lambda}(s).$

In this setting, the periodic local center manifold theorem [28, Corollary 18]
for (T-DDE) applies. To be more precise, define the _center fiber bundle_ as

$X_{0}:=\\{(t,\varphi)\in\mathbb{R}\times X\ :\ \varphi\in X_{0}(t)\\}.$

and denote for any $\delta>0$ the $\delta$-ball in $X$ centered at the origin
by $B_{\delta}(X)$. Then there exists a $C^{k}$-smooth map
$\mathcal{C}:X_{0}\to X$ and a sufficiently small $\delta>0$ such that the
manifold

$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma):=\\{\gamma_{t}+\mathcal{C}(t,\varphi)\in
X\ :\ (t,\varphi)\in X_{0}\mbox{ and }\mathcal{C}(t,\varphi)\in
B_{\delta}(X)\\}$ (11)

is a $T$-periodic $C^{k}$-smooth $(n_{0}+1)$-dimensional locally positively
invariant manifold in $X$, called the _(local) center manifold around
$\Gamma$_, defined in the vicinity of $\Gamma$ for a sufficiently small
$\delta>0$.

##  3 Periodic spectral computations for classical DDEs

In the upcoming characterization of the center manifold, it turns out that we
need a time periodic smooth basis of the center eigenspace $X_{0}(s)$.
Therefore, in subsec: time periodic smooth Jordan, we will prove that such a
periodic smooth basis exists, see thm:eigenfunctions for the main result.

To characterize the (dual) (generalized) eigenspaces by periodic (generalized)
eigenfunctions, we will study abstract ODEs on the spaces $X^{\star}$ and
$X^{\odot\star}$. As the semigroups and generators on both spaces are only
defined in a weak⋆ sense, it is no surprise that we have to study the abstract
ODEs also in a weak⋆ setting. Therefore, we recall for a moment the (partial)
weak⋆ differential operator.

[[5, Definition 4.4]] Let $E$ be a Banach space, $J\subseteq\mathbb{R}$ an
interval and $\Omega\subseteq J\times J$. We say that a function $f:J\to
E^{\star}$ is _weak ⋆ differentiable_ with _weak ⋆ derivative_
$d^{\star}f:J\to E^{\star}$ if

$\frac{d}{dt}\langle f(t),x\rangle=\langle d^{\star}f(t),x\rangle,\quad\forall
x\in E,\ t\in J.$

If in addition $d^{\star}f$ is weak⋆ continuous, then $f$ is called _weak ⋆
continuously differentiable_. Furthermore, we say that a function $g:\Omega\to
E^{\star}$ has _partial weak ⋆ derivatives_ $\partial_{t}^{\star}g:\Omega\to
E^{\star}$ and $\partial_{s}^{\star}g:\Omega\to E^{\star}$ if

$\displaystyle\frac{\partial}{\partial t}\langle g(t,s),x\rangle$
$\displaystyle=\langle\partial_{t}^{\star}g(t,s),x\rangle,\quad\forall x\in
E,\ (t,s)\in\Omega,$ $\displaystyle\frac{\partial}{\partial s}\langle
g(t,s),x\rangle$
$\displaystyle=\langle\partial_{s}^{\star}g(t,s),x\rangle,\quad\forall x\in
E,\ (t,s)\in\Omega.$

If in addition $\partial_{t}^{\star}g$ and $\partial_{s}^{\star}g$ are weak⋆
continuous, then $g$ is called _weak ⋆ continuously differentiable_. For using
spectral theory on the real Banach space $X$, we have to _complexify_ $X$ and
all discussed operators on $X$. This is not entirely trivial and is discussed
in [14, Section III.7 and Section IV.2]. To clarify, by the spectrum of a real
(unbounded) operator $L$ defined on (a subspace of) $X$, we mean the spectrum
of its complexification $L_{\mathbb{C}}$ on (a subspace of) the complexified
Banach space $X_{\mathbb{C}}$. For the ease of notation, we omit the
additional symbols.

###  3.1 Periodic smooth Jordan chains

Let us focus on a specific Floquet multiplier $\lambda\in\sigma(U(s+T,s))$ for
a fixed $s\in\mathbb{R}$. By the construction given in [14, Section IV.4], it
is possible to find a basis of $E_{\lambda}(s)$ that is in _Jordan normal
form_. That is, there exists an ordered basis
$\\{\phi_{s}^{0},\dots,\phi_{s}^{m_{\lambda}-1}\\}$ of $E_{\lambda}(s)$ called
a _Jordan chain_ such that

$(U(s+T,s)-\lambda I)\phi_{s}^{i}=\begin{cases}0,\quad&i=0,\\\
\phi_{s}^{i-1},\quad&i=1,\dots,m_{\lambda}-1,\end{cases}$ (12)

and $\phi_{s}^{i}$ should be interpreted via the translation property (2) for
all $i=0,\dots,m_{\lambda}-1$. As the map
$U_{\lambda}(\tau,s):=U(\tau,s)|_{E_{\lambda}(s)}:E_{\lambda}(s)\to
E_{\lambda}(\tau)$ is a topological isomorphism [14, Theorem XIII.3.3], we
know that $\\{\phi_{\tau}^{0},\dots,\phi_{\tau}^{m_{\lambda}-1}\\}$ is an
ordered basis of $E_{\lambda}(\tau)$, where
$\phi_{\tau}^{i}:=U_{\lambda}(\tau,s)\phi_{s}^{i}$ for all $\tau\in\mathbb{R}$
and $i=0,\dots,m_{\lambda}-1$. The following lemma shows that this specific
basis of $E_{\lambda}(\tau)$ has additional structure.

The ordered basis
$\\{\phi_{\tau}^{0},\dots,\phi_{\tau}^{m_{\lambda}-1}\\}\subseteq\mathcal{D}(A(\tau))$
consists of $C^{k+1}$-smooth functions and forms a Jordan chain for
$E_{\lambda}(\tau)$ for all $\tau\in\mathbb{R}$. Let $\tau\in\mathbb{R}$ be
given. Since $\phi_{\tau}^{i}=U_{\lambda}(\tau,s)\phi_{s}^{i}$, it is clear
from the computation

$(U(\tau+T,\tau)-\lambda I)\phi_{\tau}^{i}=U(\tau,s)(U(s+T,s)-\lambda
I)\phi_{s}^{i}=\begin{dcases}0,\quad&i=0,\\\
\phi_{\tau}^{i-1},\quad&i=1,\dots,m_{\lambda}-1,\end{dcases}$

that $\\{\phi_{\tau}^{0},\dots,\phi_{\tau}^{m_{\lambda}-1}\\}$ is a Jordan
chain for $E_{\lambda}(\tau)$. Let us now prove that
$\phi_{\tau}^{i}\in\mathcal{D}(A(\tau))$, where the domain is defined in (10),
by first showing that $\phi_{\tau}^{i}\in C^{1}([-h,0],\mathbb{R}^{n})$. Since
$\phi_{\tau}^{i}=U_{\lambda}(\tau,s)\phi_{s}^{i}$, it follows from [14,
Theorem XII.3.1] that $\phi_{\tau}^{i}$ satisfies the delay differential
equation

$\dot{\phi}^{i}(\tau)=L(\tau)\phi_{\tau}^{i}.$ (13)

Since $\phi_{s}^{i}\in C([-h,0],\mathbb{R}^{n})=X$, we know from applying the
method of steps onto (13) that its solution for $\tau>s$ is at least
continuously differentiable [18, Theorem 1.2.2]. We prove the claim now by
induction. If $i=0$, choose $m\in\mathbb{N}$ large enough to guarantee that
$\tau+mT>s+h$ because then the history $\phi_{\tau+mT}^{0}$ coincides
precisely with the solution of (13) on $[\tau+mT-h,\tau+mT]\subset[s,\infty)$,
and so $\phi_{\tau+mT}\in C^{1}([-h,0],\mathbb{R}^{n})$. It follows from

$\lambda^{m}\phi_{\tau}^{0}=U(\tau+T,\tau)^{m}\phi_{\tau}^{0}=U(\tau+mT,\tau)\phi_{\tau}^{0}=\phi_{\tau+mT}^{0},$

that $\phi_{\tau}^{0}\in C^{1}([-h,0],\mathbb{R}^{n})$ due to linearity and
the fact that $\lambda\neq 0$. The second equality holds due to [14, Corollary
XIII.2.2]. Now, assume that $\phi_{\tau}^{0},\dots,\phi_{\tau}^{i-1}$ are in
$C^{1}([-h,0],\mathbb{R}^{n})$ for some $i\in\\{1,\dots,m_{\lambda}-1\\}$.
Choose again an $m\in\mathbb{N}$ such that $\tau+mT\geq s+h$ because then by
the Jordan chain structure, there exist scalars $c_{m,l}$ for
$l\in\\{0,\dots,i\\}$ such that

$\lambda^{m}\phi_{\tau}^{i}+\sum_{l=0}^{i-1}c_{m,l}\phi_{\tau}^{l}=U(\tau+mT,\tau)\phi_{\tau}^{i}=\phi_{\tau+mT}^{i}\in
C^{1}([-h,0],\mathbb{R}^{n}).$

By the induction hypothesis $\phi_{\tau}^{0},\dots,\phi_{\tau}^{i-1}$ are all
in $C^{1}([-h,0],\mathbb{R}^{n})$ and so we conclude that
$\lambda^{m}\phi_{\tau}^{i}\in C^{1}([-h,0],\mathbb{R}^{n})$ which proves that
$\phi_{\tau}^{i}\in C^{1}([-h,0],\mathbb{R}^{n})$ since $\lambda\neq 0$. This
completes the induction.

The same arguments can be used now, with choosing $m\in\mathbb{N}$ such that
$\tau+mT\geq s+(k+1)h$ and employing the method of steps to increase
smoothness of solutions, to conclude that all maps
$\tau\mapsto\phi_{\tau}^{i}$ are all $C^{k+1}$-smooth.

It remains to show that $\phi_{\tau}^{i}$ satisfies the second condition in
the domain defined in (10). Since $\phi_{\tau}^{i}$ satisfies (13), we get
from the translation property that

$\frac{d}{d\theta}\phi_{\tau}^{i}(\theta)\bigg{|}_{\theta=0}=\frac{d}{d\theta}\phi^{i}(\tau+\theta)\bigg{|}_{\theta=0}=\dot{\phi}^{i}(\tau)=L(\tau)\phi_{\tau}^{i}.$

Hence, $\phi_{\tau}^{i}$ satisfies the second condition from (10). Let us now
take a look at the periodicity of the Jordan chain defined in (12). It is
clear from the computation

$\phi_{s+T}^{i}-\phi_{s}^{i}=\begin{dcases}(\lambda-1)\phi_{s}^{i},\quad&i=0,\\\
(\lambda-1)\phi_{s}^{i}+\phi_{s}^{i-1},\quad&i=1,\dots,m_{\lambda}-1,\end{dcases}$

that $\tau\mapsto\phi_{\tau}^{i}$ is $T$-periodic if and only if $\lambda=1$
and $i=0$. However, in the upcoming characterization of the center manifold,
we explicitly need a $T$-periodic $C^{k+1}$-smooth (generalized) eigenbasis.
To construct this basis, let us first introduce some notation. For a real or
complex Banach space $E$ and integer $l\geq 0$, we define
$C_{T}^{l}(\mathbb{R},E)$ as the Banach space consisting of $T$-periodic
$C^{l}$-smooth $E$-valued functions defined on $\mathbb{R}$ equipped with the
standard $C^{l}$-norm. The following result is a generalization from finite-
dimensional ODEs [23, Proposition III.1] towards infinite-dimensional DDEs.

Let $\lambda$ be a Floquet multiplier with $\sigma$ its associated Floquet
exponent. Then there exist $\varphi_{i}\in C_{T}^{k+1}(\mathbb{R},X)$
satisfying

$\bigg{(}\frac{d}{d\tau}-A^{\odot\star}(\tau)+\sigma\bigg{)}j\varphi_{i}(\tau)=\begin{cases}0,\quad&i=0,\\\
-j\varphi_{i-1}(\tau),\quad&i=1,\dots,m_{\lambda}-1,\end{cases}$ (14)

or equivalently

$\bigg{(}\frac{d}{d\tau}-A(\tau)+\sigma\bigg{)}\varphi_{i}(\tau)=\begin{cases}0,\quad&i=0,\\\
-\varphi_{i-1}(\tau),\quad&i=1,\dots,m_{\lambda}-1,\end{cases}$ (15)

such that the set of functions
$\\{\varphi_{0}(\tau),\dots,\varphi_{m_{\lambda}-1}(\tau)\\}$ is an ordered
basis of $E_{\lambda}(\tau)$.

Let $s\in\mathbb{R}$ be an initial starting time and consider the basis
$\\{\phi_{s}^{0},\dots,\phi_{s}^{m_{\lambda}-1}\\}\subseteq\mathcal{D}(A(s))$
of $E_{\lambda}(s)$ in Jordan normal form, see lemma:basisD(A). We show the
claim by induction on $i\in\\{0,\dots,m_{\lambda}-1\\}$. For the base case
($i=0$), consider the initial value problem

$\begin{dcases}(d^{\star}-A^{\odot\star}(\tau)+\sigma)j\varphi_{0}(\tau)=0,\quad\tau\geq
s,\\\ \varphi_{0}(s)=\phi_{s}^{0},\end{dcases}$ (16)

where $d^{\star}$ denotes the weak⋆ differential operator from
def:wkstarderivative. It follows from the differential equation in (16) that

$\displaystyle d^{\star}(j\circ e^{\sigma(\cdot-s)}\varphi_{0})(\tau)$
$\displaystyle=\sigma
e^{\sigma(\tau-s)}j\varphi_{0}(\tau)+e^{\sigma(\tau-s)}d^{\star}j\varphi_{0}(\tau)$
$\displaystyle=e^{\sigma(\tau-s)}\bigg{(}d^{\star}+\sigma\bigg{)}j\varphi_{0}(\tau)=A^{\odot\star}(\tau)j(e^{\sigma(\tau-s)}\varphi_{0}(\tau)).$

This differential equation is of the form [5, Equation (4.10)] and hence its
unique solution [5, Theorem 4.14] on $[s,\infty)$ is given by

$e^{\sigma(\tau-s)}\varphi_{0}(\tau)=U(\tau,s)\varphi_{0}(s),$ (17)

whenever $\varphi_{0}(s)\in j^{-1}\mathcal{D}(A_{0}^{\odot\star})$, because
$U$ leaves $j^{-1}\mathcal{D}(A_{0}^{\odot\star})$ invariant. Since
$\varphi_{0}(s)=\phi_{s}^{0}$ the claim follows from lemma:basisD(A) because
$\phi_{s}^{0}\in
C^{1}([-h,0],\mathbb{R}^{n})\subset\operatorname{Lip}([-h,0],\mathbb{R}^{n})=j^{-1}\mathcal{D}(A_{0}^{\odot\star})$.
Let us now prove the $T$-periodicity of $\varphi_{0}$. Choosing $\tau=s+T$ in
(17) and using (12) yields

$e^{\sigma T}\varphi_{0}(s+T)=U(s+T,s)\varphi_{0}(s)=\lambda\varphi_{0}(s).$

Because $\lambda=e^{\sigma T}$ is non-zero we get
$\varphi_{0}(s+T)=\varphi_{0}(s)$ and so $\varphi_{0}$ is $T$-periodic. Hence,
$\varphi_{0}$ extends to $\mathbb{R}$. To prove the smoothness assertion,
recall from lemma:basisD(A) that
$\tau\mapsto\phi_{\tau}^{0}=U(\tau,s)\phi_{s}^{0}$ is $C^{k+1}$-smooth, and
because $\tau\mapsto e^{-\sigma(\tau-s)}$ is analytic, it is clear from (17)
that the map $\varphi_{0}$ defined by

$\varphi_{0}(\tau)=e^{-\sigma(\tau-s)}\phi_{\tau}^{0},\quad\forall\tau\in\mathbb{R},$

is $C^{k+1}$-smooth. Hence, the weak⋆ differential operator $d^{\star}$ in
(16) can be replaced by $\frac{d}{d\tau}$ for $i=0$. By linearity, we have
that $\varphi_{0}(\tau)\in\mathcal{D}(A(\tau))$ for all $\tau\in\mathbb{R}$
which proves the base case for (14) and (15).

