steady states for the stochastic dynamics have $K$ nonzero weights (fig. 5g,
h). In this case there are ${K\choose m}$ equilibria with $m$ negative
synaptic weights and $\sum_{m=1}^{K}{K\choose m}=2^{K}-1$ such equilibria in
total. With odd $a+c<0$, any of these are stable and we observed convergence
to various of them (fig. 5g, h). For even $a+c<0$, only the flat solution with
all weights at $K^{-1/p}$ are linearly stable. In simulations, we did not
observe convergence to this solution. Instead we observed large fluctuations
characterized by prolonged excursions of individual synaptic weights (fig. 6a,
b). When $a+c\neq 1$, the dynamics of the synaptic weight norm are not closed.
With $a+c<0$ and even, the unit-norm $\ell^{p}$-sphere appeared unstable in
simulations (fig. 6c, d).

Figure 6: Large fluctuations in synaptic weights for $a+c<0$ and even. a)
Example dynamics for two different parameter sets. b) Evolution of the
synaptic weight norm over 20 realizations. c) Impact of decreasing the
learning rate.

####  A.4.2 Rank one input correlations

Let $\bm{\mathsfit{\mu}}={\bm{r}}^{a+1}$, the $(a+1)$-fold outer product of
the vector ${\bm{r}}$. This corresponds to the case of constant inputs. The
dynamics reduce to

$\tau\dot{J}_{i}={r}_{i}{J}_{i}^{c}({\bm{r}}^{T}{\bm{J}})^{a}-{J}_{i}({\bm{r}}^{T}{\bm{J}})^{a}\sum_{j}{r}_{j}{J}_{j}^{c-1}\lvert{J}_{j}\rvert^{p}$
(80)

and at steady states,

${r}_{i}{J}_{i}^{c}({\bm{r}}^{T}{\bm{J}})^{a}={J}_{i}({\bm{r}}^{T}{\bm{J}})^{a}\sum_{j}{r}_{j}{J}_{j}^{c-1}\lvert{J}_{j}\rvert^{p}$
(81)

Weights orthogonal to the input direction, ${\bm{r}}^{T}{\bm{J}}=0$, are a
steady state. Otherwise, we see that ${J}_{i}=0$ is always a steady state for
$c>0$. If ${r}_{i}=0$, then either ${J}_{i}=0$ or
$\sum_{j}{r}_{j}{J}_{j}^{c-1}\lvert{J}_{j}\rvert^{p}=0$. If ${\bm{J}}$ is a
steady state, the Jacobian is

$\tau\frac{d\dot{J}_{i}}{dJ_{k}}=\delta_{ik}(rJ)^{a}\left(cJ_{i}^{c-1}r_{i}-J_{j}^{c-1}\lvert
J_{j}\rvert^{p}r_{j}\right)+a(rJ)^{a-1}\left(J_{i}^{c}r_{i}-J_{i}J_{j}^{c-1}\lvert
J_{j}\rvert^{p}r_{j}\right)r_{k}-(c+p-1)(rJ)^{a}J_{i}J_{k}^{c-1}\lvert
J_{k}\rvert^{p}r_{k}$ (82)

where $({\bm{r}}^{T}{\bm{J}})^{0}$=1, including at ${\bm{r}}^{T}{\bm{J}}=0$.
At an orthogonal steady state, ${\bm{r}}^{T}{\bm{J}}=0$, the Jacobian
simplifies to exactly 0 so that direction defines a slow subspace of the
linearized dynamics.

By definition, ${\bm{r}}$ is an E-eigenvector of $\bm{\mathsfit{\mu}}$ with
eigenvalue
$\left\lvert\mkern-1.0mu\left\lvert{\bm{r}}\right\rvert\mkern-1.0mu\right\rvert_{2}^{2a}$
and $\bm{\mathsfit{\mu}}$ has a rank one CP decomposition in ${\bm{r}}$. So if
$(b,c)=(1,0)$, ${\bm{J}}={\bm{r}}$ is an attracting steady state of eq. 80
(theorem 1). Here we focus on the dynamics with weight-dependence. We study
the simple case of $c=1$ and a piecewise constant ${\bm{r}}$ with $n$ elements
equal to $r$, and the remaining zero. We see that in this case, the unit-norm
$n$-sphere is an equilibrium set for the dynamics and determine when it is
stable.

######  Theorem 7.

Let $\bm{\mathsfit{\mu}}\in\mathbb{R}^{K\times K\times\ldots\times
K}={\bm{r}}^{a+1}$ be a rank one tensor of order $a+1$, the $(a+1)$-fold outer
product of ${\bm{r}}$, where ${\bm{r}}\in\mathbb{R}^{K}$. Let ${\bm{r}}$ be
$n$-sparse and piecewise constant, with $n$ nonzero elements equal to $r$ and
the remaining $K-n$ elements zero. Let

$M({\bm{J}})=\sum_{i=1}^{K}J_{i}$ (83)

and name $\mathcal{S}$ the unit $\ell^{p}\,n$-sphere in $\mathbb{R}^{K}$, with
nonzero elements on the dimensions corresponding to the nonzero elements of
${\bm{r}}$. If $c=1$ then

  1. 1.

The $K-n$ elements of ${\bm{J}}$ corresponding to the zero elements of
${\bm{r}}$ have a fixed point at zero. It is stable if $M({\bm{J}})>0$ and
unstable if $M({\bm{J}})<0$.

  2. 2.

$\mathcal{S}$ is a slow manifold for the dynamics of the remaining $n$
synaptic weights. It is stable if $a$ is odd or $r>0$ and unstable if both $a$
is even and $r<0$.

###### Proof.

Let ${\bm{r}}$ be $n$-sparse and piecewise constant, with its first $n$
elements equal to $r$ and the remaining $K-n$ elements zero. Assume that
${\bm{J}}\neq 0$. We proceed in order of the claims. First consider the $K-n$
inputs where ${r}_{i}=0$. For $c=1$, their dynamics are

$\tau\dot{J}_{i}=-r^{a+1}M^{a}({\bm{J}})L({\bm{J}})J_{i}$ (84)

where

$L({\bm{J}})=\sum_{i=1}^{K}\lvert J_{i}\rvert^{p}$ (85)

$L\geq 0$ by definition with equality only at ${\bm{J}}=0$. So if
$r^{a+1}M^{a}>0$, these weights will converge to a steady state at zero. If
$r^{a+1}M^{a}<0$, these weights will diverge exponentially. If $M$ is fixed at
$0$ these weights are stable.

Second consider the dynamics of the $n$ weights with nonzero $r_{i}$, which
reduce to

$\tau\dot{J}_{i}=r^{a+1}M^{a}({\bm{J}})\left(1-L({\bm{J}})\right)J_{i}$ (86)

and the steady state condition for $J_{i}$ is that either $J_{i}=0$, $M=0$ or
$L=1$. So we have steady states for the first $n$ elements of ${\bm{J}}$ on
either the $\ell^{p}$ $n$-sphere or on the hyperplane orthogonal to ${\bf 1}$
(and the trivial steady state $J_{i}=0$). Next we examine stability for those
$n$ weights at one such point ${\bm{J}}^{*}$. From eq. 82, the Jacobian matrix
at ${\bm{J}}^{*}$ has rank one

$\tau\frac{d\dot{J}_{i}}{dJ_{k}}=-pr^{a+1}M^{a}({\bm{J}}^{*})J^{*}_{i}\lvert
J^{*}_{k}\rvert^{p}$ (87)

It has one eigenvalue $-(p/\tau)r^{a+1}M^{a}({\bm{J}}^{*})\sum_{j}J_{j}\lvert
J^{*}_{j}\rvert^{p}$, with associated eigenvector ${\bm{J}}^{*}$. The
remaining $n-1$ eigenvalues are zero, and the orthogonal complement of
${\bm{J}}^{*}$ is their slow eigenspace. Each point ${\bm{J}}^{*}$ on the
$\ell^{p}\,n$-sphere has such a slow eigenspace so the full sphere is a slow
manifold. To determine the stability of the unit-norm $n$-sphere we will
examine the dynamics of the synaptic weight norm. The dynamics of $L$ and $M$
form a closed system:

$\displaystyle\tau\dot{L}=$ $\displaystyle pr^{a+1}M^{a}L(1-L)$ (88)
$\displaystyle\tau\dot{M}=$ $\displaystyle r^{a+1}M^{a+1}(1-L)$

There are two line equilibria on $M=0$ and $L=1$ and the Jacobian determinant
is $pr^{a+3}M^{a+1}(1-L)^{2}$, which is zero on either of those line
equilibria so a linear stability analysis is uninformative. Recall that $L\geq
0$ by definition. There are three relevant cases for the dynamics. When $a$ is
odd, all factors of $r$ are positive and so is $M^{a+1}$. When $a$ is even,
the sign of $r$ impacts the sign of $\dot{M}$. We next examine the three
cases: 1) $a$ odd, 2) $a$ even and $r>0$ and 3) $a$ even and $r<0$.

Figure 7:  Dynamics of the synaptic weight norm (phase portraits). The vectors
$\dot{L},\dot{M}$ are plotted with unit norm. For each case, we show two
corresponding parameter sets. a) Case 1: $a$ odd. b) Case 2: $a$ even and
$r>0$. c) Case 3: $a$ even and $r<0$.

First take $a$ odd (fig. 7a). Then $L=1$ is attracting when $M>0$ but
repelling when $M<0$. $M$ is always increasing for $L<1$ and decreasing for
$L>1$. With $a$ even and $r>0$, $L=1$ is always attracting (fig. 7b). $M=0$ is
attracting for $L>1$ and vice versa. If $a$ is even and $r<0$, $L=1$ is always
repelling. In this case, if $L(0)>1$ the synaptic weights will explode while
if $L(0)<1$ the synaptic weights will evolve towards the stable equilibrium
$L=0,M=0$ (fig. 7c). This corresponds to ${\bm{J}}=0$. In sum, the unit-norm
solution $L=1$ can be attracting or repelling. It is attracting if $a$ is odd,
or $a$ even with $r>0$. It is repelling if $a$ is even and $r<0$. ∎


# Dynamics of a class of time-period strongly 2-cooperative system: integer-
valued Lyapunov function and embedding property of limit sets

Mengmeng Gao and Dun Zhou
School of Mathematics and Statistics
Nanjing University of Science and Technology
Nanjing, Jiangsu, 210094, P. R. China
Corresponding author, Email<EMAIL_ADDRESS>Partially supported by NSF
of China No.11971232, 12331006, 12071217.

###### Abstract

We construct an integer-valued Lyapunov function $\sigma(\cdot)$ for
generalized negative cyclic feedback system; and prove that $\sigma(\cdot)$ on
any $\omega$-limit set which generated by Poincaré mapping of bounded solution
of such strongly $2$-cooperative system is constant. Therefore, the
$\omega$-limit can be continuously embedded into a compact subset of a two
dimensional plane. Finally, a dissipative condition is given to ensure that
all orbits of such system are bounded.

##  1 Introduction

The concept of monotone dynamical systems(MDS) was introduced by M. W. Hirsch,
H. Matano in the 1980s ([12, 13, 14, 15, 16, 17, 24, 25, 26]); and has wide
range of applications in biomathematics and control theory. From the view of
geometry, the MDS has a special structure-closed convex cone, with the help of
this special structure, “most” bounded trajectories will approach a set of
equilibria, that is, so called generic quasi-convergence. Among others,
cooperative systems are typical MDS. Taking a finite-dimensional system as an
example:

$\dot{x}=f(x),\quad x\in D$ (1.1)

where $D\subset\mathbb{R}^{n}$ is a non-empty, open convex set, $f\in
C^{1}(\Omega)$. Then, (1.1) is a cooperative if and only if

$\frac{\partial f_{i}}{\partial x_{j}}(x)\geq 0,\quad i\neq j,x\in D,$ (1.2)

holds true. Moreover, if the Jacobi matrix of $f$ at every point $x\in D$ is
irreducible, then (1.1)- is a strongly cooperative system(see [36]).

Nevertheless, there are a lot of systems arising in applied science do not
belong MDS, for instance, the negative cyclic feedback system(see [21]). In
this context, a generalization of MDS was proposed, Sanchez in [31] introduced
the invariant cones of rank $k$. A non-emepty closed set
$C\subset\mathbb{R}^{n}$ is cone of rank $k$, if: (i) $x\in C$,
$\alpha\in\mathbb{R}^{n}\Rightarrow\alpha x\in C$; (ii) $\max\\{\dim
W:C\supset W\text{ linear subspace}\\}=k$. It is obvious that, MDS are systems
with cones of rank $1$. By using cones of rank $2$, Sanchez projected part of
the dynamics into planes, and then deduce the Poincaré-Bendixson property for
some orbits, that is, some compact omega-limit sets without equilibrium points
are just a periodic orbit. Later, this property was generalized to generic
Poincaré-Bendixson theory for systems with cones of rank $2$ by Feng et al.
[4].

In the current paper, we focus on a time periodic system in the following
form:

$\begin{split}\dot{x}_{1}&=f_{1}(t,x_{1},x_{2},x_{n}),\\\
\dot{x}_{i}&=f_{i}(t,x_{i-1},x_{i},x_{i+1}),\quad 2\leq i\leq n-1.\\\
\dot{x}_{n}&=f_{n}(t,x_{n-1},x_{n},x_{1}),\\\ \end{split}$ (1.3)

with the nonlinear term satisfies

$\begin{split}\delta_{i}\frac{\partial f_{i}}{\partial x_{i-1}}(t,x)>0&,\quad
1\leq i\leq n,\\\ \delta_{i+1}\frac{\partial f_{i}}{\partial x_{i+1}}(t,x)\geq
0&,\quad 2\leq i\leq n-1,\\\ \end{split}$ (1.4)

where $\delta_{i}\in\\{-1,1\\},\ x_{0}=x_{n},\ x_{n+1}=x_{1}$,
$f=(f_{1},f_{2},\cdots,f_{n})$ is a $C^{1}$-function defined on
$\mathbb{R}\times\Omega\subset\mathbb{R}\times\mathbb{R}^{n}$ and $\Omega$ is
a non-empty, open and convex set defined on $\mathbb{R}^{n}$, and there exists
$T>0$ such that $f(t+T,\cdot)\equiv f(t,\cdot)$. Such structure widely exists
in biological, control models: biological oscillators, neurological systems,
cell regulation, enzyme reactions, as well as gene transcription [5, 10, 3,
38].

For instance, consider the following four dimensional system:

$\displaystyle\dot{x}_{1}=a_{1}-a_{2}x_{1}x_{4},$ (1.5)
$\displaystyle\dot{x}_{2}=a_{3}x_{1}-a_{4}x_{2},$
$\displaystyle\dot{x}_{3}=a_{5}x_{2}-a_{6}x_{3},$
$\displaystyle\dot{x}_{4}=a_{7}x_{3}-a_{8}x_{1}x_{4},$

where $a_{i}>0,i=1,\cdots,8$ are constant. It is an integral feedback model
named antithetic controllers(see [1]). The above system characterizes the
closed-loop interconnection of an antithetic controller (reprented by the
variables $x_{1}$ and $x_{4}$) and a simple two-dimensional linear system
(represented by the variables $x_{2}$ and $x_{3}$).

(1.3)+(1.4) is a generalization of the following cyclic feedback system that
introduced by J. Mallet-Paret and H. Smith in [21]

$\displaystyle\dot{x}_{1}$ $\displaystyle=f_{1}(x_{1},x_{n}),$ (1.6)
$\displaystyle\dot{x}_{i}$ $\displaystyle=f_{i}(x_{i-1},x_{i}),\quad 2\leq
i\leq n-1.$ $\displaystyle\dot{x}_{n}$ $\displaystyle=f_{n}(x_{n-1},x_{n})$

and satisfies

$\delta_{i}\frac{\partial f_{i}}{\partial x_{i-1}}(x_{i},x_{i-1})>0,\quad
1\leq i\leq n$ (1.7)

where $\delta_{i}\in\\{-1,1\\}$.

Let $\Delta=\delta_{1}\cdots\delta_{n}$,
$x_{i}=u_{i}x_{i},u_{i}\in\\{+1,-1\\},1\leq i\leq n$, then (1.3)+(1.4) becomes
a new feedback system (see Section 5.1) that satisfies

$\displaystyle\delta_{1}\frac{\partial f_{1}}{\partial x_{n}}$
$\displaystyle>0,\ \delta_{1}\frac{\partial f_{n}}{\partial x_{1}}\geq 0,$
(1.8) $\displaystyle\frac{\partial f_{i}}{\partial x_{i-1}}$
$\displaystyle>0,\quad 2\leq i\leq n,$ $\displaystyle\frac{\partial
f_{i}}{\partial x_{i+1}}$ $\displaystyle\geq 0,\quad 2\leq i\leq n-1.$

Therefore, we can always assume, without loss of generality, that (1.3)+(1.4)
is in the form of (1.3)+(1.8). Then, if $\delta_{1}=\Delta=+1$, (1.3)+(1.8) is
a cooperative system with positive feedback; if $\delta_{1}=\Delta=-1$, it is
a non-cooperative system with negative feedback.

In a positive feedback system, the final product of a behavior will lead to
more behavior occurring in the feedback loop, amplifying the initial action.
Low frequency oscillators, birthing processes, and the expression of
premembrane protein are positive feedback systems(see [19, 9]). A negative
feedback system is a system that, when subjected to external stimuli or
disturbances, the system will suppress or weaken this external excitation or
interference through negative feedback, thereby achieving a new equilibrium
state of the system. Typical negative feedback models include interspecific
competition and cooperation systems, Lotka Volterra models, negative feedback
loops of enterobacterium transcriptional replicators, and so on (see [11,
28]). These negative feedback loops are considered the main reason for the
existence of biological oscillations.

In the case $\delta_{1}=+1$(positive feedback), (1.3)+(1.8) is a strongly
monotone dynamic system (see [36]). And hence, if (1.3)+(1.8) is an autonomous
system ($f$ is independent of $t$ [34]), the $\omega$-limit set generated by
“most” bounded trajectories is an equilibrium point. In fact, Poincaré-
Bendixson theorem is established, that is, the $\omega$-limit set of any
bounded solution of (1.3)+(1.8) contains no equilibrium should be a periodic
orbit(see [29]); moreover, generic Morse-Smale property was also obtained(see
[6, 29]). If (1.3)+(1.8) is a time $T$-periodic system(i.e.
$f(t+T,\cdot)\equiv f(t,\cdot)$), “most” bounded orbits of the system will
converge to a linearly stable $kT$-periodic orbit, where $k$ is a positive
integer(see [30]).

In the case $\delta_{1}=-1$(negative feedback), (1.3)+(1.8) is no longer a MDS
from the view of cone. In autonomous case ($f=(f_{1},\cdots,f_{n})^{T}$
independent of $t$), J. Mallet-Paret and H. Smith in [21], obtained the
Poincaré-Bendixson theorem for (1.6)+(1.7). Later, the Morse decomposition of
global attractor was studied by T. Gedeon in [8], and complicated dynamics
take place within some Morse set. If (1.6)+(1.7) admits a unique equilibrium,
then it is a globally asymptotically under suitable assumptions (see [41]).
Elkhader in [2] extended the results in (1.3)+(1.8) to a more general form,
that is, $f_{1}=f_{1}(x_{1},x_{n})$, $f_{n}=f_{n}(x_{n-1},x_{n})$ in
(1.3)+(1.4), under the assumption that $f$ is a $C^{n-1}$ function on
$\Omega$. Recently, by introducing the notion of $k$-cooperative dynamical
systems(see Section 2.2), (1.3)+(1.8) is a strongly $2$-cooperative system;
and hence, Poincaré-Bendixson theorem is proved for (1.3)+(1.8) (see [20, 40])
by projecting the $\omega$-limit set of bounded orbit into a suitable
$2$-dimensional space.

In practical problems, the nonlinear term $f$ in (1.3)+(1.8) often depends on
$t$, that is, (1.3)+(1.8) is a non-autonomous system. A natural question is
how to describe the long time behavior of the solution of (1.3)+(1.8), and can
we project the $\omega$-limit set into a two dimensional plane? We try to
answer this question and give some characterization about the global dynamics
of (1.3)+(1.8) when $f$ of $t$ is $T$-period.

We first generalize the important tool called the integer-valued Lyapunov
function $\sigma(\cdot)$ (see Definition 3.1) in [2, 21] to (1.3)+(1.8)(see
Theorem 3.1). $\sigma(\cdot)$ is only on an open and dense subset $\Lambda$ of
$\mathbb{R}^{n}$ on which it is also continuous. The deductions in Theorem 3.1
are different from that in [2, Proposition 2.1] or [21, Propsotion 1.1]; and
moreover, the form of $\sigma(\cdot)$ and it’s decreasing properties are
uniquely determined by each other (see Proposition 3.3).

As an application, we then use $\sigma(\cdot)$ to reduce the dynamics on
$\omega$-limit set of any bounded solution of (1.3)+(1.8) to the dynamics on a
compact subset of $\mathbb{R}^{2}$. Precisely speaking, assume that
$\phi(t,x^{0})$ is a forward bounded solution of (1.3)+(1.8) with an initial
value of $x^{0}$. Let $P$ be the associated Poincaré map and
$\omega^{p}(x^{0})$ be the $\omega$-limit set. Then

  * •

(see Theorem 4.1) For any two point $x,y\in\omega^{p}(x^{0})\ (x\neq y)$ and
$t\in\mathbb{R}$, there is

$\sigma(\phi(t,x)-\phi(t,y))\equiv constant;$

  * •

(see Theorem 4.2) $\omega^{p}(x^{0})$ can be continuous embedded into a
compact subset of two-dimensional plane.

We give some remarks in the following

1) From the view skew-product flows, Theorem 4.2 means that the dynamics on
$\cup_{t\in[0,T)}\omega^{p}(\phi(t,\\\ x^{0}))$ can be viewed as an
$T$-periodically forced flow on a compact subset of $\mathbb{R}^{2}$. And it
should be noted that, in general, dynamics of two-dimensional discrete
mappings still can be very complicated, for instance, the existence of Smale’s
horseshoe mapping. Nevertheless, in some specific models, the dynamics may
become simpler due to the constraints of some conditions (see Section 5.3).

2) Our results can be viewed as a partial generalization of autonomous case in
[20, 40] to time periodic case. In [20, 40], nested invariant cones were
frequently used to reduce the complexity of dynamics on $\omega$-limit. While
in this paper, we give a delicate characterization of integer-valued Lyapunov
function $\sigma(\cdot)$; and by using $\sigma(\cdot)$, we obtain the
constancy property of $\sigma(\cdot)$ on $\omega^{p}(x_{0})$ and finally
achieve our goal. In fact, based on Theorem 3.1, we can further establish the
Floquet theory for linearized equation of (1.3)+(1.4), then construct nested
invariant cones, and finally obtain the structural stability of (1.3)+(1.4) in
autonomous case. Moreover, the continuous embedding can be then improved to
Lipschitz embedding. All these will be included in our forthcoming paper.

3) The discovery of integer-valued Lyapunov function can be traced back to the
characterization of oscillating matrix eigenvectors (see [7, p.105]). Similar
functions widely exist in many other mathematical models developed in physics
and biology, including semilinear parabolic equations on one-dimensional
bounded fixed regions, tridiagonal competitive cooperative systems, Cauchy-
Riemann equation on $S^{1}$, and play very important roles in characterizing
the global dynamics of such systems (see [23, 27, 32, 37, 39]).

The paper is organized as follows. In Section 2, we introduce some basic
concept of dynamics. In Section 3, we define an integer-valued Lyapunov
function for (3.1)+(3.2), and develop its properties. In Section 4, we state
our main results and give detail proofs. In Section 5, we give a dissipative
condition, so that every orbit of (1.3)+(1.4) is bounded, and then discuss
some relationships between $1$-cooperative and $2$-cooperative systems.

##  2 Preliminaries

In this section, we list some concept for discrete dynamical systems,
$k$-cooperative systems, as well as a characterization for nature number set.