Now to complete the induction, assume that the maps
$\varphi_{0},\dots,\varphi_{i-1}\in C_{T}^{k+1}(\mathbb{R},X)$ are constructed
for some $i\in\\{1,\dots,m_{\lambda}-1\\}$ and consider the initial value
problem

$\begin{dcases}(d^{\star}-A^{\odot\star}(\tau)+\sigma)j\varphi_{i}(\tau)=-j\varphi_{i-1}(\tau),\quad\tau\geq
s,\\\ \varphi_{i}(s)=\sum_{k=0}^{i}\alpha_{ik}\phi_{s}^{k},\end{dcases}$ (18)

where $\phi_{s}^{0},\dots,\phi_{s}^{i}\in\mathcal{D}(A(s))$ are from the
Jordan chain. The goal is to find scalars $\alpha_{ik}$ such that
$\varphi_{i}$ becomes $T$-periodic. A similar computation as done for the base
case tells us by using (18) that

$d^{\star}(j\circ
e^{\sigma(\cdot-s)}\varphi_{i})(\tau)=e^{\sigma(\tau-s)}\bigg{(}d^{\star}+\sigma\bigg{)}j\varphi_{i}(\tau)=A^{\odot\star}(\tau)j(e^{\sigma(\tau-s)}\varphi_{i}(\tau))-j(e^{\sigma(\tau-s)}\varphi_{i-1}(\tau)).$
(19)

We will show that the differential equation above admits a unique solution on
$[s,\infty)$. Consider the function $w_{i}:[s,\infty)\to X$ defined by

$w_{i}(\tau):=U(\tau,s)\sum_{k=0}^{i}\frac{(s-\tau)^{k}}{k!}\varphi_{i-k}(s),\quad\forall\tau\in[s,\infty).$
(20)

Because $\phi_{s}^{0},\dots,\phi_{s}^{i}\in\mathcal{D}(A(s))$, we have from
(18) that $\varphi_{0}(s),\dots,\varphi_{i}(s)\in\mathcal{D}(A(s))$ by
linearity. It is clear that

$U(\tau,s)\varphi_{j}(s)=\sum_{k=0}^{j}\alpha_{jk}\phi_{\tau}^{k}\in\mathcal{D}(A(\tau)),\quad\forall\tau\in[s,\infty),\
j\in\\{0,\dots,i\\},$

and so it follows that $\tau\mapsto w_{i}(\tau)$ takes values in
$\mathcal{D}(A(\tau))\subseteq j^{-1}\mathcal{D}(A_{0}^{\odot\star})$ and is
$C^{k+1}$-smooth, which implies the weak⋆ differentiability of $w_{i}$.
Clearly $w_{i}(s)=\varphi_{i}(s)$ and notice that

$\displaystyle d^{\star}(j\circ w_{i})(\tau)$
$\displaystyle=A^{\odot\star}(\tau)jU(\tau,s)\sum_{k=0}^{i}\frac{(s-\tau)^{k}}{k!}\varphi_{i-k}(s)-jU(t,s)\sum_{k=0}^{i-1}\frac{(s-\tau)^{k}}{k!}\varphi_{i-k}(s)$
$\displaystyle=A^{\odot\star}(\tau)jw_{i}(\tau)-jw_{i-1}(\tau),$

and so $w_{i}$ is a solution on $[s,\infty)$ of (19). Since $w_{i-1}$ is at
least continuous, it follows from [28, Proposition 32] and by construction of
$w_{i}$ that (19) admits a unique solution $j\circ w_{i}$ on $[s,\infty)$,
where $w_{i}=e^{\sigma(\cdot-s)}\varphi_{i}$. As a consequence,
$\varphi_{i}=e^{-\sigma(\cdot-s)}w_{i}$ is $C^{k+1}$-smooth because
$\tau\mapsto e^{-\sigma(\tau-s)}$ is analytic.

Let us now turn our attention towards proving $T$-periodicity. We see from
(20) that $\varphi_{i}(s)=\varphi_{i}(s+T)$ if and only if

$(U(s+T,s)-\lambda
I)\varphi_{i}(s)=U(s+T,s)\sum_{k=1}^{i}\frac{(-1)^{k}T^{k}}{k!}\varphi_{i-k}(s).$
(21)

Recall from (18) that $\varphi_{i}(s)=\sum_{k=0}^{i}\alpha_{ik}\phi_{s}^{k}$
and retrieving (12) yields

$\displaystyle\sum_{k=1}^{i}\alpha_{ik}\phi_{s}^{k-1}$
$\displaystyle=U(s+T,s)\sum_{l=1}^{i}\sum_{k=1}^{i-l}\alpha_{i-l,k}\frac{(-1)^{l}T^{l}}{l!}\phi_{s}^{k}$
$\displaystyle=\sum_{l=1}^{i}\sum_{k=0}^{i-l}\alpha_{i-l,k}\frac{(-1)^{l}T^{l}}{l!}\begin{cases}\lambda\phi_{s}^{k},\quad&k=0,\\\
\lambda\phi_{s}^{k}+\phi_{s}^{k-1},\quad&k=1,\dots,i-1.\end{cases}$

Because the right-hand side is a known element in the subspace spanned by
$\phi_{s}^{0},\dots,\phi_{s}^{i-1}$, the $\alpha_{ik}$’s are uniquely
determined for $k=0,\dots,i$ and so we have proven that $\varphi_{i}$ is
$T$-periodic. Hence, $\tau\mapsto\varphi_{i}(\tau)$ extends to a
$C^{k+1}$-smooth solution on $\mathbb{R}$, taking values in
$\mathcal{D}(A(\tau))$. Similarly, the weak⋆ differential operator from (18)
can be replaced by $\frac{d}{d\tau}$ and so the formulas (14) and (15) hold.

Furthermore, $\varphi_{0}(\tau),\dots,\varphi_{m_{\lambda}}(\tau)$ are all
linearly independent because they are all solutions to the abstract ODE

$\bigg{(}\frac{d}{d\tau}-A^{\odot\star}(\tau)+\sigma\bigg{)}^{m_{\lambda}}j\varphi(\tau)=0,\quad\forall\tau\in\mathbb{R},$

which completes the proof. Let us now take some time to discuss the connection
between the $T$-periodicity and the translation property (2) of the
(generalized) eigenfunctions. It is clear that
$\\{\phi_{\tau}^{0},\dots,\phi_{\tau}^{m_{\lambda}-1}\\}$ is (in general) a
non-$T$-periodic basis of $E_{\lambda}(\tau)$ that has the translation
property because $\phi_{\tau}^{i}=U(\tau,s)\phi_{s}^{i}$. On the other hand,
thm:eigenfunctions shows us that
$\\{\varphi_{0}(\tau),\dots,\varphi_{m_{\lambda}-1}(\tau)\\}$ is a
$T$-periodic basis of $E_{\lambda}(\tau)$, but how is this basis related to
the translation property? Notice that the function $\varphi_{0}(\tau)\in X$
would have the translation property if and only if it satisfies the transport
equation

$\frac{\partial}{\partial\tau}\varphi_{0}(\tau)(\theta)=\frac{\partial}{\partial\theta}\varphi_{0}(\tau)(\theta),$
(22)

but a small calculation directly shows that

$\displaystyle\frac{\partial}{\partial\tau}\varphi_{0}(\tau)(\theta)$
$\displaystyle=e^{-\sigma(\tau-s)}(-\sigma+\dot{\phi}_{\tau}^{0}(\theta)),$
$\displaystyle\frac{\partial}{\partial\theta}\varphi_{0}(\tau)(\theta)$
$\displaystyle=e^{-\sigma(\tau-s)}\dot{\phi}_{\tau}^{0}(\theta),$

and so $\varphi_{0}$ satisfies (22) if and only if $\sigma=0$ i.e.
$\lambda=1$. A similar analysis for the $T$-periodic generalized
eigenfunctions shows that these never have the translation property. Hence,
the only $T$-periodic solution of (15) which satisfies the translation
property is the map $\tau\mapsto\dot{\gamma}_{\tau}$. It is however the
$T$-periodic basis
$\\{\varphi_{0}(\tau),\dots,\varphi_{m_{\lambda}-1}(\tau)\\}$ of
$E_{\lambda}(\tau)$ that is needed for the characterization of the center
manifold $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ in
sec:characterization.

In the upcoming construction of the characterization of the center manifold,
we also need the _Floquet operator_ (at time $\tau$) associated to the Floquet
multiplier $\lambda$, defined as the coordinate map
$Q_{\lambda}(\tau):\mathbb{C}^{m_{\lambda}}\to E_{\lambda}(\tau)$ by

$Q_{\lambda}(\tau)\xi:=\sum_{i=0}^{m_{\lambda}-1}\xi_{i}\varphi_{i}(\tau),\quad\forall\xi=(\xi_{0},\dots,\xi_{m_{\lambda}-1})\in\mathbb{C}^{m_{\lambda}}.$
(23)

It is clear from thm:eigenfunctions that the map $\tau\mapsto
Q_{\lambda}(\tau)$ is $T$-periodic, $C^{k+1}$-smooth and takes values in
$\mathcal{L}(\mathbb{C}^{m_{\lambda}},E_{\lambda}(\tau))$ for all
$\tau\in\mathbb{R}$. Furthermore, a direct calculation shows that

$\bigg{(}-\frac{d}{d\tau}+A^{\odot\star}(\tau)\bigg{)}j(Q_{\lambda}(\tau)\xi)=j(Q_{\lambda}(\tau)M_{\lambda}\xi),$
(24)

where $M_{\lambda}$ is the ${m_{\lambda}}\times m_{\lambda}$ Jordan matrix
defined by

$M_{\lambda}:=\begin{pmatrix}\sigma&1&\cdots&0\\\ 0&\sigma&\ddots&\vdots\\\
\vdots&\ddots&\ddots&1\\\ 0&\dots&0&\sigma\end{pmatrix}.$ (25)

This result is an extension of [23, Proposition III.3] from finite-dimensional
ODEs to infinite-dimensional DDEs. Because we are dealing with the real state
space $X=C([-h,0],\mathbb{R}^{n})$, the linear operator $M_{\lambda}$, written
in matrix form in (25), should represent a real operator, recall also
remark:complexification. Depending on the location of the Floquet multiplier
$\lambda$ in the complex plane, we have three options [23]:

  * •

If $\lambda$ is real and positive, we choose $\sigma$ and
$\varphi_{0}(\tau),\dots,\varphi_{m_{\lambda}-1}(\tau)$ real.

  * •

If $\lambda$ is not real, then its complex conjugate
$\overline{\lambda}\neq\lambda$ is also a Floquet multiplier. Hence, we choose
$\sigma$ and $\varphi_{0}(\tau),\dots,\varphi_{m_{\lambda}-1}(\tau)$ complex
and introduce $\overline{\sigma}$ and
$\overline{\varphi_{0}}(\tau),\dots,\overline{\varphi_{m_{\lambda}-1}}(\tau)$
for the complex conjugate Jordan block.

  * •

If $\lambda$ is real and negative, both methods describe above do not succeed.
Indeed, the Floquet exponents $\sigma$ are of the form
$\frac{\pi}{T}+\frac{2li\pi}{T}$ with $l\in\mathbb{Z}$. The standard way to
deal with this situation is to double the period, since if
$\lambda\in\sigma(U(s+T,s))$ is a Floquet multiplier then
$\lambda^{2}\in\sigma(U(s+2T,s))$.

To study this last case, one has to require $T$-antiperiodicity of the maps
$\varphi_{i}$ from thm:eigenfunctions. By _$T$ -antiperiodicity_, we mean that
the maps $\varphi_{i}$ are $2T$-periodic and satisfy in addition

$\varphi_{i}(\tau+T)=-\varphi_{i}(\tau),\quad\forall\tau\in\mathbb{R},\
i=0,\dots,m_{\lambda}-1.$

The existence of such $T$-antiperiodic (generalized) eigenfunctions will be
proven in the following proposition. In this antiperiodic setting, the
associated Floquet exponent $\sigma\in\mathbb{R}$ is defined as $e^{\sigma
T}=|\lambda|$.

Let $\lambda$ be a real and negative Floquet multiplier with $\sigma$ its
associated Floquet exponent. Then there exist $T$-antiperiodic maps
$\varphi_{0},\dots,\varphi_{m_{\lambda}-1}\in C_{2T}^{k+1}(\mathbb{R},X)$
satisfying (14) and (15). Furthermore, there exists a real $T$-periodic
projector $P_{\lambda}:\mathbb{R}\to\mathcal{L}(X)$ onto $E_{\lambda}(\tau)$.
Moreover, the Floquet operator satisfies
$Q_{\lambda}(\tau+T)=-Q_{\lambda}(\tau)$ and the differential equation (24),
where $M_{\lambda}$ is now a linear operator on $\mathbb{R}^{m_{\lambda}}$. To
prove the first assertion, we copy the proof of thm:eigenfunctions but in the
$2T$-periodic setting. The proof goes identical up to (17). If we set
$\tau=s+T$ in (17) we get

$|\lambda|\varphi_{0}(s+T)=e^{\sigma
T}\varphi_{0}(s+T)=U(s+T,s)\varphi_{0}(s)=\lambda\varphi_{0}(s)$

and so
$\varphi_{0}(s+T)=\operatorname{sign}(\lambda)\varphi_{0}(s)=-\varphi_{0}(s)$,
which proves the $T$-antiperiodicity of $\varphi_{0}$.

Consider now (18) and suppose that the right-hand side of the differential
equation satisfies $\varphi_{i-1}(s+T)=-\varphi_{i-1}(s)$. Our goal now is to
find the $\alpha_{ik}$ such that $\varphi_{i}$ is $T$-antiperiodic. Instead of
requiring the $T$-periodicity of $\varphi_{i}$ we now require
$T$-antiperiodicity of $\varphi_{i}$ and see that this holds if and only if

$(U(s+T,s)-\lambda
I)\varphi_{i}(s)=U(s+T,s)\sum_{k=1}^{i}\frac{(-1)^{k}T^{k}}{k!}\varphi_{i-k}(s)\\\
$

which is precisely (21). Hence, the same procedure in thm:eigenfunctions can
be followed to find the associated $\alpha_{ik}$’s uniquely and show that
$\varphi_{i}$ is $T$-antiperiodic.

The real spectral projection $P_{\lambda}(\tau)\in\mathcal{L}(X)$ onto
$E_{\lambda}(\tau)$ for all $\tau\in\mathbb{R}$ is constructed in the same way
as the Dunford integral in [28, Appendix A.2] and so $T$-periodicity follows.
For the Floquet operator, it follows from linearity and
$\varphi_{i}(\tau+T)=-\varphi_{i}(\tau)$ for all $i=0,\dots,m_{\lambda}-1$
that $Q_{\lambda}(\tau+T)=-Q_{\lambda}(\tau)$ for all $\tau\in\mathbb{R}$.