###  2.1 Basic concept in dynamical systems

######  Definition 2.1.

Let X be a topological vector space and G be a topological group. If
$\psi:G\times X\rightarrow X$ is continuous and satisfies:

  * (1)

$\psi(e,x)=x$, where $e$ is an identity element of $G$;

  * (2)

$g_{1},g_{2}\in G,\psi(g_{1},\psi(g_{2},x))=\psi(g_{1}g_{2},x)$

Then, $(X,G,\psi)$ is called a dynamical system.

Obviously, for any $g\in G$, $\psi(g,\cdot):X\rightarrow X,\
x\mapsto\psi(g,x)$ is homeomorphic. If $G=\mathbb{R}$ is an additive group of
real numbers, then $(X,\mathbb{R},\psi)$ is called a flow. If $G=\mathbb{Z}$
is an additive group of integers, $(X,\mathbb{Z},\psi)$ is called a discrete
dynamical system. Let $T:X\rightarrow X$ be a homeomorphism and
$\psi:\mathbb{Z}\times X\rightarrow X$ be such that $\psi(n,x)=T^{n}(x)$, then
$(X,\mathbb{Z},\psi)$ (or $(X,T)$) is a discrete dynamical system as well.

######  Definition 2.2.

Let $(\mathbb{R}^{n},T)$ be a discrete dynamical system. Then, the sets
$O(x)=\\{T^{n}(x)\\}_{n\in\mathbb{Z}}$,
$O^{+}(x)=\\{T^{n}(x)\\}_{n\in\mathbb{N}}$,
$O^{-}(x)=\\{T^{-n}(x)\\}_{n\in\mathbb{N}}$ are called orbit, positive half
orbit and negative half orbit of $x\in\mathbb{R}^{n}$, respectively.

######  Definition 2.3.

Let $(\mathbb{R}^{n},T)$ be a discrete dynamical system. Assume that
$O^{+}(x)$(resp. $O^{-}(x)$) is bounded, then the $\omega$-limit(resp.
$\alpha(x)$-limit ) set of $x$ is defined in the following

$\begin{split}\omega^{T}(x)&=\\{z\in\mathbb{R}^{n}\mid\exists
n_{i}\rightarrow+\infty,T^{n_{i}}(x)\rightarrow z\\};\\\
(\text{resp.}\alpha^{T}(x)&=\\{z\in\mathbb{R}^{n}\mid\exists
n_{i}\rightarrow+\infty,T^{-n_{i}}(x)\rightarrow z\\}.)\end{split}$

Obviously, $\omega^{T}(x)$, $\alpha^{T}(x)$ are non-empty, compact,
$T$-invariant sets.

######  Definition 2.4.

Let $X$ and $\widetilde{X}$ are topological spaces, $(X,T)$ and
$(\widetilde{X},\widetilde{T})$ are dynamical systems, $M$ is a compact
invariant set in $(X,T)$. If there is a homeomorphic mapping $h:M\rightarrow
h(M)\subset\widetilde{X}$, such that

$h\circ T=\widetilde{T}\circ h.$

Then, $(M,T)$ is called embedded into $(h(\widetilde{M}),\widetilde{T})$.

######  Definition 2.5.

Let $(X,T)$ be a discrete dynamical system. If there is $x\in X$ such that
$\omega^{T}(x)=X$, then the continuous mapping $T$ on $X$ is called
transitive.

######  Lemma 2.1.

Let $X$ be a $Baire$ space, and $(X,T)$ be a discrete dynamical system, then
$T$ is transitive if and only if $T^{-1}$ is transitive.

###### Proof.

See [22, Corollary I.11.5]. ∎

###  2.2 $k$-positive and $k$-cooperative systems

The concept of $k$-cooperative and $k$-positive systems was recently
introduced by E. Weiss, M. Margaliot in [40]. Consider the following
$n$-dimensional linear equation:

$\dot{x}=A(t)x,x\in\mathbb{R}^{n}$ (2.1)

where $A(t)$ is continuous in the interval under consideration. Then, (2.1) is
called $k$-positive system ($k=1,\cdots,n-1$), if it maps the set of vectors
with at most $k-1$ sign variations to itself; moreover, if $A(t)$ is
irreducible, it is then a strongly $k$-positive system. Particularly, if
$k=1$, (2.1) is then a positive linear system. System (1.1) is called a
$k$-cooperative (resp. strongly $k$-cooperative) system, if and only if that
the corresponding linear variation system is a $k$-positive (resp. strongly
$k$-positive) system.

By introducing compound matrices and Metzler matrices, a $k$-positive system
($k=2,\cdots,n-2$) equivalents to the $k$-th additive compound matrix of
$A(t)$ is Metzler (see [40, Theorem 4]). In other words, a $k$-positive system
means that $A(t)$ of (2.1) is of the following form:

  * •

$(-1)^{k-1}a_{1n}(t),(-1)^{k-1}a_{n1}(t)\geq 0$;

  * •

$a_{ij}(t)\geq 0$ for all $i,j$ with $|i-j|=1$;

  * •

$a_{ij}(t)=0$ for all $i,j$ with $1<|i-j|<n-1$.

###  2.3 A property of the natural number set

######  Lemma 2.2.

Let $\mathcal{M}$ and $\mathcal{N}$ be two infinite subsets of $\mathbb{N}$.
Then there are natural numbers $m_{1},m_{2},m_{3}\in\mathcal{M}$,
$n_{1},n_{2},n_{3}\in\mathcal{N}$ and $k_{1},k_{2},l_{1},l_{2}$, such that

$n_{3}>k_{1}m_{1}+k_{2}m_{2}+m_{1}$

and

$m_{3}=l_{1}n_{1}+l_{2}l_{2}+n_{3}-k_{1}m_{1}+k_{2}m_{2}.$

###### Proof.

See [37, Lemma 3.4]. ∎

##  3 Integer-valued Lyapunov function for a class of strongly 2-positive
system

In this section, we consider a class of strongly 2-positive system, and
introduce some properties for an integer-valued Lyapunov function of these
system. More precisly, the system we consider here is the following

$\displaystyle\dot{x}_{1}$
$\displaystyle=a_{1,1}(t)x_{1}+a_{1,2}(t)x_{2}+a_{1,n}(t)x_{n},$ (3.1)
$\displaystyle\dot{x}_{i}$
$\displaystyle=a_{i,i-1}(t)x_{i-1}+a_{i,i}(t)x_{i}+a_{i,i+1}(t)x_{i+1},\quad
2\leq i\leq n-1,$ $\displaystyle\dot{x}_{n}$
$\displaystyle=a_{n,n-1}(t)x_{n-1}+a_{n,1}(t)x_{1}+a_{n,n}(t)x_{n}.$

in which the coefficient functions with respect to $t$ are continuous over
$\mathbb{R}$. Denote $A(t)=(a_{ij}(t))_{n\times n}$ be the coefficient matrix
of (3.1), and assume that

$\displaystyle a_{1,n}\leq 0,a_{n,1}\leq 0$ (3.2) $\displaystyle
a_{i,i-1}(t)\geq 0,\quad 2\leq i\leq n,$ $\displaystyle a_{i,i+1}(t)\geq
0,\quad 1\leq i\leq n-1$
$\displaystyle\prod_{i=1}^{n}a_{i,i-1}(t)+\prod_{i=1}^{n}a_{i,i+1}(t)<0.$

Note that $A(t)$ is irreducible, and hence, (3.1)+(3.2) is a strongly
$2$-positive system.

Denote $\mathcal{O}=\\{x\mid x\in\mathbb{R}^{n}$ and $x_{i}\neq
0,i=1,2,\cdots,n\\}$. Then, similar as in [2, 21], we introduce the integer-
valued Lyapunov function $\sigma(\cdot)$ of system (3.1)+(3.2) in the
following

######  Definition 3.1.

For any $x\in\mathcal{O}$, define

$\sigma(x)=card\\{i|\delta_{i}x_{i}x_{i-1}\leq 0\\}.$

where $\delta_{1}=-1,\delta_{i}=1,\ 2\leq i\leq n$, and $card$ represents the
number of elements contained in the set.

Define

$\tilde{n}=\left\\{\begin{matrix}n,\quad\text{if $n$ is odd},\\\
n-1,\quad\text{if $n$ is even}.\end{matrix}\right.$

It is obvious that $\mathcal{O}$ is an open and dense set in $\mathbb{R}^{n}$,
$\sigma(\cdot)\in\\{1,3,5,\cdots,\tilde{n}\\}$ and continuous on
$\mathcal{O}$. Moreover, its domain of definition can be extended(by
continuity) to

$\Lambda=\\{x\mid x\in\mathbb{R}^{n},\text{if}\ x_{i}=0,\text{then}\
\delta_{i}\delta_{i+1}x_{i+1}x_{i-1}<0\\}.$

######  Definition 3.2.

For all $x\in\mathbb{R}^{n}$, define functions as:

$\sigma_{m},\ \sigma_{M}:\mathbb{R}^{n}\to\\{1,3,\cdots,\tilde{n}\\}.$

where $\sigma_{m}(x)$ and $\sigma_{M}(x)$ are the minimum and maximum values
of $\sigma(y)$ over $\mathcal{U}\cap\Lambda$, and $\mathcal{U}$ is a
sufficiently small neighborhood of $x$ in $\mathbb{R}^{n}$.

We have the following characterization about $\sigma(\cdot)$.

######  Theorem 3.1.

Let $x(t)$ be a nontrivial bounded solution of system $\eqref{linear
system}+\eqref{linear sys assumption}$, then

  1. (i)

$x(t)\in\Lambda$, except for finite points of $\mathbb{R}$;

  2. (ii)

$\sigma(x(t))$ is locally constant, where $x(t)\in\Lambda$;

  3. (iii)

if $x(t_{0})\notin\Lambda$, then $\sigma(x(t_{0}^{+}))<\sigma(x(t_{0}^{-}))$;

  4. (iv)

if $x(t)\in\Lambda$, then $(x_{i}(t),x_{i+1}(t))\neq(0,0)$(for each
$i\in\\{1,2,\cdots,n\\}$);

  5. (v)

there exists $t_{0}>0$ such that when
$t\in[t_{0},+\infty)\cup(-\infty,-t_{0}]$, $x(t)\in\Lambda$ and $\sigma(x(t))$
is constant.

######  Corollary 3.2.

Let $\phi(t,x)$ and $\phi(t,y)$ are two different bounded solutions of system
$\eqref{system 1.1}+\eqref{assume simple system 1.1}$, then

  1. (i)

$\phi(t,x)-\phi(t,y)\in\Lambda$ except for finite points of $\mathbb{R}$;

  2. (ii)

$\sigma(\phi(t,x)-\phi(t,y))$ is locally constant, where
$\phi(t,x)-\phi(t,y)\in\Lambda$;

  3. (iii)

if $\phi(t_{0},x)-\phi(t_{0},y)\notin\Lambda$, then
$\sigma(\phi(t_{0}^{+},x)-\phi(t_{0}^{+},y))<\sigma(\phi(t_{0}^{-},x)-\phi(t_{0}^{-},y))$
;

  4. (iv)

if $\phi(t,x)-\phi(t,y)\in\Lambda$, then
$(\phi(t,x)_{i},\phi(t,x)_{i+1})\neq(\phi(t,y)_{i},\phi(t,y)_{i+1})$(for each
$i\in\\{1,2,\cdots,n\\}$);

  5. (v)

there exist $t_{0}>0$ and $C_{1},C_{2}\in\\{0,1,\cdots,n-1\\},$ where $\
C_{1}\leq C_{2}$, such that

$\sigma(\phi(t,x)-\phi(t,y))=C_{1},t\geq
t_{0},\quad\sigma(\phi(t,x)-\phi(t,y))=C_{2},t\leq-t_{0}.$

######  Proposition 3.3.

Let $t\rightarrow\tilde{A}(t)=(\tilde{a}_{ij}(t))_{n\times n}$ be a continuous
matrix function in $(a,b)$, and $\tilde{\varphi}(t,x)$ be a nontrivial
solution of

$\dot{x}=\tilde{A}(t)x$

with an initial value condition $\tilde{\varphi}(t_{0},x^{0})=x^{0}$. Then the
following two are equivalent:

  1. (i)

there is an open and dense subset $\mathcal{G}$ of $(a,b)$ such that for each
$t$ in $\mathcal{G}$, $\tilde{A}(t)$ is in the form of (3.2);

  2. (ii)

$\sigma(\tilde{\varphi}(t,x_{0}))$ is decreasing with respect to $t\geq
t_{0}$, that is

$x^{0}\neq
0,x^{0}\notin\Lambda\Rightarrow\tilde{\varphi}(t_{0}+\epsilon,x^{0}),\tilde{\varphi}(t_{0}-\epsilon,x^{0})\in\Lambda,$

and

$\sigma_{m}(x^{0})=\sigma(\tilde{\varphi}(t_{0}+\epsilon,x^{0}))<\sigma(\tilde{\varphi}(t_{0}-\epsilon,x^{0}))=\sigma_{M}(x^{0})$

where $\epsilon>0$ is small enough.

######  Remark 3.1.

Proposition 3.3 was proved for positive cyclic feedback system in [6, Theorem
1], that is, $a_{1,n}\geq 0$, $a_{n,1}\geq 0$ and
$\prod_{i=1}^{n}a_{i,i-1}(t)+\prod_{i=1}^{n}a_{i,i+1}(t)>0$ in (3.2).

Based on Proposition 3.3, we can prove Theorem 3.1.

######  Theorem 3.1.

By the definition of $\sigma(\cdot)$, (i)-(iv) can be drawn directly from
Proposition 3.3. By (i) and (iii), $\sigma(x(t))$ only drops strictly a finite
number of times with respect to $t$, so (v) is work as well. ∎

###### Proof of Corollary 3.2.

Let $z(t)=\phi(t,x)-\phi(t,y)$, then $z(t)$ satisfies the following linear
equation:

$\dot{z}(t)=\tilde{A}(t)z(t),$ (3.3)

where $\tilde{A}=(\tilde{a}_{ij})_{n\times n}$,
$\tilde{a}_{ij}=\int_{0}^{1}\frac{\partial
f_{i}(t,r\phi(t,x)+(1-r)\phi(t,y))}{\partial x_{j}}dr$. Obviously, (3.3)
satisfies (3.1), then the corollary is true based on theorem 3.1. ∎

In the rest of this section, we prove Proposition 3.3. Note that if
$\sigma_{m}(x)=\tilde{n}$, then Proposition 3.3 is automatically established.
We only consider the case that $\sigma_{m}(x)<\tilde{n}-2$. To prove
Proposition 3.3, some lemmas are listed for preparations.

For each $h\in\\{1,3,\cdots,\tilde{n}-2\\}$, let
$m_{h}=\\{x\in\mathbb{R}^{n}\backslash\\{0\\}|\sigma_{m}(x)=h\\},N_{h}=\partial
m_{h}\cap m_{h},\theta_{h}=\\{x|\sigma_{m}(x)>h\\}.$

######  Lemma 3.4.

For each $h\in\\{1,3,\cdots,\tilde{n}-2\\},\theta_{h}$is an open set.

###### Proof.

Suppose on the contrary that $\theta_{h}$ is not open. Fix
$x^{0}\in\theta_{h}$ and $1\ll n\in\mathbb{N}$, then there is $x^{n}\in
B_{\frac{1}{2n}}(x^{0})$ ($B_{\frac{1}{2n}}(x^{0})$ represents a open ball
with center $x^{0}$ and radius $\frac{1}{2n}$), such that

$\sigma_{m}(x^{n})\leq h.$

For the given $x^{n}$, by the definition of $\sigma_{m}(\cdot)$, there also
exists $y^{n}\in B_{\frac{1}{2n}}(x^{n})\cap\Lambda$ such that

$\sigma(y^{n})\leq h.$

Let $n\rightarrow+\infty,$ then $y^{n}\rightarrow x^{0}$, but
$\sigma(y^{n})\leq h$, which is a contradiction to that $\sigma_{m}(x^{0})>h$.
Thus, $\theta_{h}$ is an open set. ∎

Let $x_{i_{1}},\cdots,x_{i_{k}},i_{1}<\cdots<i_{k}$, be those non-zero
components of $x$, then

$\begin{split}x\in m_{h}\Leftrightarrow\ &\text{the number of sign changes in
the sequence}\\\ &x_{i_{1}},\cdots,x_{i_{k}},-x_{i_{1}}\text{is
}h.\end{split}$

Therefore, $x\in N_{h}$ if and only if $x\in m_{h}$ and there exists $i$ such
that one of the following holds:

  * (i)

$x_{i-1}=x_{i}=0$;

  * (ii)

$x_{i}=0$ and $\left\\{\begin{aligned} x_{i-1}x_{i+1}>0,i\neq n,1\\\
x_{i-1}x_{i+1}<0,i=n,1.\end{aligned}\right.$

Moreover, we have the relationship between $\partial m_{h}$ and $N_{h}$ as
follows.

######  Lemma 3.5.

$\partial m_{h},N_{h}$ are coincide locally, that is, for each $x\in N_{h}$,
there exists $\delta_{x}>0$ such that for any $\tilde{x}\in\partial m_{h}$
with $|\tilde{x}-x|<\delta_{x}$, one has $\tilde{x}\in N_{h}$.

###### Proof.

Choose $x\in N_{h}$ and let $x_{i_{1}},\cdots,x_{i_{k}}$($i_{1}<\cdots<i_{k}$)
be non-zero components of $x$. Then there exists $\delta_{0}>0$, such that for
each $\tilde{x}\in\mathbb{R}^{n}$ with $|\tilde{x}-x|<\delta_{0}$ one has
$\tilde{x}_{i_{1}},\cdots,\tilde{x}_{i_{k}}$ share the same signs with
$x_{i_{1}},\cdots,x_{i_{k}}$. Thus, $\sigma_{m}(\tilde{x})\geq h$.

Next, we prove that if $\tilde{x}\in B_{\delta_{0}}(x)\cap\partial m_{h}$,
then

$\sigma_{m}(\tilde{x})=h\Leftrightarrow\tilde{x}\in N_{h}.$

Indeed, if this is not correct, then

$\sigma_{m}(\tilde{x})>h.$

Note that $\theta_{h}$ is an open set(see Lemma 3.4), there exists
$\tilde{\delta}>0$, such that

$\sigma_{m}(y)>h,$

for any $y\in B_{\tilde{\delta}}(\tilde{x})$, a contradiction to that
$\tilde{x}\in\partial m_{h}$. Thus, $\sigma_{m}(\tilde{x})=h$ and
$\tilde{x}\in N_{h}$. ∎

Let $N_{h,0}=\operatorname{Int}m_{h}$(interior of $m_{h}$). For
$q=1,\cdots,\tilde{n}-h-2$, define $N_{h,q}\subset m_{h}$ be the subset of
vectors $x\in m_{h}$ satisfy

$\displaystyle x\notin N_{h,p},\quad p=0,\cdots,q-1;$ $\displaystyle
x_{i}=\cdots=x_{i+q}=0\Rightarrow\left\\{\begin{aligned}
x_{i-1}x_{i+q+1}<0,\quad i+q+1\leq n,\\\ x_{i-1}x_{i+q+1-n}>0,\quad
i+q+1>n;\end{aligned}\right.$

for $q=\tilde{n}-h-1$, define

$N_{h,\tilde{n}-h-1}=N_{h}\setminus\bigcup\limits_{p=1}^{\tilde{n}-h-2}N_{h,p}\Leftrightarrow
N_{h}=N_{h,1}\cup\cdots\cup N_{h,\tilde{n}-h-1}.$

It is not hard to see that if $x\in N_{h,q}$, $x^{\prime}\in N_{h}$ with
$|x-x^{\prime}|\ll 1$, then $x^{\prime}\in N_{h,p}$ for some $p\leq q$ .

Back to (3.1)+(3.2), without loss of generality, we may always assume that
(3.2) is the following special form:

$\displaystyle a_{1,n}$ $\displaystyle<0,a_{n,1}\leq 0$ $\displaystyle
a_{i,i-1}(t)$ $\displaystyle>0,\quad 2\leq i\leq n,$ $\displaystyle
a_{i,i+1}(t)$ $\displaystyle\geq 0,\quad 1\leq i\leq n-1.$

Let $\varphi(t,x)$ be a solution of (3.1)+(3.2). For each
$q\leq\tilde{n}-h-2$, let $x\in N_{h,q}$ be the point such that if
$x_{i-1}\neq 0,x_{i}=0$, then one of the following holds:

  * (a)

$i=1,x_{n}x_{2}\leq 0$,

$x_{n}\dot{\varphi}(t,x)_{1}=x_{n}(a_{1,1}x_{1}+a_{1,2}x_{2}+a_{1,n}x_{n})=a_{1,n}x_{n}^{2}+a_{1,2}x_{n}x_{2}<0;$

  * (b)

$i\in\\{2,3,\cdots,n-1\\},x_{i-1}x_{i+1}\geq 0$,

$x_{i-1}\dot{\varphi}(t,x)_{i}=x_{i-1}(a_{i-1,i}x_{i-1}+a_{i,i}x_{i}+a_{i,i+1}x_{i+1})>0;$

  * (c)

$i=n,x_{n-1}x_{1}\leq 0$,

$x_{n-1}\dot{\varphi}(t,x)_{n}=x_{n-1}(a_{n,1}x_{1}+a_{n,n-1}x_{n-1}+a_{n,n}x_{n})>0.$

For any $x\in N_{h}$, let $I_{x}\subset\\{1,\cdots,n\\}$ be the set of indices
with the above property. Then $I_{x}$ is non-empty; moreover, we have the
following

######  Lemma 3.6.

Assume that $x,x^{\prime}\in N_{h,q}$ for some
$q\in\\{0,\cdots,\tilde{n}-h-1\\}$. If $|x-x^{\prime}|\ll 1$, then

$I_{x}\cap I_{x^{\prime}}\neq\varnothing.$

###### Proof.

Choose $x\in N_{h,q}$ and let $x_{j1},\cdots,x_{jk}$ be non-zero components of
$x$. Then for $x^{\prime}\in N_{h,q}$ with $|x-x^{\prime}|\ll 1$, one has
$x^{\prime}_{j1},\cdots,x^{\prime}_{jk}$ keep the same signs with
$x_{j1},\cdots,x_{jk}$. Suppose that $I_{x}\cap I_{x^{\prime}}=\varnothing$
and let $i\in I_{x}$, then $\ i\notin I_{x^{\prime}},\
x_{i-1}x^{\prime}_{i-1}>0$.

(a) $i=1$, then $x^{\prime}_{1}\neq 0$ or $x^{\prime}_{n}x^{\prime}_{2}>0$. If
$x^{\prime}_{n}x^{\prime}_{2}>0,x_{2}=0$, then there exists sign change in
$x^{\prime}_{n},-x^{\prime}_{2}$, which means

$\sigma_{m}(x^{\prime})>h,$

a contradiction to $x^{\prime}\in N_{h,q}$. If
$x^{\prime}_{n}x^{\prime}_{2}>0,x_{2}\neq 0$, however, this cannot happen due
to $x_{n}x_{2}<0$. Thus, $x^{\prime}_{i}\neq 0$.

Similar arguments for (b): $i\in\\{2,3,\cdots,n-1\\}$ and (c): $i=n$.

As a consequence, for any $i\in I_{x}$, there is $x^{\prime}_{i}\neq 0$, a
contradiction to that $x^{\prime}\notin N_{h,q}$. Thus, $I_{x}\cap
I_{x^{\prime}}\neq\varnothing.$ ∎

We now prove Proposition 3.3.