###  3.2 Dual periodic smooth Jordan chains

Our next goal is to repeat the construction above, but now for the adjoint
system. Let us first observe that $\sigma(U(s+T,s))=\sigma(U^{\star}(s-T,s))$
because $U(s+T,s)$ is a real operator (remark:complexification) and
$U^{\star}(s,s+T)=U^{\star}(s-T,s)$ due to [14, Corollary XIII.2.2]. Hence, it
follows from general spectral theory and [15, Proposition IV.2.18] that

$X=\mathcal{N}((\lambda I-U(s+T,s))^{k_{\lambda}})\oplus\mathcal{N}((\lambda
I-U^{\odot}(s-T,s))^{k_{\lambda}})^{\perp},\\\ $

where $\perp$ denotes the annihilator. Notice that the first sum in this
direct sum decomposition is precisely the $m_{\lambda}$-dimensional subspace
$E_{\lambda}(s)$ of $X$ which has been characterized in subsec: time periodic
smooth Jordan. The aim of this subsection is to characterize the
$m_{\lambda}$-dimensional subspace
$E_{\lambda}^{\odot}(s):=\mathcal{N}((\lambda
I-U^{\odot}(s-T,s))^{k_{\lambda}})$ of $X^{\odot}$ and therefore we start by
constructing a Jordan chain. Again, by the construction given in [14, Section
IV.4], there exists an ordered basis
$\\{\phi_{m_{\lambda}-1}^{\odot}(s),\dots,\phi_{0}^{\odot}(s)\\}$ of
$E_{\lambda}^{\odot}(s)$ called a _Jordan chain_ such that

$(U^{\odot}(s-T,s)-\lambda
I)\phi_{i}^{\odot}(s)=\begin{dcases}0,\quad&i=m_{\lambda}-1,\\\
\phi_{i+1}^{\odot}(s),\quad&i=m_{\lambda}-2,\dots,0.\end{dcases}$ (26)

Because the $T$-periodic time-dependent perturbation $B$ is Lipschitz
continuous, it follows from [5, Remark 5.10] that the backward evolutionary
system $U^{\odot}$ leaves $X^{\odot}$ invariant, which proves that the Jordan
chain in (26) is well-defined. Recall from subsec: time periodic smooth Jordan
that the bounded linear operator $U_{\lambda}(\tau,s):E_{\lambda}(s)\to
E_{\lambda}(\tau)$ is a topological isomorphism. Hence,
$U_{\lambda}^{\odot}(\tau,s):=(U_{\lambda}(s,\tau))^{\odot}:E_{\lambda}^{\odot}(s)\to
E_{\lambda}^{\odot}(\tau)$ is a topological isomorphism and so
$\\{\phi_{m_{\lambda}-1}^{\odot}(\tau),\dots,\phi_{0}^{\odot}(\tau)\\}$ is an
ordered basis of $E_{\lambda}^{\odot}(\tau)$, where
$\phi_{i}^{\odot}(\tau):=U_{\lambda}^{\odot}(\tau,s)\phi_{i}^{\odot}(s)$ for
all $\tau\in\mathbb{R}$ and $i=0,\dots,m_{\lambda}-1$. The following lemma
shows that $E_{\lambda}^{\odot}(\tau)$ has additional structure.

The ordered basis
$\\{\phi_{m_{\lambda}-1}^{\odot}(\tau),\dots,\phi_{0}^{\odot}(\tau)\\}\subseteq\mathcal{D}(A_{0}^{\star})$
consists of $C^{k+1}$-smooth functions and forms a Jordan chain for
$E_{\lambda}^{\odot}(\tau)$ for all $\tau\in\mathbb{R}$. The proof of the
Jordan chain structure is analogous to that of lemma:basisD(A). Let
$i\in\\{m_{\lambda}-1,\dots,0\\}$ be given. Because $\phi_{i}^{\odot}(s)\in
X^{\odot}$, it follows from [5, Theorem 5.8] that
$\tau\mapsto\phi_{i}^{\odot}(\tau)$ satisfies

$\frac{d}{d\tau}\langle\phi_{i}^{\odot}(\tau),\varphi\rangle=-\langle
A^{\odot\star}(\tau)j\varphi,\phi_{i}^{\odot}(\tau)\rangle,\quad\forall\varphi\in
j^{-1}\mathcal{D}(A_{0}^{\odot\star}),$ (27)

where the map $\tau\mapsto\langle\phi_{i}^{\odot}(\tau),\varphi\rangle$ is
continuously differentiable for all $\tau\leq s$. Because $U^{\odot}$ leaves
$X^{\odot}$ invariant, we know that
$\phi_{i}^{\odot}(\tau)=(c_{i}(\tau),g_{i}(\tau))\in X^{\odot}$, where
$\phi_{i}^{\odot}(\tau)(0^{+})=:c_{i}(\tau)\in\mathbb{C}^{n\star}$ and
$g_{i}(\tau)\in L^{1}([0,h],\mathbb{C}^{n\star})$ represents the derivative of
$\phi_{i}^{\odot}$ with respect to the second component. Notice that in the
representation of spaces we worked with the complexification
(remark:complexification). Furthermore, it follows that $g_{i}(\tau)(h)=0$ for
all $\tau\leq s$. As we are working in the setting of classical DDEs, we can
evaluate the pairings explicitly [14, Equation (II.5.6)] and so

$\displaystyle\langle
A^{\odot\star}(\tau)j\varphi,\phi_{i}^{\odot}(\tau)\rangle$
$\displaystyle=\langle(L(\tau)\varphi,\dot{\varphi}),(c_{i}(\tau),g_{i}(\tau))\rangle$
$\displaystyle=\langle
c_{i}(\tau)\zeta(\tau,\cdot),\varphi\rangle+\int_{0}^{h}g_{i}(\tau)(\theta)\dot{\varphi}(-\theta)d\theta$
$\displaystyle=\int_{0}^{h}d_{\theta}[c_{i}(\tau)\zeta(\tau,\theta)+g_{i}(\tau)(\theta)]\varphi(-\theta),$

where we used (4) and (2) in the second equality, partial integration for
Riemann-Stieltjes integrals and the conditions on $g$ coming from (2) in the
third equality. The $d_{\theta}$ refers to Riemann-Stieltjes integration over
the $\theta$-variable. We can also express the left-hand side of (27) as

$\frac{d}{d\tau}\langle\phi_{i}^{\odot}(\tau),\varphi\rangle=\dot{c}_{i}(\tau)\varphi(0)+\int_{0}^{h}\frac{\partial
g_{i}(\tau)(\theta)}{\partial\tau}\varphi(-\theta)d\theta=\int_{0}^{h}d_{\theta}\bigg{[}\frac{\partial\phi_{i}^{\odot}(\tau)(\theta)}{\partial\tau}\bigg{]}\varphi(-\theta).$

Hence, (27) is equivalent to

$\int_{0}^{h}d_{\theta}\bigg{[}c_{i}(\tau)\zeta(\tau,\theta)+\bigg{(}\frac{\partial}{\partial\tau}+\frac{\partial}{\partial\theta}\bigg{)}\phi_{i}^{\odot}(\tau)(\theta)\bigg{]}\varphi(-\theta)=0,\quad\forall\varphi\in
j^{-1}\mathcal{D}(A_{0}^{\odot\star}),$

Clearly, if we can show that $\phi_{i}^{\odot}$ satisfies

$\bigg{(}\frac{\partial}{\partial\tau}+\frac{\partial}{\partial\theta}\bigg{)}\phi_{i}^{\odot}(\tau)(\theta)+c_{i}(\tau)\zeta(\tau,\theta)=0,\quad\forall\tau\leq
s,\ \theta\in(0,h],$ (28)

then (27) is satisfied and the result follows from the uniqueness of (27) when
the initial condition $\phi_{i}^{\odot}(s)$ at starting time $s$ is specified.
The inhomogeneous transport equation (28) has the unique solution

$\phi_{i}^{\odot}(\tau)(\theta)=c_{i}(\tau-\theta)-\int_{0}^{\theta}c_{i}(\tau+v-\theta)\zeta(\tau+v-\theta,v)dv,$
(29)

when the initial condition $c_{i}$ is specified. To determine $c_{i}$ from
$\phi_{i}^{\odot}(s)$, we will have a look at the map $\theta\mapsto
g_{i}(\tau)(\theta)=\frac{\partial}{\partial\theta}\phi_{i}^{\odot}(\tau)(\theta)$
that has to satisfy $g_{i}(\tau)(h)=0$ for all $\tau\leq s$. Differentiating
(29) with respect to $\theta$ by employing the Leibniz integral rule gives

$\displaystyle g_{i}(\tau)(\theta)$
$\displaystyle=-\dot{c}_{i}(\tau-\theta)-c_{i}(\tau)\zeta(\tau,\theta)+\int_{0}^{\theta}\dot{c}_{i}(\tau+v-\theta)\zeta(\tau+v-\theta,v)dv$
$\displaystyle+\int_{0}^{\theta}c_{i}(\tau+v-\theta)D_{1}\zeta(\tau+v-\theta,v)dv.$

Using partial integration for Riemann-Stieltjes integrals on the first
integral above leads eventually to

$g_{i}(\tau)(\theta)=-\dot{c}_{i}(\tau-\theta)-\int_{0}^{\theta}c_{i}(\tau+v-\theta)d_{2}\zeta(\tau+v-\theta,v).$

To simplify the expression even more, recall that $g_{i}(\tau-\theta+h)(h)=0$
for all $\tau\in\mathbb{R}$ and $\theta\in(0,h]$ that satisfy
$\tau-\theta+h\leq s$. This is equivalent to

$0=-\dot{c}_{i}(\tau-\theta)-\int_{0}^{h}c_{i}(\tau+v-\theta)d_{2}\zeta(\tau+v-\theta,v),$
(30)

and so $g_{i}$ simplifies to

$g_{i}(\tau)(\theta)=\int_{\theta}^{h}c_{i}(\tau-\theta+v)d_{2}\zeta(\tau-\theta+v,v).$
(31)

As $g_{i}(\tau-h)(h)=0$ for all $\tau\leq s+h$, it follows from (30) that that
$c_{i}$ satisfies

$\dot{c}_{i}(\tau)=-\int_{0}^{h}c_{i}(\tau+v)d_{2}\zeta(\tau+v,v),\quad\forall\tau\leq
s-h,$ (32)

which is an advance differential equation that has to be solved backward in
time with an initial condition that still has to be specified. To determine
the initial condition, recall that $\phi_{i}^{\odot}(s)$ is known and so it
follows from (29) that $c_{i}$ has to satisfy

$c_{i}(s-\theta)=\phi_{i}^{\odot}(s)(\theta)+\int_{0}^{\theta}c_{i}(s+v-\theta)\zeta(s+v-\theta,v)dv,\quad\theta\in[0,h].$

Perform the change of variables:
$s-\theta=-\xi,z_{i}(\xi)=c_{i}(-\xi),\psi(\xi)=\phi_{i}^{\odot}(s)(s+\xi)$
and define the map $K$ as $K(\xi-v):=\zeta(v-\xi,v)$. This yields the linear
Volterra integral equation of the second kind

$z_{i}(\xi)=\psi(\xi)+\int_{-s}^{\xi}z_{i}(\eta)K(\xi-\eta)d\eta,\quad\forall\xi\in(-s,-s+h].$
(33)

Because $\psi\in\operatorname{AC}((-s,-s+h],\mathbb{C}^{n\star})$ and $K\in
L^{1}((0,h],\mathbb{C}^{n\star})$ as the kernel $\zeta$ is $C^{k}$-smooth in
the first component and of bounded variation of the second component, it
follows from [17, Theorem 2.3.5] that (33) has a unique solution
$\tilde{z}_{i}=z_{i}(-s+h+\cdot)\in\operatorname{AC}([-h,0],\mathbb{C}^{n\star})\subset
X$, where we define $\tilde{z}_{i}(0):=\lim_{t\downarrow 0}z_{i}(-s+h+t)$.
Perform now the change of variables $\tau=-t,z_{i}(t)=c_{i}(-t)$ and
$\tilde{\zeta}(t,v)=\zeta(v-t,v)$ in (32) because then it is clear that (32)
can be transformed into

$\begin{dcases}\dot{z}_{i}(t)=\int_{0}^{h}z_{i}(t-v)d_{2}\tilde{\zeta}(t,v),\quad&t\geq-s+h,\\\
z_{i}(-s+h+\cdot)=\tilde{z}_{i},\quad&\tilde{z}_{i}\in X,\end{dcases}$ (34)

which is a delay differential equation with initial condition $\tilde{z}_{i}$
on starting time $-s+h$. By the Jordan chain structure in (26), we can use
similar arguments on (34) as in the proof of lemma:basisD(A) to conclude that
$z_{i}$ is $C^{k+1}$-smooth and extends to $\mathbb{R}$. Hence, $c_{i}$ is
$C^{k+1}$-smooth, and it follows then from (31) that $\tau\mapsto g_{i}(\tau)$
is $C^{k+1}$-smooth, and the map $\theta\mapsto g_{i}(\tau)(\theta)$ has
bounded variation for all $\theta\in(0,h]$ because of the $C^{k+1}$-smoothness
of $c_{i}$ and the $C^{k}$-smoothness of $\zeta$ in the first component. As
$g_{i}(\tau)(0)=0$ for all $\tau\in\mathbb{R}$, we have shown that
$g_{i}(\tau)\in\operatorname{NBV}([0,h],\mathbb{C}^{n\star})$ and so
$\phi_{i}^{\odot}(\tau)\in\mathcal{D}(A_{0}^{\star})$ for all
$\tau\in\mathbb{R}$. Hence, we obtain the $C^{k+1}$-smoothness of
$\tau\mapsto\phi_{i}^{\odot}(\tau)$ from (29). Let us now take a look at the
periodicity of the Jordan chain for the adjoint system. It is clear from the
computation

$\phi_{i}^{\odot}(s-T)-\phi_{i}^{\odot}(s)=\begin{dcases}(\lambda-1)\phi_{m_{\lambda}-1}^{\odot}(s),\quad&i=m_{\lambda}-1,\\\
(\lambda-1)\phi_{i}^{\odot}(s)+\phi_{i+1}^{\odot}(s),\quad&i=m_{\lambda}-2,\dots,0,\end{dcases}$

that $\tau\mapsto\phi_{i}^{\odot}(\tau)$ is $T$-periodic if and only if
$\lambda=1$ and $i=m_{\lambda}-1$. However, similarly as for the (generalized)
eigenfunctions, we explicitly need a $C^{k+1}$-smooth $T$-periodic basis of
$E_{\lambda}^{\odot}(s)$ for the computation of the critical normal
coefficients in an upcoming paper.