###### Proof of Proposition 3.3.

“(i) $\Rightarrow$ (ii)”

Assume that $t_{0}\in\mathcal{G}$, we first prove that

$\sigma_{m}(x_{0})=\sigma(\tilde{\varphi}(t_{0}+\epsilon,x_{0}))$

for $0<\epsilon\ll 1$. And this equivalents to show that: for each $x^{0}\in
m_{h}$($h\in\\{1,3,\cdots,\tilde{n}-2\\}$), there exists $\epsilon>0$ such
that for all $t\in(t_{0},t_{0}+\epsilon)$,

$\tilde{\varphi}(t,x)\in\operatorname{Int}m_{h}\Leftrightarrow\tilde{\varphi}(t,x)\in\Lambda,\sigma(\tilde{\varphi}(t,x))=h.$
(3.4)

Given $q\in\\{1,\cdots,\tilde{n}-h-1\\}$ and $x\in N_{h,q}$. For any $i\in
I_{x}$, one has

$\begin{cases}x_{i-1}\dot{\tilde{\varphi}}(t_{0},x)_{i}>0,&i\in\\{2,3,\cdots,n\\},\\\
x_{i-1}\dot{\tilde{\varphi}}(t_{0},x)_{i}<0,&i=1.\end{cases}$

Let $0<t-t_{0}\ll 1,\tilde{x}=\tilde{\varphi}(t,x)$, then $\tilde{x}_{i}\neq
0$ for all $i\in I_{x}$. Therefore, $I_{x}\cap I_{\tilde{x}}=\varnothing$. If
$q=1$, then $\tilde{x}\in\Lambda$, and (3.4) is established. For
$q\in\\{2,3,\cdots,\tilde{n}-h-1\\}$, suppose that (3.4) is established for
all $p<q$. We then prove that (3.4) is established for $p=q$.

Suppose on the contrary that there exists $x\in N_{h,q}$ with no $\epsilon>0$
satisfies

$\tilde{\varphi}(t,x)\in\operatorname{Int}m_{h},t\in(t_{0},t_{0}+\epsilon).$

Note that

$I_{x}\cap
I_{\tilde{\varphi}(t,x)}=\varnothing,\,\sigma_{m}(\tilde{\varphi}(t,x))\geq
h,$

for $0<t-t_{0}\ll 1$. By virtue of Lemma 3.6, $\tilde{\varphi}(t,x)\notin
N_{h,q}$. Observe that
$\tilde{\varphi}(t,x)\notin\cup_{i=q+1}^{n-h-1}N_{h,i}$,

$\sigma_{m}(\tilde{\varphi}(t,x))>h,\quad 0<t-t_{0}\ll 1.$

In other words, there exists $0<\epsilon\ll 1$ such that

$\sigma_{m}(\tilde{\varphi}(t,x))>h,\quad t\in(t_{0},t_{0}+\epsilon).$

By Lemma 3.4 and the continuity of $\tilde{\varphi}(t,x)$, one can choose
$y\in\bigcup\limits_{i=1}^{q-1}N_{h,i}$ with $|y-x|\ll 1$, such that

$\sigma_{m}(\tilde{\varphi}(t_{0}+\frac{\epsilon}{2},y))>h.$ (3.5)

Since $y\in\bigcup\limits_{i=1}^{q-1}N_{h,i}$, there exists
$0<\epsilon_{y}<\frac{\epsilon}{2}$ such that

$\tilde{\varphi}(t,y)\in\operatorname{Int}m_{h},\quad
t\in(t_{0},t_{0}+\epsilon_{y}).$ (3.6)

Combine (3.5), (3.6) and the continuity of $\tilde{\varphi}$, there are
$\bar{t}\in(t_{0}+\epsilon_{y},t_{0}+\frac{\epsilon}{2}),\delta>0$, such that

$\tilde{\varphi}(\bar{t},y)\in\partial
m_{h},\sigma_{m}(\tilde{\varphi}(t,y)>h,t\in(\bar{t},\bar{t}+\delta).$ (3.7)

It then follows from Lemma 3.5 that

$\tilde{\varphi}(\bar{t},y)\in N_{h}.$

Thus, (3.7) implies that $\tilde{\varphi}(\bar{t},y)\in N_{h,q}$.

We now further restrict $y$ to satisfies:

$\displaystyle(i)\quad x_{i-1}y_{i}>0,$ $\displaystyle\text{ if }\ i\in
I_{x}\cap\\{2,3,\cdots,n\\},$ $\displaystyle(ii)\quad x_{i-1}y_{i}<0,$
$\displaystyle\text{ if }\ i\in I_{x}\cap\\{1\\}.$

Recall that $I_{\phi(\bar{t},y)}\cap I_{x}\neq\varnothing$, we claim that:
there exist $i_{0}\in I_{x},\bar{\bar{t}}\in(t_{0},\bar{t}]$ such that

$\displaystyle x_{i_{0}-1}\dot{\tilde{\varphi}}(\bar{\bar{t}},y)_{i_{0}}\leq
0,$ $\displaystyle\text{ if }\ i_{0}\neq 1,$ $\displaystyle
x_{i_{0}-1}\dot{\tilde{\varphi}}(\bar{\bar{t}},y)_{i_{0}}\geq 0,$
$\displaystyle\text{ if }\ i_{0}=1.$

Indeed, if the claim is not hold, then for any $i\in
I_{x},t\in(t_{0},\bar{t}]$, one has

$\begin{cases}x_{i-1}\dot{\tilde{\varphi}}(t,y)_{i}>0,&i\in\\{2,3,\cdots,n\\},\\\
x_{i-1}\dot{\tilde{\varphi}}(t,y)_{i}<0,&i=1.\end{cases}$

And hence, $\tilde{\varphi}(t,y)_{i}$ is monotone with respect to $t$, that
is, $\phi(t,y)_{i}\neq 0$. Thus, $I_{x}\cap I_{\phi(\bar{t},y)}=\varnothing$,
which is a contradiction. The claim is proved.

By continuity of $\tilde{\varphi}$, for $i\in I_{x}$, one has:

$\displaystyle(i)\quad x_{i-1}\dot{\tilde{\varphi}}(t,x)_{i}>0,$
$\displaystyle\text{ if }\ i\in\\{2,3,\cdots,n\\},0<t-t_{0}\ll 1,$
$\displaystyle(ii)\quad x_{i-1}\dot{\tilde{\varphi}}(t,x)_{i}<0,$
$\displaystyle\text{ if }\ i=1,0<t-t_{0}\ll 1.$

By continuity of $\dot{\tilde{\varphi}}$, we can further get $i\in I_{x}$:

$\displaystyle(i)\quad x_{i-1}\dot{\tilde{\varphi}}(t,y)_{i}>0,$
$\displaystyle\text{ if }i\in\\{2,3,\cdots,n\\},0<t-t_{0}\ll 1,$
$\displaystyle(ii)\quad x_{i-1}\dot{\tilde{\varphi}}(t,y)_{i}<0,$
$\displaystyle\text{ if }\ i=1,0<t-t_{0}\ll 1.$

Choose $t=\bar{\bar{t}}$, then it is a contradiction to the claim.

Consequently, (3.4) is also correct for $x\in N_{h,q},t\in\mathcal{G}$.

Let
$M_{h}=\\{x\in\mathbb{R}^{n}\backslash\\{0\\}|\sigma_{M}(x)=h\\},\tilde{N}_{h}=\partial
M_{h}\cap M_{h}$. By appropriate modification of the above proof, one can
obtain that for each $x^{0}\in M_{h}$, there exists $\epsilon_{0}>0$ such that
for all $t\in(t_{0}-\epsilon_{0},t_{0})$,

$\tilde{\varphi}(t,x)\in\operatorname{Int}M_{h}\Leftrightarrow\tilde{\varphi}(t,x)\in\Lambda,\sigma(\tilde{\varphi}(t,x))=h.$
(3.8)

To complete the proof of (i) $\Rightarrow$ (ii), it is sufficient to check
that (3.4) and (3.8) also hold for some $\epsilon_{0}>0$, when
$t_{0}\in(a,b)\setminus\mathcal{G}$. For this purpose, fix
${\color[rgb]{1,0,0}\epsilon_{0}}>0$ and let $\tilde{A}^{k}(t)$ be a sequence
of continuous functions in the form of (3.2), such that
$\tilde{A}^{k}(t)\rightarrow\tilde{A}(t)$ uniformly in
$[t_{0}-\epsilon_{0},t_{0}+\epsilon_{0}]$, as $k\rightarrow\infty$. Let
$\tilde{\varphi}^{k}(t,x)$ be the solution of $\dot{x}=\tilde{A}^{k}(t)x$
through $x$ at time $t_{0}$.

By the Gronwall inequality,
$\tilde{\varphi}^{k}(t,x)\rightarrow\tilde{\varphi}(t,x)$ as
$k\rightarrow\infty$, uniformly in $[t_{0}-\epsilon_{0},t_{0}+\epsilon_{0}]$.
As we have proven that: for $\dot{x}=\tilde{A}^{k}(t)x$, if $x\neq
0,x\notin\Lambda,t_{1}\in[t_{0}-\epsilon_{0},t_{0}]$,$t_{2}\in[t_{0},t_{0}+\epsilon_{0}]$,
then

$\sigma_{m}(\tilde{\varphi}^{k}(t_{2},x))\leq\sigma_{m}(x)<\sigma_{M}(x)\leq\sigma_{M}(\tilde{\varphi}^{k}(t_{1},x)).$
(3.9)

Note that
$\\{y\in\mathbb{R}^{n}|\sigma_{m}(x^{{}^{\prime}})\leq\sigma_{m}(x)\\}$ and
$\\{y\in\mathbb{R}^{n}|\sigma_{M}(x^{{}^{\prime}})\geq\sigma_{M}(x)\\}$ are
closed sets,

$\sigma_{m}(\tilde{\varphi}(t_{2},x))\leq\sigma_{m}(x)<\sigma_{M}(x)\leq\sigma_{M}(\tilde{\varphi}(t_{1},x)).$
(3.10)

We claim that: one can choose $\epsilon_{0}$ small enough, such that

$\tilde{\varphi}(t_{0}-\epsilon,x),\tilde{\varphi}(t_{0}+\epsilon,x)\in\Lambda,$
(3.11)

and

$\sigma_{m}(x)=\sigma(\tilde{\varphi}(t_{0}+\epsilon,x))<\sigma(\tilde{\varphi}(t_{0}-\epsilon,x))=\sigma_{M}(x),$
(3.12)

for all $0<\epsilon<\epsilon_{0}$. Suppose on the contrary that, there exists
$\epsilon_{n}\to 0$ such that

$\tilde{\varphi}(t_{0}-\epsilon_{n},x)\notin\Lambda.$

For simplicity, we assume that $\epsilon_{n}\downarrow 0$, and note that
$\mathcal{G}$ is an open and dense subset of $(a,b)$, there exist
$t_{n}\in(t_{0}-\epsilon_{n-1},t_{0}-\epsilon_{n})$ such that
$\tilde{\varphi}(t_{n},x)\in\Lambda$; moreover, by (3.10)

$\begin{split}\sigma(\tilde{\varphi}(t_{n+1},x))&=\sigma_{m}(\tilde{\varphi}(t_{n+1},x))\leq\sigma_{m}(\tilde{\varphi}(t_{0}-\epsilon_{n},x))\\\
&<\sigma_{M}(\tilde{\varphi}(t_{0}-\epsilon_{n},x))\leq\sigma_{M}(\tilde{\varphi}(t_{n},x))\\\
&=\sigma(\tilde{\varphi}(t_{n},x))\end{split}$

which is a contradiction to that $\sigma(\cdot)$ is a finite valued function.
Thus, one can choose $\epsilon_{0}$ small enough such that

$\tilde{\varphi}(t_{0}-\epsilon,x)\in\Lambda,\quad 0<\epsilon<\epsilon_{0}.$

Similarly arguments can also lead to

$\tilde{\varphi}(t_{0}+\epsilon,x)\in\Lambda,\quad 0<\epsilon<\epsilon_{0}.$

It is not hard to see that there is $h\in\\{1,3\cdots,\tilde{n}\\}$ such that

$\sigma(\tilde{\varphi}(t_{0}+\epsilon,x))=h,\quad 0<\epsilon<\epsilon_{0}.$

By (3.10), $h\leq\sigma_{m}(x)$. Note that
$\tilde{\varphi}(t_{0}+\epsilon,x)\to x$, as $\epsilon\to 0$, by the
definition of $\sigma_{m}(\cdot)$, $h\geq\sigma_{m}(x)$. As a consequence,

$\sigma(\tilde{\varphi}(t_{0}+\epsilon,x))=\sigma_{m}(x),\quad
0<\epsilon<\epsilon_{0}.$

Similarly, one can also get

$\sigma(\tilde{\varphi}(t_{0}-\epsilon,x))=\sigma_{M}(x),\quad
0<\epsilon<\epsilon_{0}.$

We have checked (3.12), and the claim is proved.

“(ii) $\Rightarrow$ (i)”.

Choose $x^{0}\in\mathbb{R}^{n}$ be such that $x^{0}_{i}=0$, $x^{0}_{j}\neq 0$,
$j\neq i$ and

$\begin{cases}x^{0}_{i-1}x^{0}_{i+1}>0,\text{ if }2\leq i\leq n-1\\\
x^{0}_{i-1}x^{0}_{i+1}>0,\text{ if
}i=1,n\quad(x^{0}_{0}=x^{0}_{n},x^{0}_{n+1}=x^{0}_{1})\end{cases}$

Then, $x^{0}\notin\Lambda$ and $\sigma_{m}(x^{0})=\sigma_{M}(x^{0})-2$.

Let $y\in\mathbb{R}^{n}$ be with $|y-x^{0}|\ll 1$, then $y\in\Lambda$, if and
only if $y_{i}\neq 0$, and one of the following holds true:

  1. (a)

if $2\leq i\leq n,\quad\left\\{\begin{aligned}
\sigma(y)=\sigma_{m}(x^{0})\Leftrightarrow x^{0}_{i-1}y_{i}>0,\\\
\sigma(y)=\sigma_{M}(x^{0})\Leftrightarrow
x^{0}_{i-1}y_{i}<0;\end{aligned}\right.$

  2. (b)

if $i=1,\quad\left\\{\begin{aligned}
\sigma(y)=\sigma_{m}(x^{0})\Leftrightarrow x^{0}_{i-1}y_{i}<0,\\\
\sigma(y)=\sigma_{M}(x^{0})\Leftrightarrow
x^{0}_{i-1}y_{i}>0.\end{aligned}\right.$

Assume that $\tilde{\varphi}(t_{0},x)=x$, then

$\begin{cases}(\tilde{A}(t_{0})x^{0})_{i}x^{0}_{i-1}\geq 0,\quad 2\leq i\leq
n,\\\ (\tilde{A}(t_{0})x^{0})_{i}x^{0}_{i-1}\leq 0,\quad i=1.\end{cases}$

That is,

$\begin{cases}x^{0}_{i-1}(\tilde{a}_{i,i-1}(t_{0})x^{0}_{i-1}+\tilde{a}_{i,i+1}(t_{0})x^{0}_{i+1}+\sum\limits_{j\notin\mathcal{I}_{i}}\tilde{a}_{ij}(t_{0})x^{0}_{j})\geq
0,i\in\\{2,\cdots,n\\}\\\
x^{0}_{i-1}(\tilde{a}_{i,i-1}(t_{0})x^{0}_{i-1}+\tilde{a}_{i,i+1}(t_{0})x^{0}_{i+1}+\sum\limits_{j\notin\mathcal{I}_{i}}\tilde{a}_{ij}(t_{0})x^{0}_{j})\leq
0,i=1.\end{cases}$

(where $\mathcal{I}_{i}=\\{i-1,i+1\\}$), which implies that for all
$t\in(a,b)$,

$\left\\{\begin{aligned} &\tilde{a}_{ij}(t)=0,\quad\text{for}\ 1<|i-j|<n-1,\\\
&\tilde{a}_{i,i+1}(t)\geq 0,\quad\text{for}\ i=1,\cdots,n-1\\\
&\tilde{a}_{i-1,i}(t)\geq 0,\quad\text{for}\ i=2,\cdots,n\\\
&\tilde{a}_{1,n}(t),\tilde{a}_{n,1}(t)\leq 0\end{aligned}\right.$

Let $\mathcal{G}$ be the subset of $(a,b)$ such that

$\prod_{i=1}^{n}\tilde{a}_{i,i-1}(t)+\prod_{i=1}^{n}\tilde{a}_{i,i+1}(t)<0.$

Then, by continuity of $\tilde{A}(t)$, $\mathcal{G}$ is open. Suppose that
$\mathcal{G}$ is not dense, then there is an interval
$(\overline{a},\overline{b})\subset(a,b)$ and indices $h,k$ such that

$\tilde{a}_{h,h-1}(t)=\tilde{a}_{k,k+1}(t)=0,\quad
t\in(\overline{a},\overline{b}).$

Define $I_{hk}$ in the following:

$\mathcal{I}_{hk}=\left\\{\begin{aligned} &\\{i|h\leq i\leq
k\\},\quad\text{if}\ h\leq k,\\\ &\\{i|i>h\ or\ i\leq k\\},\quad\text{if}\
h>k.\end{aligned}\right.$

Choose $x\in\mathbb{R}^{n}\setminus\\{0\\}$ be such that $x_{j}=0$, for all
$j\in\mathcal{I}_{hk}$. Then,

$\tilde{\varphi}(t,x)_{j}=0,\quad j\in\mathcal{I}_{hk},\
t\in(\overline{a},\overline{b}),$

a contradiction to (ii). This proves (ii) $\Rightarrow$ (i).

The proof of Proposition 3.3 is completed.

∎

##  4 Embedding property of limit sets

As an application of $\sigma(\cdot)$, we here give a characterization for the
asymptotic behaviors of bounded solutions of (1.3)+(1.8).

Let $\phi(t,x^{0})$ be a solution of (1.3)+(1.8) with an initial value
$x^{0}$. Denote $P(x^{0})=\phi(T,x^{0})$ be the associated Poincaré map.
Assume $O^{+}(x^{0})=\\{y|P^{n}(x^{0}),n\in\mathbb{N}\\}$ is a bounded set in
$\mathbb{R}^{n}$. Then, our main results in this section are the following

######  Theorem 4.1.

For any $x,y\in\omega^{p}(x^{0})\ (x\neq y)$ and $t\in\mathbb{R}$, one has

$\phi(t,x)-\phi(t,y)\in\Lambda\ \text{and }\sigma(\phi(t,x)-\phi(t,y))\equiv
constant.$

######  Theorem 4.2.

There is a homeomorphic mapping

$h:\omega^{p}(x^{0})\rightarrow h(\omega^{p}(x^{0}))\subset\mathbb{R}^{2}.$

###  4.1 Proof of Theorem 4.1

Similar to the deduction for [37, Theorem A], the proof of Theorem 4.1 is
divided into the following Propositions 4.3, 4.4 and 4.5. Note that $\sigma$
is defined only on $\Lambda$, the result in [37, Theorem A] can not be applied
to Theorem 4.1 directly.

######  Proposition 4.3.

Assume $\widetilde{x}\in\omega^{p}(x^{0})$, then

$\phi(t,x)-\phi(t,y)\in\Lambda,\mbox{and}\ \sigma(\phi(t,x)-\phi(t,y))\equiv
constant,$

for all $x,y\in\overline{O(\widetilde{x})}(x\neq y)\mbox{and}\
t\in\mathbb{R}$.

######  Proposition 4.4.

Assume $x^{0}\notin\omega^{p}(x^{0}),\ x,y\in\omega^{p}(x^{0})(x\neq y).$ If
$x\notin\alpha^{p}(x)$ or $y\notin\alpha^{p}(y)$, then

$\sigma(\phi(t,x)-\phi(t,y))\equiv constant,$

for all $t\in\mathbb{R}$.

######  Proposition 4.5.

Assume $x^{0}\notin\omega^{p}(x^{0}),\ x,y\in\omega^{p}(x^{0})(x\neq y).$ If
$x\in\alpha^{p}(x)$ and $y\in\alpha^{p}(y)$, then

$\sigma(\phi(t,x)-\phi(t,y))\equiv constant,$

for all $t\in\mathbb{R}$.

###### Proof of Theorem 4.1.

Choose $x^{0}\in\mathbb{R}^{n}$ such that $\overline{O^{+}(x^{0})}$ is
bounded. If $x^{0}\in\omega^{p}(x^{0})$, then the theorem is proved by
Proposition 4.3; if $x^{0}\notin\omega^{p}(x^{0})$, then the theorem is proved
by Propositions 4.4 and 4.5. ∎

In the following, we will give detailed proofs for Propositions 4.3, 4.4, 4.5.

###### Proof of Proposition 4.3.

For any $x,y\in\omega^{p}(x^{0})(x\neq y)$, by virtue of Corollary 3.2(v),
there exist $C_{1},C_{2}\in\mathbb{N}$, $t_{0}>0$ such that

$\displaystyle\sigma(\phi(t,x)-\phi(t,y))$ $\displaystyle=C_{1},\quad t\geq
t_{0},$ $\displaystyle\sigma(\phi(t,x)-\phi(t,y))$ $\displaystyle=C_{2},\quad
t\leq-t_{0}.$

Particularly, take $t=n_{0}T>t_{0}$, then

$\phi(n_{0}T,x)-\phi(n_{0}T,y),\quad\phi(-n_{0}T,x)-\phi(-n_{0}T,y)\in\Lambda.$

The rest of the proof can be divided into the following two cases.

(1) $x,y\in O(\widetilde{x}),x\neq y$. At this point, there exist
$m,n\in\mathbb{Z}$ such that
$\phi(mT,\widetilde{x})=x,\phi(nT,\widetilde{x})=y$, thus,

$\displaystyle\phi(t,x)-\phi(t,y)=$
$\displaystyle\phi(t,\phi(mT,\widetilde{x}))-\phi(t,\phi(nT,\widetilde{x}))$
$\displaystyle=$
$\displaystyle\phi(t+mT,\widetilde{x})-\phi(t+nT,\widetilde{x}).$

Let $I(t)=\phi(t+mT,\widetilde{x})-\phi(t+nT,\widetilde{x})$. Noticing that
$\widetilde{x}\in\omega^{p}(x^{0})$, there exists
$\\{l_{k}\\}\subset\mathbb{N}$, such that
$\phi(l_{k}T,x^{0})\rightarrow\widetilde{x}$, that is,

$\displaystyle\|I(n_{0}T)-[\phi((n_{0}+l_{k})T,\phi(mT,x^{0}))-\phi((n_{0}+l_{k})T,\phi(nT,x^{0})]\|\ll
1,\quad k\gg 1$

where $\|\cdot\|$ represents the 2-norm on $\mathbb{R}^{2}$. Recall that
$\Lambda$ is an open set, then

$\displaystyle\phi((n_{0}+l_{k})T,\phi(mT,x^{0}))-\phi((n_{0}+l_{k})T,\phi(nT,x^{0}))\in\Lambda,\quad
k\gg 1,$ (4.1)
$\displaystyle\sigma(\phi((n_{0}+l_{k})T,\phi(mT,x^{0}))-\phi((n_{0}+l_{k})T,\phi(nT,x^{0}))=C_{1},\quad
k\gg 1.$

Similarly,

$\displaystyle\phi((-n_{0}+l_{k})T,\phi(mT,x^{0}))-\phi((-n_{0}+l_{k})T,\phi(nT,x^{0})\in\Lambda,k\gg
1,$ (4.2)
$\displaystyle\sigma(\phi((-n_{0}+l_{k})T,\phi(mT,x^{0}))-\phi((-n_{0}+l_{k})T,\phi(nT,x^{0}))=C_{2},k\gg
1.$

By Corollary 3.2(v), there exists $C\in\mathbb{N}$ such that

$\displaystyle\phi(t+l_{k}T,\phi(mT,x^{0}))-\phi(t+l_{k}T,\phi(nT,x^{0}))\in\Lambda,\quad$
$\displaystyle\sigma(\phi(t+l_{k}T,\phi(mT,x^{0}))-\phi(t+l_{k}T,\phi(nT,x^{0})))=C,$

for $k\gg 1$. Combining (4.1) and (4.2), $C_{1}=C_{2}=C$, and by Corollary
3.2(iii)

$\phi(t,x)-\phi(t,y)\in\Lambda,\ \text{for all }t\in\mathbb{R}.$

(2) $x,y\in\overline{O(\widetilde{x})}$. Note that there exist
$\\{m_{s}\\},\\{n_{s}\\}\subset\mathbb{Z}$ such that
$\phi(m_{s}T,\widetilde{x})\rightarrow x,\
\phi(n_{s}T,\widetilde{x})\rightarrow y$,then exists $s_{0}>0$ such that