Let $\lambda$ be a Floquet multiplier with $\sigma$ its associated Floquet
exponent. Then there exist $\varphi_{i}^{\odot}\in
C_{T}^{k+1}(\mathbb{R},X^{\odot})$ satisfying

$\bigg{(}\frac{d}{d\tau}+A^{\star}(\tau)-\sigma\bigg{)}\varphi_{i}^{\odot}(\tau)=\begin{dcases}0,\quad&i=m_{\lambda}-1,\\\
\varphi_{i+1}^{\odot}(\tau),\quad&i=m_{\lambda}-2,\dots,0\end{dcases}$ (35)

such that the set of functions
$\\{\varphi_{m_{\lambda}-1}^{\odot}(\tau),\dots,\varphi_{0}^{\odot}(\tau)\\}$
is an ordered basis of $E_{\lambda}^{\odot}(\tau)$. Let $s\in\mathbb{R}$ be a
starting time and consider the order basis
$\\{\phi_{m_{\lambda}-1}^{\odot}(s),\dots,\phi_{0}^{\odot}(s)\\}$ of
$E_{\lambda}^{\odot}(s)$ in Jordan normal form. We show the claim by induction
on $i\in\\{m_{\lambda}-1,\dots,0\\}$. For the base case ($i=m_{\lambda}-1)$,
consider the initial value problem

$\begin{dcases}(d^{\star}+A^{\star}(\tau)-\sigma)\varphi_{m_{\lambda}-1}^{\odot}(\tau)=0,\quad\tau\leq
s,\\\
\varphi_{m_{\lambda}-1}^{\odot}(s)=\phi_{m_{\lambda}-1}^{\odot}(s),\end{dcases}$
(36)

where $\phi_{m_{\lambda}-1}^{\odot}(s)$ is the first basis vector of
$E_{\lambda}^{\odot}(s)$. It follows from the differential equation in (36)
that

$\displaystyle
d^{\star}(e^{\sigma(s-\cdot)}\varphi_{m_{\lambda}-1}^{\odot})(\tau)$
$\displaystyle=-\sigma
e^{\sigma(s-\tau)}\varphi_{m_{\lambda}-1}^{\odot}(\tau)+e^{\sigma(s-\tau)}d^{\star}\varphi_{m_{\lambda}-1}^{\odot}(\tau)$
$\displaystyle=e^{\sigma(s-\tau)}\bigg{(}d^{\star}-\sigma\bigg{)}\varphi_{m_{\lambda}-1}^{\odot}(\tau)=-A^{\star}(\tau)[e^{\sigma(s-\tau)}\varphi_{m_{\lambda}-1}^{\odot}(\tau)].$

This differential equation is of the form [5, Equation (5.8)] and so (36)
admits a unique solution [5, Theorem 5.8] on $(-\infty,s]$ given by

$e^{\sigma(s-\tau)}\varphi_{m_{\lambda}-1}^{\odot}(\tau)=U^{\odot}(\tau,s)\varphi_{m_{\lambda}-1}^{\odot}(s),\quad\forall\tau\leq
s,$ (37)

whenever $\varphi_{m_{\lambda}-1}^{\odot}(s)\in\mathcal{D}(A_{0}^{\star})$,
because $U^{\odot}$ leaves $\mathcal{D}(A_{0}^{\star})$ invariant. Since
$\varphi_{m_{\lambda}-1}^{\odot}(s)=\phi_{m_{\lambda}-1}^{\odot}(s)$, the
claim follows from lemma:adjoint D(Astar). Let us now prove the
$T$-periodicity of $\varphi_{m_{\lambda}-1}^{\odot}$. If we set $\tau=s-T$ in
(37) and use (26), we get

$e^{\sigma
T}\varphi_{m_{\lambda}-1}^{\odot}(s-T)=U^{\odot}(s-T,s)\varphi_{m_{\lambda}-1}^{\odot}(s)=\lambda\varphi_{m_{\lambda}-1}^{\odot}(s).$

Because $\lambda=e^{\sigma T}$ is non-zero we get
$\varphi_{m_{\lambda}-1}^{\odot}(s-T)=\varphi_{m_{\lambda}-1}^{\odot}(s)$ and
so $\varphi_{m_{\lambda}-1}^{\odot}$ is $T$-periodic. Hence,
$\varphi_{m_{\lambda}-1}^{\odot}$ extends to $\mathbb{R}$. To prove the
smoothness assertion, recall from lemma:adjoint D(Astar) that
$\tau\mapsto\phi_{m_{\lambda}-1}^{\odot}(\tau)=U^{\odot}(\tau,s)\phi_{m_{\lambda}-1}^{\odot}(s)$
is $C^{k+1}$-smooth, and because $\tau\mapsto e^{-\sigma(\tau-s)}$ is
analytic, it is clear from (37) that the map $\varphi_{m_{\lambda}-1}^{\odot}$
defined by

$\varphi_{m_{\lambda}-1}^{\odot}(\tau)=e^{-\sigma(s-\tau)}\phi_{m_{\lambda}-1}^{\odot}(\tau),\quad\forall\tau\in\mathbb{R},$

is $C^{k+1}$-smooth. Hence, the weak⋆ differential operator $d^{\star}$ in
(35) can be replaced by $\frac{d}{d\tau}$ for $i=m_{\lambda}-1$. By linearity
and lemma:Astar, we have that
$\varphi_{m_{\lambda}-1}^{\odot}(\tau)\in\mathcal{D}(A^{\star}(\tau))$, which
proves the base case for (35).

Assume now that the maps
$\varphi_{m_{\lambda}-1}^{\odot},\dots,\varphi_{i+1}^{\odot}\in
C_{T}^{k+1}(\mathbb{R},X^{\odot})$ are constructed for some $m_{\lambda}-2\geq
i\geq 0$ and consider the initial value problem

$\begin{dcases}(d^{\star}+A^{\star}(\tau)-\sigma)\varphi_{i}^{\odot}(\tau)=\varphi_{i+1}^{\odot}(\tau),\quad\tau\leq
s,\\\
\varphi_{i}^{\odot}(s)=\sum_{k=i}^{m_{\lambda}-1}\alpha_{ik}\phi_{k}^{\odot}(s).\end{dcases}$
(38)

The goal is to find scalars $\alpha_{ik}$ such that $\varphi_{i}^{\odot}$
becomes $T$-periodic. Notice that the differential equation from (38) is
equivalent to

$d^{\star}(e^{\sigma(s-\cdot)}\varphi_{i}^{\odot})(\tau)=e^{\sigma(s-\tau)}\bigg{(}d^{\star}-\sigma\bigg{)}\varphi_{i}^{\odot}(\tau)=-A^{\star}(\tau)[e^{\sigma(s-\tau)}\varphi_{i}^{\odot}(\tau)]+e^{\sigma(s-\tau)}\varphi_{i+1}^{\odot}(\tau).$
(39)

Hence, it suffices to prove that the abstract differential equation above
together with the initial condition in (38) admits a unique solution on
$(-\infty,s]$. Consider the function $w_{i}^{\odot}:(-\infty,s]\to X^{\odot}$
defined by

$w_{i}^{\odot}(\tau):=U^{\odot}(\tau,s)\sum_{l=i}^{m_{\lambda}-1}\frac{(\tau-t)^{l-i}}{(l-i)!}\varphi_{l}^{\odot}(t),\quad\forall\tau\in(-\infty,s].$
(40)

Since
$\phi_{m_{\lambda}-1}^{\odot}(s),\dots,\phi_{i}^{\odot}(s)\in\mathcal{D}(A^{\star}(s))$
we get from (38) that
$\varphi_{m_{\lambda}-1}^{\odot}(s),\dots,\varphi_{i}^{\odot}(s)\in\mathcal{D}(A^{\star}(s))$.
It is clear that

$U^{\odot}(\tau,s)\varphi_{j}^{\odot}(s)=\sum_{k=j}^{m_{\lambda}-1}\alpha_{ik}\phi_{k}^{\odot}(\tau)\in\mathcal{D}(A^{\star}(\tau)),\quad\forall\tau\in(-\infty,s],\
j\in\\{m_{\lambda}-1,\dots,i\\},$

and so it follows from lemma:adjoint D(Astar) that $\tau\mapsto
w_{i}^{\odot}(\tau)$ takes values in $\mathcal{D}(A^{\star}(\tau))$ and is
$C^{k+1}$-smooth, which implies weak⋆ differentiability of $w_{i}^{\odot}$.
Clearly $w_{i}^{\odot}(s)=\varphi_{i}^{\odot}(s)$ and notice that

$\displaystyle d^{\star}w_{i}^{\odot}(\tau)$
$\displaystyle=-A^{\star}(\tau)U^{\odot}(\tau,s)\sum_{l=i}^{m_{\lambda}-1}\frac{(\tau-s)^{l-i}}{(l-i)!}\varphi_{l}^{\odot}(s)+U^{\odot}(\tau,s)\sum_{l=i+1}^{m_{\lambda}-1}\frac{(\tau-s)^{l-i-1}}{(l-i-1)!}\varphi_{l}^{\odot}(s)$
$\displaystyle=-A^{\star}(\tau)w_{i}(\tau)+w_{i+1}^{\odot}(\tau),$

and so $w_{i}^{\odot}$ is a solution on $(-\infty,s]$ of (39). Since
$w_{i+1}^{\odot}$ is at least continuous, it follows from prop:solutionadjoint
and by construction of $w_{i}^{\odot}$ that (39) admits a unique solution
$w_{i}^{\odot}$ on $(-\infty,s]$ where
$w_{i}=e^{\sigma(s-\cdot)}\varphi_{i}^{\odot}$. As a consequence,
$\varphi_{i}^{\odot}=e^{-\sigma(s-\cdot)}w_{i}^{\odot}$ and because
$\tau\mapsto e^{-\sigma(\tau-s)}$ is analytic, we have that
$\varphi_{i}^{\odot}$ is $C^{k+1}$-smooth.

Let us now turn our attention towards proving $T$-periodicity. If we set
$\tau=s-T$ in (40), we see that
$\varphi_{i}^{\odot}(s-T)=\varphi_{i}^{\odot}(s)$ if and only if

$(U^{\odot}(s-T,s)-\lambda
I)\varphi_{i}^{\odot}(s)=U^{\odot}(s-T,s)\sum_{l=i+1}^{m_{\lambda}-1}\frac{(-1)^{l-i+1}}{(l-i)!}\varphi_{l}^{\odot}(s).$
(41)

Recall from (38) that
$\varphi_{i}^{\odot}(s)=\sum_{k=i}^{m_{\lambda}-1}\alpha_{ik}\phi_{k}^{\odot}(s)$
and retrieving (26) yields

$\displaystyle\sum_{k=i}^{m_{\lambda}-2}\alpha_{ik}\phi_{k+1}^{\odot}(s)$
$\displaystyle=U^{\odot}(s-T,s)\sum_{l=i+1}^{m_{\lambda}-1}\frac{(-1)^{l-i+1}T^{l-i}}{(l-i)!}\varphi_{l}^{\odot}(s)$
$\displaystyle=U^{\odot}(s-T,s)\sum_{l=i+1}^{m_{\lambda}-1}\sum_{k=l}^{m_{\lambda}-1}\alpha_{lk}\frac{(-1)^{l-i+1}T^{l-i}}{(l-i)!}\phi_{k}^{\odot}(s)$
$\displaystyle=\sum_{l=i+1}^{m_{\lambda}-1}\sum_{k=l}^{m_{\lambda}-1}\alpha_{lk}\frac{(-1)^{l-i+1}T^{l-i}}{(l-i)!}\begin{cases}\lambda\phi_{k}^{\odot}(s),\quad&k=m_{\lambda}-1,\\\
\lambda\phi_{k}^{\odot}(s)+\phi_{k+1}^{\odot}(s),\quad&k=m_{\lambda}-2,\dots,i+1.\end{cases}$

Because the right-hand side is a known element in the subspace spanned by
$\phi_{m_{\lambda}-1}^{\odot}(s),\dots,\phi_{i+1}^{\odot}(s)$, the
$\alpha_{ik}$’s are uniquely determined for $k=m_{\lambda}-2,\dots,i$ and so
we have proven that $\varphi_{i}^{\odot}$ is $T$-periodic and so extends to
$\mathbb{R}$.

Furthermore,
$\varphi_{m_{\lambda}-1}^{\odot}(\tau),\dots,\varphi_{0}^{\odot}(\tau)$ are
all linearly independent because they are all solutions to the abstract ODE

$\bigg{(}\frac{d}{d\tau}+A^{\odot}(\tau)-\sigma\bigg{)}^{m_{\lambda}}\varphi^{\odot}(\tau)=0,\quad\forall\tau\in\mathbb{R},$

which completes the proof.

Similar as for the (generalized) eigenfunctions associated to a strictly
negative real Floquet multiplier $\lambda$, we need a $T$-antiperiodic version
of thm:adjoint eigenfunctions for the adjoint (generalized) eigenfunctions.

Let $\lambda$ be a real strictly negative Floquet multiplier with $\sigma$ its
associated Floquet exponent. Then there exist $T$-antiperiodic maps
$\varphi_{m_{\lambda}-1}^{\odot},\dots,\varphi_{0}^{\odot}\in
C_{2T}^{k+1}(\mathbb{R},X^{\odot})$ satisfying (35).

We copy the proof of thm:adjoint eigenfunctions but in the $2T$-periodic
setting. The proof goes identical up to (37). If we set $\tau=s-T$ in (37) we
get

$|\lambda|\varphi_{m_{\lambda}-1}^{\odot}(s-T)=e^{\sigma
T}\varphi_{m_{\lambda}-1}^{\odot}(s-T)=U^{\odot}(s-T,s)\psi_{m_{\lambda}-1}(s)=\lambda\varphi_{m_{\lambda}-1}^{\odot}(s)$

and so
$\varphi_{m_{\lambda}-1}^{\odot}(s-T)=\operatorname{sign}(\lambda)\varphi_{m_{\lambda}-1}^{\odot}(s)=-\varphi_{m_{\lambda}-1}^{\odot}(s).$
This proves the $T$-antiperiodicity of $\varphi_{m_{\lambda}-1}^{\odot}$.

Consider now (38) and suppose that the right-hand side of the differential
equation satisfies $\varphi_{i+1}^{\odot}(s-T)=-\varphi_{i+1}^{\odot}(s)$. The
goal is to find $\alpha_{ik}$ such that $\varphi_{i}^{\odot}$ is
$T$-antiperiodic. Instead of requiring the $T$-periodicity of
$\varphi_{i}^{\odot}$ we require now that
$\varphi_{i}^{\odot}(s-T)=-\varphi_{i}^{\odot}(s)$ and so

$(U^{\odot}(t-T,t)-\lambda
I)\varphi_{i}^{\odot}(t)=U^{\odot}(t-T,t)\sum_{l=i+1}^{m_{\lambda}-1}\frac{(-1)^{l-i+1}}{(l-i)!}\varphi_{l}^{\odot}(t),$

which is precisely (41). The same procedure in thm:adjoint eigenfunctions can
be followed to find the associated $\alpha_{ik}$’s uniquely and obtain
$\varphi_{i}^{\odot}(s-T)=\varphi_{i}^{\odot}(s)$.

##  4 Characterization of the center manifold and normal forms

In this section, we study the dynamics of (DDE) on the center manifold
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ near the _nonhyperbolic_ cycle
$\Gamma$, meaning that there are, except of the trivial Floquet multiplier,
other Floquet multipliers present on the unit circle in the complex plane, or
the trivial Floquet multiplier has an algebraic multiplicity higher than one.
Recall from sec:introduction that there are three generic codimension one
bifurcation of limit cycles: the fold bifurcation, where the trivial Floquet
multiplier has an algebraic multiplicity $2$ and geometric multiplicity 1, the
period-doubling bifurcation, where there is a simple Floquet multiplier at
$-1$ and the Neimark-Sacker bifurcation where there is a simple complex
conjugate pair of Floquet multipliers present on the unit circle.

To study these bifurcations, we first separate the trivial Floquet multiplier
from the rest of the dynamics in subsec: separating. Afterwards, in subsec:
normal form, we prove the existence of a special coordinate system on the
center manifold and provide in addition the periodic (critical) normal forms.
These results are an extension of the work by Iooss [22, 23] from finite-
dimensional ODEs to infinite-dimensional DDEs. As a consequence, the periodic
normal forms obtained in thm:normalformI, thm:normalformII and
thm:normalformIII for DDEs are exactly the same as obtained by Iooss in ODEs,
see [23, Theorem III.7, Theorem III.10 and Theorem III.13]. Instead of being
interested in only codim 1 bifurcations of limit cycles, the provided
framework is also suited to study bifurcations of limit cycles of higher
codimension, see for example [33, Table 1] for the periodic normal forms for
some codim 2 bifurcations of limit cycles in ODEs, and hence DDEs. It is
nevertheless helpful to keep these three codim one bifurcations in mind.

Before we start proving the characterization and periodic normal forms, let us
first review some results from [28] on the topological direct sum
decompositions of $X$ and $X^{\odot\star}$. It turns out from [28, Hypothesis
1] that we can decompose $X$ as

$X=X_{-}(\tau)\oplus X_{0}(\tau)\oplus
X_{+}(\tau),\quad\forall\tau\in\mathbb{R},$ (42)

where $X_{-}(\tau)$ and $X_{+}(\tau)$ denote the _stable eigenspace_ and
_unstable eigenspace_ (at time $\tau$) respectively, see [28, Section 3.6] for
their definitions. Furthermore, it turns out from [28, Appendix A.1] that we
can lift the decomposition (42) towards a decomposition in $X^{\odot\star}$ as

$X^{\odot\star}=X^{\odot\star}_{-}(\tau)\oplus X^{\odot\star}_{0}(\tau)\oplus
X^{\odot\star}_{+}(\tau),\quad\forall\tau\in\mathbb{R},$ (43)

where $X^{\odot\star}_{0}(\tau)=j(X_{0}(\tau))$ and
$X^{\odot\star}_{+}(\tau)=j(X_{+}(\tau))$, see [28, Appendix A.2] for more
information.