$\displaystyle\sigma(\phi(n_{0}T,\phi(m_{s_{0}}T,\widetilde{x}))-\phi(n_{0}T,\phi(n_{s_{0}}T,\widetilde{x})))=C_{1},$
$\displaystyle\sigma(\phi(-n_{0}T,\phi(m_{s_{0}}T,\widetilde{x}))-\phi(-n_{0}T,\phi(n_{s_{0}}T,\widetilde{x})))=C_{2}.$

Let $x^{*}=\phi(m_{s_{0}}T,\widetilde{x}),\
y^{*}=\phi(n_{s_{0}}T,\widetilde{x}).$ According to (1), we know that
$\sigma(\phi(t,x^{*})-\phi(t,y^{*}))=C$, therefore,

$C_{1}=C_{2}=C.$

Again by Corollary 3.2(iii)

$\phi(t,x)-\phi(t,y)\in\Lambda,\ \text{for all }t\in\mathbb{R}.$

The proof of Proposition 4.3 is completed. ∎

###### Proof of Proposition 4.4.

Assume that there exist
$x^{{}^{\prime}},y^{{}^{\prime}}\in\omega^{p}(x^{0})(x^{{}^{\prime}}\neq
y^{{}^{\prime}})$ but $x^{{}^{\prime}}\notin\alpha^{p}(x^{{}^{\prime}})$ (the
proof of $y^{\prime}\notin\alpha^{p}(y^{\prime})$ is similar), and exists
$t_{1}\in\mathbb{R}$ such that $\phi(t_{1},x)-\phi(t_{1},y)\notin\Lambda$.
Without loss of generality, take $t_{1}=0$. Based on Corollary 3.2(v), let
$n_{0}\in\mathbb{N}$ be such that $n_{0}T>t_{0}$, then

$\phi(n_{0}T,x^{{}^{\prime}})-\phi(n_{0}T,y^{{}^{\prime}}),\quad\phi(-n_{0}T,x^{{}^{\prime}})-\phi(-n_{0}T,y^{{}^{\prime}})\in\Lambda,$

and

$\sigma(\phi(-n_{0}T,x^{{}^{\prime}})-\phi(-n_{0}T,y^{{}^{\prime}}))>\sigma(\phi(n_{0}T,x^{{}^{\prime}})-\phi(n_{0}T,y^{{}^{\prime}})).$

Let $x=\phi(-n_{0}T,x^{{}^{\prime}}),y=\phi(-n_{0}T,y^{{}^{\prime}})$, we have

$\begin{split}x-y,\ \phi(2n_{0}T,x)-\phi(2n_{0}T,y)\in\Lambda,\\\
\sigma(x-y)>\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).\end{split}$ (4.3)

Moreover, $x\in O(x^{{}^{\prime}})$, and thus, $x\notin\alpha^{p}(x)$.

Due to $x-y\in\Lambda$, and $\alpha^{p}(x)\subset\overline{O(x)},\
\alpha^{p}(x)\times\\{x\\}\subset\overline{O(x)}\times\overline{O(x)}$ are
compact set, there are open sets $U,V,U_{1},\cdots,U_{k}$ with $x\in U,\ y\in
V,\ \alpha^{p}(x)\subset\bigcup_{i=1}^{k}U_{i}$, such that

$\overline{x}-\overline{y}\in\Lambda\quad\text{for all
}(\overline{x},\overline{y})\in U_{i}\times U\,\text{or }U\times V.$ (4.4)

Note that $\phi(2n_{0}T,x)-\phi(2n_{0}T,y)\in\Lambda$, choose $U,V$ smaller if
necessary, then

$\displaystyle\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y))=\sigma(\phi(2n_{0}T,U)-\phi(2n_{0}T,V)),$
(4.5)

where $\phi(2n_{0}T,U)-\phi(2n_{0}T,V)$ denotes the subtraction of elements in
sets $\phi(2n_{0}T,U)$ and $\phi(2n_{0}T,V)$.

Recall that $x\in\omega^{p}(x^{0})$, then $U\cap O^{+}(x^{0})\neq\varnothing$.
Take $x^{*}\in U\cap O^{+}(x^{0})$, it then follows from
$\alpha^{p}(x)\subset\bigcup_{i=1}^{k}U_{i},\
y\in\omega^{p}(x^{0})=\omega^{p}(x^{*})$, that there exists $n_{1}>n_{0}$ such
that

$\phi(n_{1}T,x^{*})\in V,\phi(-n_{1}T,x)\in U_{i},\ 1\leq i\leq k.$

Therefore,

$\sigma(\phi(-n_{1}T,x)-x)=\sigma(\phi(-n_{1}T,x)-x^{*}),\
\sigma(x-\phi(n_{1}T,x^{*}))=\sigma(x-y).$

Combining Corollary 3.2(ii),(iii),

$\sigma(\phi(-n_{1}T,x)-x)\geq\sigma(x-y).$ (4.6)

Note also that $(x^{*},\phi(n_{1}T,x^{*}))\in U\times V$,

$\sigma(\phi(2n_{0}T,x^{*})-\phi((2n_{0}+n_{1})T,x^{*}))=\sigma(\phi(2n_{0}T,U)-\phi(2n_{0}T,V)).$

Combining (4.4)-(4.5),

$\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y))=\sigma(\phi(2n_{0}T,x^{*})-\phi((2n_{0}+n_{1})T,x^{*})).$

then according to (4.3) we have

$\sigma(x-y)>\sigma(\phi(2n_{0}T,x^{*})-\phi((2n_{0}+n_{1})T,x^{*})).$ (4.7)

Note that $x\in\omega^{p}(x^{*})$, there exists a sequence
$\\{l_{k}\\}\subset\mathbb{N}$, such that $\phi(l_{k}T,x^{*})\rightarrow x$
and

$\phi(l_{k}T,x^{*})-\phi((n_{1}+l_{k})T,x^{*})\in\Lambda.$

That is, there also exists $L>2n_{0}+n_{1}$, such that

$\sigma(\phi(LT,x^{*})-\phi((n_{1}+L)T,x^{*}))=\sigma(x-\phi(n_{1}T,x)).$

It then follows by Corollary 3.2(ii),(iii) and (4.7) that

$\sigma(x-y)>\sigma(x-\phi(n_{1}T,x)).$

Furthermore, by (4.6),

$\sigma(\phi(-n_{1}T,x)-x)>\sigma(x-\phi(n_{1}T,x)),$

which contradicts to the case $\widetilde{x}=x$ in Proposition 4.3. The proof
is completed.

∎

To show Proposition 4.5, we give the following lemma for preparation.

######  Lemma 4.6.

Let
$x,y\in\omega^{p}(x^{0}),x\in\omega^{p}(x),y\in\omega^{p}(y),x\notin\omega^{p}(y),y\notin\omega^{p}(x)$
and $x-y\in\Lambda$, then there exist open sets $U,V$ of $x,y$ and a natural
number $N$, such that

$\displaystyle\sigma(\phi(0^{+},x^{*})-\phi(0^{+},y^{*}))$
$\displaystyle\geq\sigma(\phi((mT)^{+},x)-\phi((mT)^{+},y^{*})),$ (4.8)
$\displaystyle\sigma(\phi(0^{+},x^{{}^{\prime}})-\phi(0^{+},y^{{}^{\prime}}))$
$\displaystyle\geq\sigma(\phi((mT)^{+},x^{{}^{\prime}})-\phi((mT)^{+},y)),\forall
m\geq N,$ (4.9)

for all $(x^{*},y^{*})\in
U\times\omega^{p}(y),(x^{{}^{\prime}},y^{{}^{\prime}})\in\omega^{p}(x)\times
V$. Meanwhile, one has

$\overline{x}-\overline{y}\in\Lambda,\quad\sigma(\overline{x}-\overline{y})\equiv
constant,$

for all $(\overline{x},\overline{y})\in U\times V$.

###### Proof.

We only prove that (4.8) is correct, while the proof of (4.9) is similar.

Choose $y^{*}\in\omega^{p}(y)$, then $x\notin\omega^{p}(y)$. According to
Corollary 3.2(v), one can choose $n_{y^{*}}=n_{0}(x,y^{*})\in\mathbb{N}$ with
$n_{y^{*}}T>t_{0}$, such that
$\phi(n_{y^{*}}T,x)-\phi(n_{y^{*}}T,y^{*})\in\Lambda$. Thus, there exist
$U_{y^{*}},V_{y^{*}}$ such that

$\sigma(\phi(n_{y^{*}}T,x)-\phi(n_{y^{*}}T,y^{*}))=\sigma(\phi(n_{y^{*}}T,U_{y^{*}})-\phi(n_{y^{*}}T,V_{y^{*}})),$

where $\phi(n_{y^{*}}T,U_{y^{*}})-\phi(n_{y^{*}}T,V_{y^{*}})$ represents the
subtraction of elements in between $\phi(n_{y^{*}}T,U_{y^{*}})$ and
$\phi(n_{y^{*}}T,V_{y^{*}})$. By virtue of the compactness of $\omega^{p}(y)$,
there exists a finite subset $\\{y^{1},y^{2},\cdots,y^{l}\\}$ of
$\omega^{p}(y)$ such that $V_{y^{i}}(i=1,2,\cdots,l)$ is an open cover of
$\omega^{p}(y)$. Let:

$U=\bigcap\limits_{i=1}^{l}U_{y^{i}},\quad N=\max_{1\leq i\leq l}n_{y^{i}}.$

Then, one has $y^{*}\in V_{y^{i}},i\in\\{1,2,\cdots,l\\}$ for any
$(x^{*},y^{*})\in U\times\omega(y)$. Therefore,

$\displaystyle\sigma(\phi(0^{+},x^{*})-\phi(0^{+},y^{*}))$
$\displaystyle\geq\sigma(\phi((n_{y^{i}}T),x^{*})-\phi((n_{y^{i}}T),y^{*}))$
$\displaystyle=\sigma(\phi(n_{y^{i}}T),x)-\phi((n_{y^{i}}T),y^{*}))$
$\displaystyle\geq\sigma(\phi((mT)^{+},x)-\phi((mT)^{+},y^{*})).$

Similarly, we can find $V$ and probably different $N\in\mathbb{N}$, which
satisfy (4.9) on $\omega^{p}(x)\times V$. By Corollary 3.2(ii),(iii), one can
find $N$ large enough such that both (4.8) and (4.9) hold together.

Observe that $x-y\in\Lambda$, if necessary, smaller $U,V$ can be taken such
that $\sigma(\overline{x}-\overline{y})=constant$ for all
$(\overline{x},\overline{y})\in U\times V$. ∎

We now prove Proposition 4.5.