###  4.1 Separating the dynamics of the periodic orbit

The coordinate system and normal forms we will present consist of two parts
and is inspired by [22, 23]. The first part expresses the dynamics along
$\Gamma$ by a time-dependent phase and the other part expresses the dynamics
transverse to $\Gamma$ in terms of this phase. The normal forms depend on the
location and multiplicities of the Floquet multipliers on the unit circle. Let
us first separate the dynamics of the periodic orbit via coordinates along
this phase and transverse to this phase.

Recall that $X_{0}(\tau)$ is a $(n_{0}+1)$-dimensional subspace of $X$ for all
$\tau\in\mathbb{R}$. For each $\lambda\in\Lambda_{0}$, we know that the
(generalized) eigenspace $E_{\lambda}(\tau)$ has a ($T$ or 2$T$)-periodic
basis that satisfies the conditions from thm:eigenfunctions or
prop:eigenfunctions2, depending on the location of $\lambda$ on the unit
circle. Recall that the trivial Floquet multiplier is always present on the
unit circle and $\dot{\gamma}_{\tau}$ is the associated $T$-periodic
eigenfunction of $U(\tau+T,\tau)$. We choose $\varphi_{0}(\tau)$ to be
$\dot{\gamma}_{\tau}$ and denote by $\tilde{X}_{0}(\tau)$ the space spanned by
$\varphi_{1}(\tau),\dots,\varphi_{n_{0}}(\tau)$ that forms a ($T$ or
$2T$)-periodic $C^{k+1}$-smooth basis as presented in thm:eigenfunctions or
prop:eigenfunctions2. Define for any $\tau\in\mathbb{R}$ the bounded linear
operator $\tilde{Q}_{0}(\tau):\mathbb{R}^{n_{0}}\to\tilde{X}_{0}(\tau)$ as

$\tilde{Q}_{0}(\tau)\xi:=\sum_{i=1}^{n_{0}}\xi_{i}\varphi_{i}(\tau),\quad\forall\xi=(\xi_{1},\dots,\xi_{n_{0}})\in\mathbb{R}^{n_{0}}.$
(44)

With this notation, it is clear that the Floquet operator (at time $\tau$)
associated to $\Lambda_{0}$ is defined as
$Q_{0}(\tau):\mathbb{R}\times\mathbb{R}^{n_{0}}\to X_{0}(\tau)$ has the action

$Q_{0}(\tau)(\xi_{0},\xi):=\xi_{0}\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)\xi,\quad\forall(\xi_{0},\xi)\in\mathbb{R}\times\mathbb{R}^{n_{0}}.$

The $({n_{0}}+1)\times({n_{0}}+1)$ matrix $M_{0}$ from takes the form

$M_{0}=\left(\begin{array}[]{c|c}0&\star\ \cdots\ 0\\\
\hline\cr\vspace{-4pt}0\\\ \vdots&\tilde{M}_{0}\\\ 0\end{array}\right),$ (45)

where $\star\in\\{0,1\\}$ depends on the algebraic multiplicity of the trivial
Floquet multiplier.

###  4.2 Characterization and normal form theorems

Depending on the algebraic multiplicity of the trivial Floquet multiplier and
the location of the other Floquet multipliers on the unit circle, the periodic
normal forms will have a different shape and therefore three different normal
form theorems will be presented.

The main idea to prove the existence of suitable coordinates on
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is to use the invariance
property of $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ around the periodic
orbit $\Gamma$ to the fullest. We try to parametrize the history
$x_{t}\in\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ in the vicinity of the
periodic orbit $\Gamma$ as

$x_{t}=\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi),$ (46)

where $\tau$ is a function of $t$, expresses the dynamics along $\Gamma$ by a
time-dependent phase and $\xi$ is a function of $\tau$ that expresses the
dynamics transverse to $\Gamma$ in terms of this phase. Such a coordinate
system is visualized for a two-dimensional local center manifold around
$\Gamma$ in fig:CM.

Figure 1: Illustration of two-dimensional center manifolds
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ together with the coordinate
system $(\tau,\xi)$. The left figure represents the case when
$-1\not\in\Lambda_{0}$ and then $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$
is locally diffeomorphic to a cylinder in a neighborhood of $\Gamma$, see
thm:normalformII. The right figure represents the case when $-1\in\Lambda_{0}$
and then $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is locally
diffeomorphic to a Möbius band in a neighborhood of $\Gamma$, see
thm:normalformIII.

The only unknown in (46) is the nonlinear operator
$H:\mathbb{R}\times\mathbb{R}^{n_{0}}\to X$ and to obtain a Taylor expansion
of this operator, we use again the invariance property of
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$. To be more precise, if we take
an initial condition
$x_{s}=\varphi\in\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$, then we must
have that $x_{t}\in\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ for all $t$
in the time domain of definition, say $I_{\varphi}\subseteq\mathbb{R}$ with
$s\in I_{\varphi}$. By [7, Theorem 3.6] we know that the history
$x_{t}\in\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ satisfies the abstract
ODE

$\frac{d}{dt}j(x_{t})=A_{0}^{\odot\star}j(x_{t})+G(x_{t}),\quad t\in
I_{\varphi},$ (47)

where $G(\psi)=F(\psi)r^{\odot\star}$ for all $\psi\in X$ and $F\in
C^{k+1}(X,\mathbb{R}^{n})$ for some $k\geq 1$ is the right-hand side of (DDE).
The idea is to show the existence of each $q$th order term in the Taylor
expansion of $H$ for $q=2,\dots,k$ by using (47) and the invariance property
of $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$.

First we consider the case where the trivial Floquet multiplier has algebraic
multiplicity $1$ and there is no Floquet multiplier located at $-1$. This is
for example the case in the Neimark-Sacker bifurcation. [Normal Form I] Assume
that the algebraic multiplicity of the trivial Floquet multiplier is one and
that $-1$ is not a Floquet multiplier. Then there exist $C^{k}$-smooth maps
$H:\mathbb{R}\times\mathbb{R}^{n_{0}}\to
X,p:\mathbb{R}\times\mathbb{R}^{n_{0}}\to\mathbb{R}$ and
$P:\mathbb{R}\times\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{0}}$ such that the
history $x_{t}\in\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ may be
represented as

$x_{t}=\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi),\quad t\in I,$

where the time dependence of the coordinates $(\tau,\xi)$ describing the
dynamics of (DDE) on $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is defined
by the normal form

$\begin{cases}\dfrac{d\tau}{dt}=1+p(\tau,\xi)+\mathcal{O}(|\xi|^{k+1}),\\\
\\\\[-8.0pt]
\dfrac{d\xi}{d\tau}=\tilde{M}_{0}\xi+P(\tau,\xi)+\mathcal{O}(|\xi|^{k+1}).\end{cases}$

Here the functions $H,p$, and $P$ are $T$-periodic in $\tau$ and at least
quadratic in $\xi$. The $\mathcal{O}$-terms are also $T$-periodic in $\tau$.
Moreover, $p$ and $P$ are polynomials in $\xi$ of degree less than or equal to
$k$ such that

$\frac{d}{d\tau}p(\tau,e^{-\tau\tilde{M}_{0}^{\star}}\xi)=0\ \mbox{ and }\
\frac{d}{d\tau}\bigg{(}e^{\tau\tilde{M}_{0}^{\star}}P(\tau,e^{-\tau\tilde{M}_{0}^{\star}}\xi)\bigg{)}=0,$

for all $\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$. The proof of this
theorem is quite long and technical. Essentially, it is a careful
generalization of [23, Theorem III.7]. Therefore, we first sketch the idea of
the proof and break it up into several steps. The final goal is to
characterize the map $H$ by its Taylor expansion. In Step 1 of the proof, we
assume this Taylor expansion and start in Step 2 with collecting terms in
powers of $\xi^{q}$ for $q=0,\dots,k$ from both sides of the resulted
equation, obtained from the invariance property of the center manifold. We get
for $q=2,\dots,k$ an equation for the coefficient $H_{q}$ and we must show
that this can be uniquely solved. This will be done via decomposing $H_{q}$
into the decomposition provided in (42) together with the separation made in
subsec: separating, see Step 3. Hence, we get for each $q=2,\dots,k$ the terms
$H_{q}^{00},\tilde{H}_{q}^{0},H_{q}^{-}$ and $H_{q}^{+}$ and then we prove the
existence of each of these terms in Step 4 $(H_{q}^{+})$, Step 5 $(H_{q}^{-})$
and Step 6 ($H_{q}^{00}$ and $\tilde{H}_{q}^{0}$). The provided normal forms
are partially derived in step 6 of the proof in combination with [23, Theorem
III.7].

[Proof of thm:normalformI] We follow the outlined route of the proof as
indicated above.

Step 1: Taylor expansion. Let us write (DDE) in the form of (47) and notice
that

$G(\gamma_{\tau}+\varphi)=G(\gamma_{\tau})+B(\tau)\varphi+\sum_{q=2}^{k}G_{q}(\tau,\varphi^{(q)})+\mathcal{O}(\|\varphi\|_{\infty}^{k+1}),\quad\forall\varphi\in
X,$ (48)

where $B(\tau)\varphi=[DF(\gamma_{\tau})\varphi]r^{\odot\star}$ is the time-
dependent bounded linear perturbation and the nonlinear terms are given by
$G_{q}(\tau,\varphi^{(q)}):=\frac{1}{q!}D^{q}F(\gamma_{\tau})(\varphi^{(q)})r^{\odot\star},$
where $D^{q}F(\gamma_{\tau}):X^{q}\to\mathbb{R}^{n}$ is the $q$th order
Fréchet derivative evaluated at $\gamma_{\tau}$ for $q=2,\dots,k$ and
$\varphi^{(q)}:=(\varphi,\dots,\varphi)\in X^{q}:=X\times\dots\times X$. We
also expand the maps $H,p$ and $P$ as

$\displaystyle H(\tau,\xi)$
$\displaystyle=\sum_{q=2}^{k}H_{q}(\tau,\xi^{(q)})+\mathcal{O}(|\xi|^{k+1}),\quad
p(\tau,\xi)=\sum_{q=2}^{k}p_{q}(\tau,\xi^{(q)}),\quad
P(\tau,\xi)=\sum_{q=2}^{k}P_{q}(\tau,\xi^{(q)}),$

with the coefficients $H_{q}(\tau,\xi^{(q)})\in
X,p_{q}(\tau,\xi^{(q)})\in\mathbb{R}$ and
$P_{q}(\tau,\xi^{(q)})\in\mathbb{R}^{n_{0}}$, where
$\xi^{(q)}:=(\xi,\dots,\xi)\in[\mathbb{R}^{n_{0}}]^{q}$. As already announced,
we will use the invariance property of
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ to show existence of the
coefficients $H_{q}(\tau,\xi^{(q)})$ for all $q=2,\dots,k$. Hence, we compare
the expansions of

$\frac{d}{dt}j(\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi))=j\bigg{(}\dot{\gamma}_{\tau}+\frac{\partial\tilde{Q}_{0}(\tau)}{\partial\tau}\xi+\frac{\partial
H(\tau,\xi)}{\partial\tau}+\bigg{(}\tilde{Q}_{0}(\tau)+D_{\xi}H(\tau,\xi)\bigg{)}\frac{d\xi}{d\tau}\bigg{)}\frac{d\tau}{dt}$

and

$A_{0}^{\odot\star}j(x_{t})+G(x_{t})=A_{0}^{\odot\star}j(\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi))+G(\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi))$

by substituting

$\frac{d\tau}{dt}=1+p(\tau,\xi)+\mathcal{O}(|\xi|^{k+1})\ \mbox{ and }\
\frac{d\xi}{d\tau}=\tilde{M}_{0}\xi+P(\tau,\xi)+\mathcal{O}(|\xi|^{k+1}).$

Using the expansions of $H,p$ and $P$ together with (48), where now $\varphi$
must be substituted by $\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi)$, we get

$\displaystyle
j\bigg{[}\dot{\gamma}_{\tau}+\frac{\partial\tilde{Q}_{0}(\tau)}{\partial\tau}\xi+\sum_{q=2}^{k}\frac{\partial
H_{q}(\tau,\xi^{(q)})}{\partial\tau}+\bigg{(}\tilde{Q}_{0}(\tau)+\sum_{q=2}^{k}D_{\xi}H_{q}(\tau,\xi^{(q)})\bigg{)}\bigg{(}\tilde{M}_{0}+\sum_{q=2}^{k}P_{q}(\tau,\xi^{(q)})\bigg{)}\bigg{]}$
$\displaystyle\bigg{(}1+\sum_{q=2}^{k}p_{q}(\tau,\xi^{(q)})\bigg{)}+\mathcal{O}(|\xi|^{k+1})$
$\displaystyle=A_{0}^{\odot\star}j(\gamma_{\tau})+G(\gamma_{\tau})+A^{\odot\star}(\tau)j\bigg{(}\tilde{Q}_{0}(\tau)\xi+\sum_{q=2}^{k}H_{q}(\tau,\xi^{(q)})\bigg{)}$
$\displaystyle+\sum_{q=2}^{k}G_{q}\bigg{(}\tau,\bigg{[}\tilde{Q}_{0}(\tau)\xi+\sum_{p=2}^{k}H_{q}(\tau,\xi^{(p)})\bigg{]}^{(q)}\bigg{)}+\mathcal{O}(|\xi|^{k+1}).$

Step 2: Collecting terms. Let us now compare the terms in powers of $\xi$ on
both side of this equation. Collecting the $\xi^{0}$-terms give us

$\frac{d}{d\tau}j({\gamma}_{\tau})=A_{0}^{\odot\star}j(\gamma_{\tau})+G(\gamma_{\tau}),$

which means that $\gamma$ is a solution (47). This was already known since
$\gamma$ is a periodic solution of (DDE). The $\xi^{1}$-terms give us

$\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(\tilde{Q}_{0}(\tau)\xi)=j(\tilde{Q}_{0}(\tau)\tilde{M}_{0}\xi),$
(49)

which is exactly the result established in (24), but now for all Floquet
multipliers on the unit circle and this characterizes the linear part. After
collecting the $\xi^{(2)}$-terms, we get the expression

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{2}(\tau,\xi^{(2)}))$
$\displaystyle=j(D_{\xi}H_{2}(\tau,\xi^{(2)})\tilde{M}_{0}\xi+p_{2}(\tau,\xi^{(2)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)P_{2}(\tau,\xi^{(2)}))-R_{2}(\tau,\xi^{(2)}),$

where $R_{2}(\tau,\xi^{(2)})=G_{2}(\tau,(\tilde{Q}_{0}(\tau)\xi)^{(2)})$.
Finally, after collecting the $\xi^{(q)}$-terms for $q=3,\dots,k$ one obtains

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}(\tau,\xi^{(q)}))$
$\displaystyle=j(D_{\xi}H_{q}(\tau,\xi^{(q)})\tilde{M}_{0}\xi+p_{q}(\tau,\xi^{(q)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)P_{q}(\tau,\xi^{(q)}))-R_{q}(\tau,\xi^{(q)}),$
(50)

where $R_{q}(\tau,\xi^{(q)})$ depends on
$G_{q^{\prime}}(\tau,\xi^{(q^{\prime})})$ for $2\leq q^{\prime}\leq q$ and
$j(H_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$,
$j(p_{q^{\prime}}(\tau,\xi^{(q^{\prime})})\dot{\gamma}_{\tau})$ and
$j(\tilde{Q}_{0}(\tau)P_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$ for
$q^{\prime}=2,\dots,q-1$.

Step 3: Projecting on subspaces. We want to project (4.2) onto the spaces
$\mathbb{R}j\dot{\gamma}_{\tau},j(\tilde{X}_{0}(\tau))$ and
${X}_{\pm}^{\odot\star}(\tau)$ to show the existence of $H_{q}$ separately on
each individual space. Because $X$ can be decomposed as in
$\eqref{eq:decomposition X hyp}$ where
$X_{0}(\tau)=\mathbb{R}\dot{\gamma}_{\tau}\oplus\tilde{X}_{0}(\tau)$ for any
$\tau\in\mathbb{R},$ we can decompose for any $q=2,\dots,k$ the function
$H_{q}$ as

$\displaystyle
H_{q}(\tau,\xi^{(q)})=H_{q}^{00}(\tau,\xi^{(q)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)\tilde{H}_{q}^{0}(\tau,\xi^{(q)})+H_{q}^{-}(\tau,\xi^{(q)})+H_{q}^{+}(\tau,\xi^{(q)}),$

where $H_{q}^{\pm}(\tau,\xi^{(q)})=P_{\pm}(\tau)H_{q}(\tau,\xi^{(q)})\in
X_{\pm}(\tau)$ together with $H_{q}^{00}(\tau,\xi^{(q)})\in\mathbb{R}$ and
$\tilde{H}_{q}^{0}(\tau,\xi^{(q)})\in\mathbb{R}^{n_{0}}$ for all
$\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$. It follows from (43) that
for any $\tau\in\mathbb{R}$ there holds

$R_{q}(\tau,\xi^{(q)})=R_{q}^{00}(\tau,\xi^{(q)})j\dot{\gamma}_{\tau}+j(\tilde{Q}_{0}(\tau)\tilde{R}_{q}^{0}(\tau,\xi^{(q)}))+R_{q}^{-}(\tau,\xi^{(q)})+R_{q}^{+}(\tau,\xi^{(q)}),$

where
$R_{q}^{\pm}(\tau,\xi^{(q)})=P_{\pm}^{\odot\star}(\tau)R_{q}(\tau,\xi^{(q)})\in
X_{\pm}^{\odot\star}(\tau)$ together with
$R_{q}^{00}(\tau,\xi^{(q)})\in\mathbb{R}$ and
$\tilde{R}_{q}^{0}(\tau,\xi^{(q)})\in\mathbb{R}^{n_{0}}$ for all
$\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$. The definition of the
spectral projector $P_{\pm}^{\odot\star}$ can be found in [28, Appendix A].
Substituting these decompositions into (4.2) yields for the left-hand side of
this equation

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}(\tau,\xi^{(q)}))$
$\displaystyle=-j\bigg{(}\frac{\partial
H_{q}^{00}(\tau,\xi^{(q)})}{\partial\tau}\dot{\gamma}_{\tau}+H_{q}^{00}(\tau,\xi^{(q)})\ddot{\gamma}_{\tau}\bigg{)}$
$\displaystyle+A^{\odot\star}(\tau)j(H_{q}^{00}(\tau,\xi^{(q)})\dot{\gamma}_{\tau})$
$\displaystyle-j\bigg{(}\frac{\partial\tilde{Q}_{0}(\tau)}{\partial\tau}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})+\tilde{Q}_{0}(\tau)\frac{\partial\tilde{H}_{q}^{0}(\tau,\xi^{(q)})}{\partial\tau}\bigg{)}$
$\displaystyle+A^{\odot\star}(\tau)j(\tilde{Q}_{0}(\tau)\tilde{H}_{q}^{0}(\tau,\xi^{(q)}))$
$\displaystyle+\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}^{-}(\tau,\xi^{(q)})+H_{q}^{+}(\tau,\xi^{(q)})),$

where we twice used the product rule for differentiation. Since
$\tau\mapsto\dot{\gamma}_{\tau}$ is a $T$-periodic eigenfunction we get from
thm:eigenfunctions that
$(-\frac{d}{d\tau}+A^{\odot\star}(\tau))j\dot{\gamma}_{\tau}=0$. Using this in
combination with (49), we arrive at

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}(\tau,\xi^{(q)}))$
$\displaystyle=j\bigg{(}-\frac{\partial
H_{q}^{00}(\tau,\xi^{(q)})}{\partial\tau}\dot{\gamma}_{\tau}\bigg{)}$ (51)
$\displaystyle+j\bigg{(}\tilde{Q}_{0}(\tau)\bigg{(}-\frac{\partial}{\partial\tau}+\tilde{M}_{0}\bigg{)}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})\bigg{)}$
$\displaystyle+\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}^{-}(\tau,\xi^{(q)})+H_{q}^{+}(\tau,\xi^{(q)}))$

and this must be equal to the right-hand side of (4.2). Let us first show
existence of $H_{q}^{\pm}$ via projecting on the spaces
$X_{\pm}^{\odot\star}(\tau)$. On these subspaces, we get the equation

$\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}^{\pm}(\tau,\xi^{(q)}))=j(D_{\xi}H_{q}^{\pm}(\tau,\xi^{(q)})\tilde{M}_{0}\xi)-R_{q}^{\pm}(\tau,\xi^{(q)}).$

Substituting $\tau=\theta$ and
$\xi=e^{(\theta-\tau)\tilde{M}_{0}}\xi=:\tilde{\xi}$ leads to

$-\frac{\partial}{\partial\theta}j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))+A^{\odot\star}(\theta)j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))-j(D_{\tilde{\xi}}H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)})\tilde{M}_{0}\xi)=-R_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}).$

When the operator $-U^{\odot\star}(\tau,\theta)$ acts on both side of the
equation, we obtain

$\displaystyle-U^{\odot\star}(\tau,\theta)\bigg{[}-\frac{\partial}{\partial\theta}j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))+A^{\odot\star}(\theta)j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))-j(D_{\tilde{\xi}}H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)})\tilde{M}_{0}\xi)\bigg{]}$
$\displaystyle=U^{\odot\star}(\tau,\theta)R_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}).$
(52)

Let us focus on the left-hand-side of this equation. It follows from [5,
Theorem 5.5] that

$-U^{\odot\star}(\tau,\theta)A^{\odot\star}(\theta)j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))=-[{\partial_{\theta}^{\star}}U^{\odot\star}(\tau,\theta)]j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)})).$

Filling this result back into (4.2) and using the partial weak⋆ derivative
operator yields

$U^{\odot\star}(\tau,\theta)[{\partial_{\theta}^{\star}}j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))]+[{\partial_{\theta}^{\star}}U^{\odot\star}(\tau,\theta)]j(H_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}))=U^{\odot\star}(\tau,\theta)R_{q}^{\pm}(\theta,\tilde{\xi}^{(q)}),$

where we have used the product rule for differentiation, but essentially in
the dual pairings due to the partial weak⋆ derivative. Using the product rule
again and recalling that $\tilde{\xi}=e^{(\theta-\tau)\tilde{M}_{0}}\xi$, we
get the identity

$\partial_{\theta}^{\star}[U^{\odot\star}(\tau,\theta)j(H_{q}^{\pm}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}})^{(q)}))]=U^{\odot\star}(\tau,\theta)R_{q}^{\pm}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}})^{(q)}).$

Using the definition of the weak⋆ derivative, we get for all $x^{\odot}\in
X^{\odot}$ that

$\frac{\partial}{\partial\theta}\langle
j(U(\tau,\theta)H_{q}^{\pm}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})),x^{\odot}\rangle=\langle
U^{\odot\star}(\tau,\theta)R_{q}^{\pm}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)}),x^{\odot}\rangle.\\\
$ (53)

Step 4: Existence of $H_{q}^{+}$. Let us first find an expression for
$H_{q}^{+}(\tau,\xi^{(q)})$. As $X_{+}(s)$ is finite-dimensional, $U(\tau,s)$
extends to all $\tau,s\in\mathbb{R}$ on the subspace $X_{+}(s)$. So let
$s\geq\tau$ be given and integrate (53) over the interval $[\tau,s]$ to obtain

$\displaystyle\langle j(H_{q}^{+}(\tau,\xi^{(q)})),x^{\odot}\rangle$
$\displaystyle=\langle
j(U(\tau,s)H_{q}^{+}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})),x^{\odot}\rangle$
$\displaystyle-\int_{\tau}^{s}\langle
U^{\odot\star}(\tau,\theta)R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)}),x^{\odot}\rangle
d\theta.$ (54)

Let us focus on the first term of the right-hand side. Notice that

$H_{q}^{+}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})=\sum_{|\alpha|=q}\frac{1}{\alpha!}P_{+}(s)H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})$

where $H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})\in X$. Then, we
get

$U(\tau,s)H_{q}^{+}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})=\sum_{|\alpha|=q}\frac{1}{\alpha!}U(\tau,s)P_{+}(s)H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})$

and using the exponential trichotomy property of the forward evolutionary
system [28, Hypothesis 1], there is a $b>0$ such that for a given
$\varepsilon>0$ there exists a $K_{\varepsilon}>0$ with the property

$\|U(\tau,s)H_{q}^{+}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})\|_{\infty}\leq
K_{\varepsilon}e^{b(\tau-s)}\sum_{|\alpha|=q}\frac{1}{\alpha!}\|H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})\|_{\infty},$

where the number $N$ from [28, Hypothesis 1] is absorbed in the
$K_{\varepsilon}$ constant. Since the diagonal elements of the matrix
$\tilde{M}_{0}$ have real part zero, $e^{(s-\tau)\tilde{M}_{0}}\xi$ is a
polynomial in $\xi$ and so
$\|H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})\|_{\infty}$ can
grow at most polynomially for $s\to\pm\infty$. With this in mind, we get

$\displaystyle|\langle
j(U(\tau,s)H_{q}^{+}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})),x^{\odot}\rangle|$
$\displaystyle\leq
K_{\varepsilon}e^{b(\tau-s)}\|x^{\odot}\|\sum_{|\alpha|=q}\frac{1}{\alpha!}\|H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})\|_{\infty}$
$\displaystyle\leq
M_{\varepsilon}e^{b(\tau-s)}\max_{|\alpha|=q}\|H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})\|_{\infty}$
$\displaystyle\to 0,\quad\mbox{ as }s\to\infty.$

Using this convergence, taking the limit in (4.2) yields

$\displaystyle\langle j(H_{q}^{+}(\tau,\xi^{(q)})),x^{\odot}\rangle$
$\displaystyle=\langle\int_{\tau}^{\infty}-U^{\odot\star}(\tau,\theta)R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})d\theta,x^{\odot}\rangle,$
(55)

if we can show that for any $x^{\odot}\in X^{\odot}$ and fixed
$\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$ the map
$g_{q,\tau,\xi}^{+}:[\tau,\infty)\to\mathbb{R}$ defined by
$g_{q,\tau,\xi}^{+}(\theta)=\langle-U^{\odot\star}(\tau,\theta)R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)}),x^{\odot}\rangle$
is in $L^{1}([\tau,\infty),\mathbb{R})$. Let
$\tau\in\mathbb{R},\xi\in\mathbb{R}^{n_{0}}$ and $x^{\odot}\in X^{\odot}$ be
given. From [28, Hypothesis 1 and Proposition 18] we get

$\displaystyle\int_{\tau}^{\infty}|g_{q,\tau,\xi}^{+}(\theta)|d\theta$
$\displaystyle\leq
K_{\varepsilon}N\|x^{\odot}\|e^{b\tau}\int_{\tau}^{\infty}e^{-b\theta}\|R_{q}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})\|d\theta.$

Recall that $e^{(\theta-\tau)\tilde{M}_{0}}\xi$ is a polynomial in $\xi$ and
that $R_{q}(\tau,\xi^{(q)})$ depends on
$G_{q^{\prime}}(\tau,\xi^{(q^{\prime})})$ for $2\leq q^{\prime}\leq q$ and
$j(H_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$,
$j(p_{q^{\prime}}(\tau,\xi^{(q^{\prime})})\dot{\gamma}_{\tau})$ and
$j(\tilde{Q}_{0}(\tau)P_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$ for
$q^{\prime}=2,\dots,q-1$. Since $G_{q^{\prime}}$ is periodic in the first
variable and evaluated at a polynomial, $G_{q^{\prime}}$ grows at most
polynomially for $2\leq q^{\prime}\leq q$. As we can assume that
$H_{q^{\prime}}$ is $T$-periodic in the first variable for
$q^{\prime}=2,\dots,q-1$ (we will show this later for $q^{\prime}=q$) and
evaluated at a polynomial it follows that
$j(H_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$ grows at most polynomially for
$q^{\prime}=2,\dots,q-1$. As $p_{q^{\prime}}$ and $P_{q^{\prime}}$ are
$T$-periodic in the first variable for $q^{\prime}=2,\dots,q-1$ (we will show
this later for $q^{\prime}=q$), it follows that
$j(p_{q^{\prime}}(\tau,\xi^{(q^{\prime})})\dot{\gamma}_{\tau})$ and
$j(\tilde{Q}_{0}(\tau)P_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$ grow at most
polynomially. To conclude, there exists a polynomial
$r_{q,\tau,\xi}^{+}:\mathbb{R}\to\mathbb{R}$ such that
$\|R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})\|\leq
r_{q,\tau,\xi}^{+}(\theta)$ for all $\theta\geq\tau$. Hence,

$\int_{\tau}^{\infty}|g_{q,\tau,\xi}^{+}(\theta)|d\theta\leq
K_{\varepsilon}N\|x^{\odot}\|e^{b\tau}\int_{\tau}^{\infty}e^{-b\theta}r_{q,\tau,\xi}^{+}(\theta)d\theta<\infty,$
(56)

because the map $[\tau,\infty)\ni\theta\mapsto
e^{-b\theta}g_{q,\tau,\xi}^{+}(\theta)\in\mathbb{R}$ decays fast enough to
zero as $\theta\to\infty$. We have proven that the weak⋆ integral in (55)
exists. Because
$R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})\in
j(X_{+}(\theta))$ and (55) holds for any $x^{\odot}\in X^{\odot}$, we obtain

$\displaystyle j(H_{q}^{+}(\tau,\xi^{(q)}))$
$\displaystyle=j\int_{\tau}^{\infty}-U(\tau,\theta)j^{-1}R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})d\theta.$

By $\odot$-reflexivity we have that $j$ is an isomorphism on
$j(X)=X^{\odot\odot}$ and hence

$H_{q}^{+}(\tau,\xi^{(q)})=-\int_{\tau}^{\infty}U(\tau,\theta)j^{-1}R_{q}^{+}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})d\theta$
(57)

can be evaluated as a standard Riemann integral. It can easily be checked that
$H_{q}^{+}$ is $T$-periodic in the first variable because $P_{+}^{\odot\star}$
is $T$-periodic and $R_{q}$ is $T$-periodic in the first variable. Let us now
prove the continuity of the map $H_{q}^{+}$. As $U^{\odot\star}(t,\tau)$
restricted to $j(X_{+}(\tau))$ is invertible, we can adjust the proof from
[28, Lemma 1] to prove continuity of the limiting function
$v(\tau,\infty,\tau,g)$ (notation from [28, Lemma 1]) for a continuous
function $g:[\tau,\infty)\to X^{\odot\star}$ under the assumption that
$H_{q}^{+}$ is bounded in norm. As it is proved in (56) that $H_{q}^{+}$ is
bounded in norm and noticing that $P_{+}^{\odot\star}$ and $R_{q}$ are
continuous for all $q\in\\{1,\dots,k\\}$, the result follows.

Step 5: Existence of $H_{q}^{-}$. Now, we can look for an explicit expression
of $H_{q}^{-}(\tau,\xi^{(q)})$. Integrating (53) over $[s,\tau]$ for a fixed
$s\in\mathbb{R}$, yields for any $x^{\odot}\in X^{\odot}$, due to the
definition of the weak⋆ integral

$\displaystyle\langle j(H_{q}^{-}(\tau,\xi^{(q)})),x^{\odot}\rangle$
$\displaystyle=\langle
j(U(\tau,s)H_{q}^{-}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})),x^{\odot}\rangle$
$\displaystyle+\int_{\tau}^{s}\langle
U^{\odot\star}(\tau,\theta)R_{q}^{-}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)}),x^{\odot}\rangle
d\theta.\ $ (58)

Similar to the $H_{q}^{+}$-case, we want to show that the first term goes to
zero, but now as $s\to-\infty$. Recall that
$\|H_{s}^{\alpha}((e^{(s-\tau)\tilde{M}_{0}}\xi)^{\alpha})\|_{\infty}$ can
grow at most polynomially for $s\to\pm\infty$ and so due to the exponential
trichotomy of the forward evolutionary system [28, Hypothesis 1], there exists
an $a<0$ such that for a given $\varepsilon>0$ there is a $M_{\varepsilon}>0$
with the property

$\displaystyle|\langle
j(U(\tau,s)H_{q}^{-}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})),x^{\odot}\rangle|$
$\displaystyle\leq
M_{\varepsilon}e^{a(\tau-s)}\max_{|\alpha|=q}\|H_{s}^{\alpha}(s,(e^{(s-\tau)\tilde{M}_{0}}\xi)^{(q)})\|_{\infty}$
$\displaystyle\to 0,\quad\mbox{ as $s\to-\infty$},$

where the other constants are already absorbed in $M_{\varepsilon}$. We
conclude that

$\langle
j(H_{q}^{-}(\tau,\xi^{(q)})),x^{\odot}\rangle=\langle\int_{-\infty}^{\tau}U^{\odot\star}(\tau,\theta)R_{q}^{-}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})d\theta,x^{\odot}\rangle,$
(59)

if we are able to show that for any $x^{\odot}\in X^{\odot}$ and fixed
$\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$ that the map
$g_{q,\tau,\xi}^{-}:(-\infty,\tau]\to\mathbb{R}$ defined by
$g_{q,\tau,\xi}^{-}(\theta)=\langle
U^{\odot\star}(\tau,\theta)R_{q}^{-}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)}),x^{\odot}\rangle$
is in $L^{1}((-\infty,\tau],\mathbb{R})$. The exponential trichotomy implies
that for a given $\varepsilon>0$ one can find a $K_{\varepsilon}>0$ such that

$\displaystyle\int_{\tau}^{\infty}|g_{q,\tau,\xi}^{-}(\theta)|d\theta$
$\displaystyle\leq
K_{\varepsilon}N\|x^{\odot}\|e^{a\tau}\int_{-\infty}^{\tau}e^{-a\theta}\|R_{q}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})\|d\theta.$

From the same reasoning as in the $H_{q}^{+}$-case, there exists a polynomial
$r_{q,\tau,\xi}^{-}:\mathbb{R}\to\mathbb{R}$ that satisfies the estimate
$\|R_{q}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})\|\leq
r_{q,\tau,\xi}^{-}(\theta)$ for all $\theta\leq\tau$. Hence,

$\int_{\tau}^{\infty}|g_{q,\tau,\xi}^{-}(\theta)|d\theta\leq
K_{\varepsilon}N\|x^{\odot}\|e^{a\tau}\int_{-\infty}^{\tau}e^{-a\theta}r_{q,\tau,\xi}^{-}(\theta)d\theta<\infty,$
(60)

because the map $\theta\mapsto e^{-a\theta}r_{q,\tau,\xi}^{-}(\theta)$ decays
fast enough to zero as $\theta\to-\infty$. Hence, $g_{q,\tau,\xi}^{-}\in
L^{1}((-\infty,\tau],\mathbb{R})$ and so the weak⋆ integral exists. Since (59)
holds for all $x^{\odot}\in X^{\odot}$ we get

$H_{q}^{-}(\tau,\xi^{(q)})=j^{-1}\int_{-\infty}^{\tau}U^{\odot\star}(\tau,\theta)R_{q}^{-}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})d\theta,$
(61)

if we can prove that the weak⋆ integral takes values in $j(X)$. Notice that we
proved in (60) that $H_{q}^{-}$ is bounded in norm. With the notation from
[28, Lemma 1] we have that
$j(H_{q}^{-}(\tau,\xi^{(q)}))=v(\tau,\tau,-\infty,g)$ with the continuous map
$g$ defined by
$g(\theta)=P_{-}^{\odot\star}(\theta)R_{q}(\theta,(e^{(\theta-\tau)\tilde{M}_{0}}\xi)^{(q)})$
for all $\theta\in(-\infty,\tau]$ since $P_{-}^{\odot\star}$ and $R_{q}$ are
continuous for all $q\in\\{1,\dots,k\\}$. It follows from [28, Lemma 1] that
$H_{q}$ takes values in $j(X)$ and is continuous. It is not difficult to show
that $H_{q}^{-}$ is $T$-periodic in the first variable because
$P_{-}^{\odot\star}$ is $T$-periodic and $R_{q}$ is $T$-periodic in the first
variable.

Step 6: Existence of $H_{q}^{00}$ and $\tilde{H}_{q}^{0}$. To obtain
$H_{q}^{00}(\tau,\xi^{(q)})$ and $H_{q}^{0}(\tau,\xi^{(q)})$, we project (4.2)
onto $\mathbb{R}j\dot{\gamma}_{\tau}$ and $j(\tilde{X}_{0}(\tau))$. Since $j$
is an isomorphism on $j(X)=X^{\odot\odot}$ we get from combining (4.2) and
(51) that the coefficients must satisfy

$\displaystyle-\frac{\partial
H_{q}^{00}(\tau,\xi^{(q)})}{\partial\tau}-D_{\xi}H_{q}^{00}(\tau,\xi^{(q)})\tilde{M}_{0}\xi$
$\displaystyle=p_{q}(\tau,\xi^{(q)})-R_{q}^{00}(\tau,\xi^{(q)}),$
$\displaystyle-\frac{\partial\tilde{H}_{q}^{0}(\tau,\xi^{(q)})}{\partial\tau}+\tilde{M}_{0}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})-D_{\xi}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})\tilde{M}_{0}\xi$
$\displaystyle=P_{q}(\tau,\xi^{(q)})-{R}_{q}^{0}(\tau,\xi^{(q)}).$

These are precisely the equations obtained in [23, Theorem III.7] and hence
from the results of [23, Theorem III.7], the provided normal forms follow. In
addition, it is proven in [23, Theorem III.7] that
$H_{q}^{00},\tilde{H}_{q}^{0},p_{q},P_{q}$ are continuous and so we conclude
that $H,p$ and $P$ are $C^{k}$-smooth maps.

Recall that the map $\tau\mapsto\gamma_{\tau}$ is $T$-periodic and
$C^{k}$-smooth. Furthermore, from (23) in combination with (44) we have that
$\tau\mapsto\tilde{Q}_{0}(\tau)$ is $T$-periodic and $C^{k}$-smooth. It also
follows from previous theorem that $(\tau,\xi)\mapsto H(\tau,\xi)$ is
$T$-periodic in the first component and $C^{k}$-smooth. Hence,
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ can be also written as

$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)=\\{\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi)\
:\ \tau\in\mathbb{R}\mbox{ and }\xi\in\mathbb{R}^{n_{0}}\\}\subset X,$ (62)

and has exactly the same properties as the description of
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ given in (11). Hence,
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is the center manifold for
(DDE) around the periodic orbit $\Gamma$ whenever the Floquet multiplier
$\lambda$ fulfills the requirements of thm:normalformI. This center manifold
is $T$-periodic in the sense that for any $\xi\in\mathbb{R}^{n_{0}}$ the map
$\mathbb{R}\ni\tau\mapsto\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi)\in
X$ is $T$-periodic.

Next we consider the case where the trivial Floquet multiplier has algebraic
multiplicity larger than $1$ and there is no Floquet multiplier located at
$-1$. This is for example the case in the fold bifurcation.

[Normal Form II] Assume that the algebraic multiplicity of the trivial Floquet
multiplier is more than one and that $-1$ is not a Floquet multiplier. Then
there exist $C^{k}$-smooth maps $H:\mathbb{R}\times\mathbb{R}^{n_{0}}\to
X,p:\mathbb{R}\times\mathbb{R}^{n_{0}}\to\mathbb{R}$ and
$P:\mathbb{R}\times\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{0}}$ such that the
history $x_{t}\in\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ may be
represented as

$x_{t}=\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi),\quad t\in I,$

where the time dependence of the coordinates $(\tau,\xi)$ describing the
dynamics of (DDE) on $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ is defined
by the normal form

$\begin{cases}\dfrac{d\tau}{dt}=1+\xi_{1}p(\tau,\xi)+\mathcal{O}(|\xi|^{k+1}),\\\
\\\\[-8.0pt]
\dfrac{d\xi}{d\tau}=\tilde{M}_{0}\xi+P(\tau,\xi)+\mathcal{O}(|\xi|^{k+1}).\end{cases}$

Here the functions $H,p$ and $P$ are $T$-periodic in $\tau$ and at least
quadratic in $\xi$. The $\mathcal{O}$-terms are also $T$-periodic in $\tau$.
Moreover, $p$ and $P$ are polynomials in $\xi$ of degree less than or equal to
$k$ such that

$\frac{d}{d\tau}p(\tau,e^{-\tau\tilde{M}_{0}^{\star}}\xi)=0\mbox{ and
}\frac{d}{d\tau}\bigg{(}e^{\tau\tilde{M}_{0}^{\star}}P(\tau,e^{-\tau\tilde{M}_{0}^{\star}}\xi)\bigg{)}=0,$

for all $\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$. Notice the
appearance of the $\xi_{1}$-term in the normal form description. This is due
to the fact that the $\star$ in (45) is now replaced by $1$ instead of $0$
compared to thm:normalformI. The proof of this theorem is very similar to that
of thm:normalformI.

[Proof of thm:normalformII] We proceed in the same way as the proof of
thm:normalformI and start by comparing the expansions of

$\frac{d}{dt}j(\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi))=j\bigg{(}\dot{\gamma}_{\tau}+\frac{\partial\tilde{Q}_{0}(\tau)}{\partial\tau}\xi+\frac{\partial
H(\tau,\xi)}{\partial\tau}+\bigg{(}\tilde{Q}_{0}(\tau)+D_{\xi}H(\tau,\xi)\bigg{)}\frac{d\xi}{d\tau}\bigg{)}\frac{d\tau}{dt}$

and

$A_{0}^{\odot\star}j(\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi))+G(\gamma_{\tau}+\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi))$

by substituting

$\frac{d\tau}{dt}=1+\xi_{1}+p(\tau,\xi)+\mathcal{O}(|\xi|^{k+1})\ \mbox{ and
}\frac{d\xi}{d\tau}=\tilde{M}_{0}\xi+P(\tau,\xi)+\mathcal{O}(|\xi|^{k+1}).$

We copy the same notation from the proof of thm:normalformI and use the
expansions of $H,p$ and $P$ together with (48) where now $\varphi$ must be
substituted by $\tilde{Q}_{0}(\tau)\xi+H(\tau,\xi)$. Eventually,

$\displaystyle
j\bigg{[}\dot{\gamma}_{\tau}+\frac{\partial\tilde{Q}_{0}(\tau)}{\partial\tau}\xi+\sum_{q=2}^{k}\frac{\partial
H_{q}(\tau,\xi^{(q)}))}{\partial\tau}+\bigg{(}\tilde{Q}_{0}(\tau)+\sum_{q=2}^{k}D_{\xi}H_{q}(\tau,\xi^{(q)})\bigg{)}\bigg{(}\tilde{M}_{0}+\sum_{q=2}^{k}P_{q}(\tau,\xi^{(q)})\bigg{)}\bigg{]}$
$\displaystyle\bigg{(}1+\xi_{1}+\sum_{q=2}^{k}p_{q}(\tau,\xi^{(q)})\bigg{)}+\mathcal{O}(|\xi|^{k+1})$
$\displaystyle=A_{0}^{\odot\star}j(\gamma_{\tau})+G(\gamma_{\tau})+A^{\odot\star}(\tau)j\bigg{(}\tilde{Q}_{0}(\tau)\xi+\sum_{q=2}^{k}H_{q}(\tau,\xi^{(q)})\bigg{)}$
$\displaystyle+\sum_{q=2}^{k}G_{q}\bigg{(}\tau,\bigg{[}\tilde{Q}_{0}(\tau)\xi+\sum_{p=2}^{k}H_{q}(\tau,\xi^{(p)})\bigg{]}^{(q)}\bigg{)}+\mathcal{O}(|\xi|^{k+1}).$

Let us now compare the terms in powers of $\xi$ on both side of the equation.
The $\xi^{0}$-terms give us

$\frac{d}{d\tau}j({\gamma}_{\tau})=A_{0}^{\odot\star}j(\gamma_{\tau})+G(\gamma_{\tau}),$

which means that $\gamma$ is a solution (47). The $\xi^{1}$-terms tell us

$\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(\tilde{Q}_{0}(\tau)\xi)=j((\tilde{Q}_{0}(\tau)\tilde{M}_{0}+\gamma_{\tau}\Pi_{1})\xi),$
(63)

which is exactly the result established in (23), but now for all Floquet
multipliers on the unit circle and characterizes the linear part. Here
$\Pi_{1}:\mathbb{R}^{n_{0}}\to\mathbb{R}$ is the projection on the first
component, defined as $\Pi_{1}\xi:=\xi_{1}$, where
$\xi=(\xi_{1},\dots,\xi_{n_{0}})$. After collecting the $\xi^{(2)}$-terms, we
get

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{2}(\tau,\xi^{(2)}))$
$\displaystyle=j(D_{\xi}H_{2}(\tau,\xi^{(2)})\tilde{M}_{0}\xi+p_{2}(\tau,\xi^{(2)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)P_{2}(\tau,\xi^{(2)}))-R_{2}(\tau,\xi^{(2)}),$

where
$R_{2}(\tau,\xi^{(2)})=G_{2}(\tau,(\tilde{Q}_{0}(\tau)\xi)^{2})-\xi_{1}(\frac{d\tilde{Q}_{0}(\tau)}{d\tau}\xi+\tilde{Q}_{0}(\tau)\tilde{M}_{0}\xi)$.
Finally, after collecting the $\xi^{(q)}$-terms for $q=3,\dots,k$, we get

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}(\tau,\xi^{(q)}))$
(64)
$\displaystyle=j(D_{\xi}H_{q}(\tau,\xi^{(q)})\tilde{M}_{0}\xi+p_{q}(\tau,\xi^{(q)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)P_{q}(\tau,\xi^{(q)}))-R_{q}(\tau,\xi^{(q)}),$
(65)

where $R_{q}(\tau,\xi^{(q)})$ depends on
$G_{q^{\prime}}(\tau,\xi^{(q^{\prime})})$ for $2\leq q^{\prime}\leq q$ and
$j(H_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$,
$j(p_{q^{\prime}}(\tau,\xi^{(q^{\prime})})\dot{\gamma}_{\tau})$ and
$j(\tilde{Q}_{0}(\tau)P_{q^{\prime}}(\tau,\xi^{(q^{\prime})}))$ for
$q^{\prime}=2,\dots,q-1$.

We want to project (64) onto the spaces
$\mathbb{R}\dot{\gamma}_{\tau},\tilde{X}_{0}(\tau)$ and ${X}_{\pm}(\tau)$ to
show the existence of $H_{q}$ separately on each individual space. We
decompose for any $q=2,\dots,k$ the functions $H_{q}$ and $R_{q}$ as

$\displaystyle H_{q}(\tau,\xi^{(q)})$
$\displaystyle=H_{q}^{00}(\tau,\xi^{(q)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)\tilde{H}_{q}^{0}(\tau,\xi^{(q)})+H_{q}^{-}(\tau,\xi^{(q)})+H_{q}^{+}(\tau,\xi^{(q)})$
$\displaystyle R_{q}(\tau,\xi^{(q)})$
$\displaystyle=R_{q}^{00}(\tau,\xi^{(q)})\dot{\gamma}_{\tau}+\tilde{Q}_{0}(\tau)\tilde{R}_{q}^{0}(\tau,\xi^{(q)})+R_{q}^{-}(\tau,\xi^{(q)})+R_{q}^{+}(\tau,\xi^{(q)}),$

where $H_{q}^{\pm}(\tau,\xi^{(q)})=P_{\pm}(\tau)H_{q}(\tau,\xi^{(q)})\in
X_{\pm}(\tau)$ and
$R_{q}^{\pm}(\tau,\xi^{(q)})=P_{\pm}^{\odot\star}(\tau)R_{q}(\tau,\xi^{(q)})\in
X_{\pm}^{\odot\star}(\tau)$ with coordinates
$H_{q}^{00}(\tau,\xi^{(q)}),R_{q}^{00}(\tau,\xi^{(q)})\in\mathbb{R}$ and
$\tilde{H}_{q}^{0}(\tau,\xi^{(q)}),\tilde{R}_{q}^{0}(\tau,\xi^{(q)})\in\mathbb{R}^{n_{0}}$
for all $\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n_{0}}$.

Carrying out the calculations in the same way as the proof of thm:normalformII
and recalling that
$(-\frac{d}{d\tau}+A^{\odot\star}(\tau))j\dot{\gamma}_{\tau}=0$ together with
(63), we obtain

$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}(\tau,\xi^{(q)}))$
$\displaystyle=j\bigg{(}-\frac{\partial
H_{q}^{00}(\tau,\xi^{(q)})}{\partial\tau}\dot{\gamma}_{\tau}+\Pi_{1}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})\dot{\gamma}_{\tau}\bigg{)}$
$\displaystyle+j\bigg{(}\tilde{Q}_{0}(\tau)\bigg{(}-\frac{\partial}{\partial\tau}+\tilde{M}_{0}\bigg{)}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})\bigg{)}$
$\displaystyle+\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}^{-}(\tau,\xi^{(q)})+H_{q}^{+}(\tau,\xi^{(q)}))$

and this must be equal to the right-hand side of (64). To obtain the
coefficients, we project onto the spaces
$\mathbb{R}j\dot{\gamma}_{\tau},j(\tilde{X}_{0}(\tau)),j(X_{+}(\tau))$ and
$X_{-}^{\odot\star}(\tau)$. This yields the equations

$\displaystyle-\frac{\partial
H_{q}^{00}(\tau,\xi^{(q)})}{\partial\tau}D_{\xi}H_{q}^{00}(\tau,\xi^{(q)})\tilde{M}_{0}\xi$
$\displaystyle=p_{q}(\tau,\xi^{(q)})-\Pi_{1}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})-R_{q}^{00}(\tau,\xi^{(q)})$
$\displaystyle-\frac{\partial\tilde{H}_{q}^{0}(\tau,\xi^{(q)})}{\partial\tau}+\tilde{M}_{0}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})-D_{\xi}\tilde{H}_{q}^{0}(\tau,\xi^{(q)})\tilde{M}_{0}\xi$
$\displaystyle=P_{q}(\tau,\xi^{(q)})-\tilde{R}_{q}^{0}(\tau,\xi^{(q)})$
$\displaystyle\bigg{(}-\frac{\partial}{\partial\tau}+A^{\odot\star}(\tau)\bigg{)}j(H_{q}^{\pm}(\tau,\xi^{(q)}))$
$\displaystyle=j(D_{\xi}H_{q}^{\pm}(\tau,\xi^{(q)})\tilde{M}_{0}\xi)-R_{q}^{\pm}(\tau,\xi^{(q)}).$

We see that the equations for the $X_{\pm}^{\odot\star}(\tau)$-component are
the same as in the proof of thm:normalformI. Hence, we obtain $H_{q}^{\pm}$ as
in (57) and (61) respectively. To solve the remaining part of this hierarchy
of equations, notice these equations are solvable in exactly the same way as
the proof of thm:normalformI and the proposed normal forms follow. One should
make the observation that $\tilde{H}_{q}^{0}$ has to be computed before
$H_{q}^{00}$.

Under these assumptions on the Floquet multipliers, we have also proven that
$\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)$ can also be parametrized as
(62).

The last normal form theorem is more involved because we have to deal with the
Floquet multiplier $-1$ that induces $T$-antiperiodic maps due to
prop:eigenfunctions2. Introduce the decomposition

$\tilde{X}_{0}(\tau)=\tilde{X}^{\prime}_{0}(\tau)\oplus\tilde{X}^{\prime\prime}_{0}(\tau),$

where $\tilde{X}^{\prime}_{0}(\tau)$ is spanned by $T$-periodic maps
$\varphi_{0}(\tau),\dots,\varphi_{n_{0}^{\prime}}(\tau)$ and where
$\tilde{X}^{\prime\prime}_{0}(\tau)$ is spanned by $2T$-periodic maps
$\varphi_{n_{0}^{\prime}+1}(\tau),\dots,\varphi_{n_{0}^{\prime}+n_{0}^{\prime\prime}}(\tau)$
and $n_{0}^{\prime}+n_{0}^{\prime\prime}=n_{0}$, corresponding to all
(generalized) eigenfunctions of the monodromy operator belonging to the
Floquet multiplier $-1$. Define the symmetry
$\tilde{S}_{0}:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{0}}$ as

$(\xi^{\prime},\xi^{\prime\prime})=\xi\mapsto\tilde{S}_{0}\xi=(\xi^{\prime},-\xi^{\prime\prime}),$

then we have the following theorem, which is for example the case in the
period-doubling bifurcation.

[Normal form III] Assume that $-1$ is a Floquet multiplier. Then the results
of thm:normalformI or thm:normalformII, depending on the location and
algebraic multiplicity of other the Floquet multipliers on the unit circle
hold with the following modification: the maps $H,p$ and $P$ are $2T$-periodic
in the first component and additionally satisfy

$H(\tau+T,\xi)=H(\tau,\tilde{S}_{0}\xi)$

and

$p(\tau+T,\xi)=p(\tau,\tilde{S}_{0}\xi),\quad
P(\tau+T,\xi)=P(\tau,\tilde{S}_{0}\xi),$

for all $\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{m}$. The proof of this
theorem is analogous to that of thm:normalformI or thm:normalformII but in a
$2T$-periodic setting. Hence, we obtain the results from thm:normalformI or
thm:normalformII, depending on the location and algebraic multiplicity of the
Floquet multipliers on the unit circle where now the maps $H,p$ and $P$ being
$2T$-periodic in $\tau$. It remains to show the additional symmetries on the
maps $H,p$ and $P$. Because the structure of the parts in the normal form are
similar to that of the ODE case, treated in [23, Theorem III.13] this part
will be omitted since the proof is identical by making the substitution of
$\tau\mapsto\gamma(\tau)$ towards $\tau\mapsto\gamma_{\tau}$.

Via this theorem we obtain a $2T$-periodic $(n_{0}+1)$-dimensional
$C^{k}$-smooth manifold $\mathcal{W}_{\operatorname{loc}}^{c}(\Gamma)\subset
X$ that is a center manifold for (DDE) around the periodic orbit $\Gamma$.

##  5 Conclusion and outlook

We have proven that the periodic normal forms to study bifurcations of limit
cycles suit naturally in the framework of classical DDEs. This task has been
accomplished by proving two principal results: the existence of a periodic
smooth basis for the center eigenspace (thm:eigenfunctions) and the existence
of a special coordinate system on the center manifold (thm:normalformI,
thm:normalformII and thm:normalformIII) in which the local dynamics can be
described. A paper providing computational formulas to study all codimension
one bifurcations of limit cycles in classical DDEs, along the lines of the
periodic normalization method [26, 33], is in preparation.

In this paper, we restricted ourselves to the setting of classical DDEs.
However, our proof on the existence of a periodic smooth finite-dimensional
center manifold near a nonhyperbolic cycle in [28] is given in the general
context of dual perturbation theory (sun-star calculus). As a consequence, the
results extend to a much broader class of delay equations, such as for example
renewal equations [10, 14, 2]. The natural question arises if the results from
this paper can also be generalized towards the general context of sun-star
calculus. We already see some difficulties in the linear part because the
proof on the existence of the periodic smooth (adjoint) (generalized)
eigenfunctions rely on lemma:basisD(A) and lemma:adjoint D(Astar). Both lemmas
are based on the smoothness property of delay differential equations, which is
a result that is not present in the general setting of delay equations.
Nevertheless, if one requires only weak⋆ continuous differentiability of the
forward and backward evolutionary systems, we believe that the proven normal
form theorems can still be applied (in a weak⋆ sense) to the general context
of dual perturbation theory, but of course under certain assumptions.

If one is interested in bifurcations of limit cycles for systems consisting of
infinite delay [12] or abstract DDEs [24, 25, 32, 31] that describe for
example neural fields, it is known that $\odot$-reflexivity is in general lost
[32, Theorem 12], and therefore the center manifold theorem for nonhyperbolic
cycles from [28] does not directly apply. However, we believe that this
technical difficulty can be resolved by employing similar techniques as in
[25]. We are convinced that these techniques can also be used to prove the
existence of the periodic normal forms in the setting of abstract DDEs and
systems consisting of infinite delay because the proof of the periodic normal
forms are written in the general setting of sun-star calculus.

## Acknowledgements

The authors would like to thank Prof. Odo Diekmann (Utrecht University) and
Prof. Stephan van Gils (University of Twente) for helpful discussions and
suggestions.

##  Appendix A Variation-of-constants formula for the adjoint problem

In this section of the appendix, we will prove that solutions of an
inhomogeneous perturbed abstract ordinary differential equation are precisely
given by an abstract integral equation. This result is important in the proof
of thm:adjoint eigenfunctions.

Let $J\subseteq\mathbb{R}$ be an interval and suppose that
$(s,t)\in\Omega_{J}^{\star}$. Consider an inhomogeneous perturbation $f:J\to
X^{\star}$ on the generator $A^{\star}(s)$ to the adjoint problem [5, Equation
(5.8)]. This yields the initial value problem

$\begin{dcases}d^{\star}u(s)=-A^{\star}(s)u(s)+f(s),\quad&s\leq t,\\\
u(t)=\psi,\quad&\psi\in X^{\odot},\end{dcases}$ (66)

which suggests the variation-of-constants formula

$u(s)=U^{\odot}(s,t)\psi+\int_{t}^{s}U^{\star}(s,\tau)f(\tau)d\tau,\quad\psi\in
X^{\odot},$ (67)

for $t\leq s$, where the integral must be interpreted as a weak⋆ integral.
This suggestion, with the additional assumptions on $f$, will be verified in
this section. Let us first turn our attention towards the weak⋆ integral
appearing in (67).

Let $g:J\to X^{\star}$ be a continuous function and denote the set
$\\{(s,r,t)\in J^{3}\ :\ s\leq r\leq t\\}$ by $\Theta_{J}^{\star}$. Then the
map $v(\cdot,\cdot,\cdot,g):\Theta_{J}^{\star}\to X^{\star}$ defined as the
weak⋆ integral

$v(s,r,t,g):=\int_{t}^{s}U^{\star}(r,\tau)g(\tau)d\tau,\quad\forall(s,r,t)\in\Theta_{J}^{\star},$

is continuous and takes values in $X^{\odot}$. The statement of the theorem is
a dual version of the first part of [28, Lemma 1]. The proof is along the same
lines as the proof in [28, Lemma 1], where one just has to work with the
backward evolutionary system $U^{\star}$. See also [8, Lemma 3.1] for a
semigroup analogue of this lemma. As we have proven that the weak⋆ integral,
appearing in (67) is well-defined, we can turn our attention towards the
verification of the variation-of-constants formula. The proof is inspired from
[28, Proposition 37] and [25, Proposition 21].

Let $f:J\to X^{\star}$ be a continuous function. If $u$ is a solution of (66)
on $J$ then $u$ is given by (67). Let $(s,t)\in\Omega_{J}^{\star}$ with $t>s$
be arbitrary. Define the function $w:[s,t]\to X^{\star}$ as
$w(\tau):=U^{\star}(s,\tau)u(\tau)$ for all $\tau\in[s,t]$. We claim that $w$
is weak⋆ differentiable with weak⋆ derivative

$d^{\star}w(\tau)=U^{\star}(s,\tau)d^{\star}u(\tau)+U^{\star}(s,\tau)A^{\star}(\tau)u(\tau).$
(68)

To show this claim, let $\tau\in[s,t]$ and $x\in X$ be given. For any
$h\in\mathbb{R}$ such that $\tau+h\in[s,t]$ we have that

$\displaystyle\langle w(\tau+h)-w(\tau),x\rangle$ $\displaystyle=\langle
U^{\star}(s,\tau+h)u(\tau+h)-U^{\star}(s,\tau)u(\tau),x\rangle$
$\displaystyle=\langle
U^{\star}(s,\tau+h)[u(\tau+h)-u(\tau)],x\rangle+\langle[U^{\star}(s,\tau+h)-U^{\star}(s,\tau)]u(\tau),x\rangle$
$\displaystyle=\langle
u(\tau+h)-u(\tau),U(\tau+h,s)x\rangle+\langle[U^{\star}(s,\tau+h)-U^{\star}(s,\tau)]u(\tau),x\rangle$

Because $U$ is a strongly continuous forward evolutionary system, we have that
$U(\tau+h,s)x\to U(\tau,s)x$ in norm as $h\to 0$. The definition of the weak⋆
derivative implies

$\frac{1}{h}(u(\tau+h)-u(\tau))\to d^{\star}u(\tau)\quad\mbox{
weakly${}^{\star}$ as }h\to 0,$

if we can show that the difference quotients remains bounded in the limit.
Since $u$ is a solution to (66), we know that $u$ is weak⋆ continuously
differentiable and so locally Lipschitz continuous by [25, Remark 16]. Because
$[s,t]$ is compact, $u$ is Lipschitz continuous on $[s,t]$ and so the
difference quotient remains bounded in the limit. Combining these two facts,
we get

$\frac{1}{h}\langle u(\tau+h)-u(\tau),U(\tau+h,s)x\rangle\to\langle
d^{\star}u(\tau),U(s,\tau)x\rangle\quad\mbox{ as }h\to 0.$

Furthermore, since
$u(\tau)\in\mathcal{D}(A^{\star}(\tau))=\mathcal{D}(A_{0}^{\star})$, it
follows from [5, Theorem 5.7] that

$\frac{1}{h}\langle[U^{\star}(s,\tau+h)-U^{\star}(s,\tau)]u(\tau),x\rangle\to\langle
U^{\star}(s,\tau)A^{\star}(\tau)u(\tau),x\rangle\quad\mbox{ as }h\to 0.$

Hence, we get

$\frac{1}{h}\langle w(\tau+h)-w(\tau),x\rangle\to\langle
U^{\star}(s,\tau)d^{\star}u(\tau)+U^{\star}(s,\tau)A^{\star}(\tau)u(\tau),x\rangle\quad\mbox{
as }h\to 0,$

which proves (68). Substituting the differential equation from (66) into (68)
yields

$d^{\star}w(\tau)=U^{\star}(s,\tau)f(\tau)$

and so $d^{\star}w$ is weak⋆ continuous since $f$ was assumed to be norm
continuous. For every $x\in X$ we have

$\displaystyle\langle u(s)-U^{\star}(s,t)u(t),x\rangle$ $\displaystyle=\langle
w(s),x\rangle-\langle w(s),x\rangle=\int_{t}^{s}\langle
d^{\star}w(\tau),x\rangle
d\tau=\langle\int_{t}^{s}U^{\star}(s,\tau)f(\tau)d\tau,x\rangle.$

Since $x$ and $(s,t)\in\Omega_{J}^{\star}$ were arbitrary, we conclude that

$u(s)-U^{\star}(s,t)u(t)=\int_{t}^{s}U^{\star}(s,\tau)f(\tau)d\tau,$

or equivalently

$u(s)=U^{\star}(s,t)\psi+\int_{t}^{s}U^{\star}(s,\tau)f(\tau)d\tau,$

since $u(t)=\psi$ by assumption. The continuity and range of $f$ ensures from
lemma:wkstarintegral that the weak⋆ integral takes values in $X^{\odot}$.
Since $\psi\in X^{\odot}$, we have that

$u(s)=U^{\odot}(s,t)\psi+\int_{t}^{s}U^{\star}(s,\tau)f(\tau)d\tau,$

which completes the proof.

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