###### Proof of Proposition 4.5.

Assume that there exist $x^{{}^{\prime}},y^{{}^{\prime}}\in\omega^{p}(x^{0})\
(x^{{}^{\prime}}\neq y^{{}^{\prime}})$ satisfy
$x^{\prime}\in\alpha^{p}(x^{\prime}),y^{\prime}\in\alpha^{p}(y^{\prime})$, and
there is $t_{1}\in\mathbb{R}$, such that
$\phi(t_{1},x^{\prime})-\phi(t_{1},y^{\prime})\notin\Lambda$. Without loss of
generality, we set $t_{1}=0$. By Corollary 3.2(v), one can choose
$n_{0}\in\mathbb{N}$ with $n_{0}T>t_{0}$, such that

$\displaystyle\phi(n_{0}T,x^{{}^{\prime}})-\phi(n_{0}T,y^{{}^{\prime}}),\quad\phi(-n_{0}T,x^{{}^{\prime}})-\phi(-n_{0}T,y^{{}^{\prime}})\in\Lambda,$
$\displaystyle\sigma(\phi(-n_{0}T,x^{{}^{\prime}})-\phi(-n_{0}T,y^{{}^{\prime}}))>\sigma(\phi(n_{0}T,x^{{}^{\prime}})-\phi(n_{0}T,y^{{}^{\prime}})).$

If $x^{{}^{\prime}}\in\alpha^{p}(y^{{}^{\prime}})(similar\ to\
y^{{}^{\prime}}\in\alpha^{p}(x^{{}^{\prime}})$), then
$x^{{}^{\prime}}\in\overline{O(y^{{}^{\prime}})}$, which is a contradiction to
Proposition 4.3. Thus, we may assume that
$x^{{}^{\prime}}\notin\alpha^{p}(y^{{}^{\prime}}),y^{{}^{\prime}}\notin\alpha^{p}(x^{{}^{\prime}})$.
Let $x^{*}=\phi(-n_{0}T,x^{{}^{\prime}}),y^{*}=\phi(-n_{0}T,y^{{}^{\prime}})$,
we have

$\sigma(x^{*}-y^{*})>\sigma(\phi(2n_{0}T,x^{*})-\phi(2n_{0}T,y^{*})),$

Apparently,

$\displaystyle x^{*}-y^{*},\phi(2n_{0}T,x^{*})-\phi(2n_{0}T,y^{*})\in\Lambda,$
$\displaystyle x^{*}\notin\alpha^{p}(y^{*}),y^{*}\notin\alpha^{p}(x^{*}).$

Let $X=\alpha^{p}(x^{*})$ (similar to $\alpha^{p}(y^{*})$), then $X$ is a
$Baire$ space. By Definition 2.5, the continuous mapping $P^{-1}$ is
transitive on $X$. It then follows from Lemma 2.1, that $P$ is also transitive
on $X$. Thus, there exists $x^{{}^{\prime\prime}}\in X=\alpha^{p}(x^{*})$ such
that $\alpha^{p}(x^{*})=\omega^{p}(x^{{}^{\prime\prime}})$; and moreover,
there exists a sequence $\\{n_{k}\\}\subset\mathbb{N}$ such that
$P^{n_{k}}x^{{}^{\prime\prime}}\rightarrow x^{*}$, as $n_{k}\to\infty$. For
$k\gg 1$, let $x=P^{n_{k}}x^{{}^{\prime\prime}}$, then
$\omega^{p}(x)=\alpha^{p}(x^{*})$. Therefore, there exist $x,y$ close enough
to $x^{*},y^{*}$, such that
$x\in\omega^{p}(x),x\notin\omega^{p}(y),y\in\omega^{p}(y),y\notin\omega^{p}(x),x-y\in\Lambda$
and

$\sigma(x-y)>\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).$ (4.10)

We now claim that: there exist appropriate $k\in\mathbb{N}$,
$\\{x^{1},x^{2},\cdots,x^{k}\\}\in\omega^{p}(x)$, and
$\\{y^{1},y^{2},\cdots,y^{k}\\}\\\ \in\omega^{p}(y)$, such that

$\displaystyle\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i}))$
$\displaystyle\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)),\quad 1\leq i\leq
k,$ (4.11) $\displaystyle\sigma(x-y)$ $\displaystyle=\sigma(x^{k}-y^{k}).$
(4.12)

If the claim is correct, set $i=k$ , then

$\sigma(x-y)\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)),$

which is a contradiction to (4.10), Proposition 4.5 is then proved.

We now prove that the claim is true. For this purpose, denote

$\displaystyle\mathcal{M}=$ $\displaystyle\\{m\in\mathbb{N};\
m\geq\max(N,2n_{0}),\ and\ \phi(mT,x)\in U\\},$ $\displaystyle\mathcal{N}=$
$\displaystyle\\{n\in\mathbb{N};\ n\geq\max(N,2n_{0}),\ and\ \phi(nT,y)\in
V\\},$

where $N,U,V$ is as defined in Lemma 4.6.

Note that $x\in\omega^{p}(x),y\in\omega^{p}(y)$, then $\mathcal{M}$ and
$\mathcal{N}$ are infinite sets. As a consequence, there are natural numbers,
$k_{1},k_{2},l_{1},l_{2}$,$m_{1},m_{2},m_{3}$,$n_{1},n_{2},n_{3}$ as in Lemma
2.2; and we define sequences as follows:

$\displaystyle x^{i}=\phi(m_{1}T,x),\quad y^{i}=\phi(im_{1}T,y),\quad 1\leq
i\leq k_{1},$ $\displaystyle x^{k_{1}+j}=\phi(m_{2}T,x),\quad
y^{k_{1}+j}=\phi((jm_{2}+k_{1}m_{1})T,y),\quad 1\leq j\leq k_{2},$
$\displaystyle x^{k_{1}+k_{2}+1}=\phi((n_{3}-k_{1}m_{1}-k_{2}m_{2})T,x),\quad
y^{k_{1}+k_{2}+1}=\phi(n_{3}T,y),$ $\displaystyle
x^{k_{1}+k_{2}+i+1}=\phi((in_{1}+n_{3}-k_{1}m_{1}-k_{2}m_{2})T,x),\quad
y^{k_{1}+k_{2}+i+1}=\phi(n_{1}T,y),\quad 1\leq i\leq l_{1},$ $\displaystyle
x^{k_{1}+k_{2}+l_{1}+j+1}=\phi((jn_{2}+l_{1}n_{1}+n_{3}-k_{1}m_{1}-k_{2}m_{2})T,x),\quad$
$\displaystyle y^{k_{1}+k_{2}+l_{1}+j+1}=\phi(n_{2}T,y),\quad 1\leq j\leq
l_{2}.$

We now prove the following inequation by induction:

$\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i}))\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)),\,1\leq
i\leq k_{1}+k_{2}+l_{1}+l_{2}+1.$ (4.13)

(1) If $i=1$, then

$\displaystyle\sigma(\phi(0^{+},x^{1})-\phi(0^{+},y^{1}))$
$\displaystyle=\sigma(\phi((m_{1}T)^{+},x)-\phi((m_{1}T)^{+},y)$
$\displaystyle\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).$

Then, (4.13) is correct for $i=1$.

(2) Suppose that there is $i\in\\{1,2,\cdots,k_{1}+k_{2}-1\\},$ such that

$\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i}))\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).$

Note that

$\displaystyle x^{i+1}-y^{i+1}$
$\displaystyle=\phi(m_{1}T,x)-\phi(m_{1}T,y^{i}),\quad 1\leq i<k_{1}$
$\displaystyle x^{i+1}-y^{i+1}$
$\displaystyle=\phi(m_{2}T,x)-\phi(m_{2}T,y^{i}),\quad k_{1}\leq
i<k_{1}+k_{2}$

Let $x^{*}=x^{i},y^{*}=y^{i},m=m_{s},s=1,2$, by virtue of (4.8),

$\begin{split}\sigma(\phi(0^{+},x^{i+1})-\phi(0^{+},y^{i+1}))&=\sigma(\phi((m_{s}T)^{+},x)-\phi((m_{s}T)^{+},y^{i}))\\\
&\leq\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i})\\\
&\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).\end{split}$

Therefore,

$\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i}))\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)),\quad
1\leq i\leq k_{1}+k_{2}.$

(3) If $i=k_{1}+k_{2}+1$, then

$\displaystyle
x^{k_{1}+k_{2}+1}-y^{k_{1}+k_{2}+1}=\phi((n_{3}-k_{1}m_{1}-k_{2}m_{2})T,x)-\phi((n_{3}-k_{1}m_{1}-k_{2}m_{2})T,y^{k_{1}+k_{2}}).$

Let
$x^{*}=x^{k_{1}+k_{2}},y^{*}=y^{k_{1}+k_{2}},m=n_{3}-k_{1}m_{1}-k_{2}m_{2}(m_{1}>N)$,
by (4.8)

$\displaystyle\sigma(\phi(0^{+},x^{k_{1}+k_{2}+1})-\phi(0^{+},y^{k_{1}+k_{2}+1}))$
$\displaystyle=\sigma(\phi((mT)^{+},x))-\phi((mT)^{+},y^{k_{1}+k_{2}})$
$\displaystyle\leq\sigma(\phi(0^{+},x^{k_{1}+k_{2}})-\phi(0^{+},y^{k_{1}+k_{2}})$
$\displaystyle\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).$

Thus,

$\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i}))\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)),\quad\
i=k_{1}+k_{2}+1.$

Similarly, as to $1\leq i\leq l_{1}+l_{2}$, let
$x^{{}^{\prime}}=x^{i},y^{{}^{\prime}}=y^{i},m=n_{s},s=1,2$, by (4.9) we have

$\sigma(\phi(0^{+},x^{k_{1}+k_{2}+1+i})-\phi(0^{+},y^{k_{1}+k_{2}+1+i}))\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)).$

In conclusion,

$\sigma(\phi(0^{+},x^{i})-\phi(0^{+},y^{i}))\leq\sigma(\phi(2n_{0}T,x)-\phi(2n_{0}T,y)),\quad
1\leq i\leq k,$

where $k=k_{1}+k_{2}+l_{1}+l_{2}+1.$

By virtue of Lemma 4.6,

$\sigma(\bar{x}-\bar{y})\equiv constant,$

for any $(\bar{x},\bar{y})\in U\times V$. Observe that
$x^{k}=\phi(m_{3}T,x)\in U,y^{k}=\phi(n_{2}T,y)\in V$, (4.12) is correct as
well.

We have already checked that the claim is true. The proof of Proposition 4.5
is completed. ∎

###  4.2 Proof of Theorem 4.2

###### Proof of Theorem 4.2.

Define a mapping $h$ on $\omega^{p}(x^{0})$ by

$\omega^{p}(x^{0})\ni x\mapsto(x_{1},x_{2})\in\mathbb{R}^{2}.$

The mapping is one-to-one, otherwise, there exist
$x^{1},x^{2}\in\omega^{p}(x^{0}),x^{1}\neq x^{2}$, such that

$h(x^{1})=h(x^{2}),$

that is,

$x^{1}-x^{2}=(0,0,\ast,\cdots)\notin\Lambda,$

which is contradictory to Theorem 4.1.

Thus, for any $y\in h(\omega^{p}(x^{0}))\subset\mathbb{R}^{2},$ there is only
one $x\in\omega^{p}(x^{0})$ such that $y=h(x).$ Therefore, one can define

$\tilde{\varphi}(k,y)=\tilde{\varphi}(k,h(x))=h(\phi(kT,x))=(\phi(kT,x)_{1},\phi(kT,x)_{2}),\quad
k\in\mathbb{Z}.$

Let $M=\omega^{p}(x^{0})$, then
$h(M)=h(\omega^{p}(x^{0}))\subset\mathbb{R}^{2}$, we prove that
$(h(M),\mathbb{Z},\tilde{\varphi})$ is a dynamical system.

(1) $\tilde{\varphi}(k,\cdot)|_{h(M)}$ is a continuous function for any
$k\in\mathbb{Z}$.

Suppose on the contrary that $\tilde{\varphi}(k,\cdot)$ is discontinuous at
some $y^{*}\in h(M)$. Then, there exist $x^{*}\in M$ and a pair of sequence
$\\{x^{n}\\}\subset M$, $\\{y^{n}\\}\subset h(M)$ such that

$\displaystyle\tilde{\varphi}(k,y^{n})=h(\phi(kT,x^{n}))=(\phi(kT,x^{n})_{1},\phi(kT,x^{n})_{2}),$
$\displaystyle\tilde{\varphi}(k,y^{*})=h(\phi(kT,x^{*}))=(\phi(kT,x^{*})_{1},\phi(kT,x^{*})_{2}),$

and

$\phi(kT,x^{n})\rightarrow\phi(kT,x^{*}),\quad\tilde{\varphi}(k,y^{n})\nrightarrow
y^{*}.$

Therefore, there exists a subsequence $y^{n_{l}}\rightarrow
y^{{}^{\prime}},y^{{}^{\prime}}\neq y^{*}.$ While, $x^{n_{l}}\to x^{*}$ is
equivalent to that

$\begin{split}&(\phi(kT,x^{n_{l}})_{1},\phi(kT,x^{n_{l}})_{2},\cdots)\to(\phi(kT,x^{*})_{1},\phi(kT,x^{*})_{2},\cdots)\\\
\Rightarrow&\tilde{\varphi}(k,y^{n_{l}})=(\phi(kT,x^{n_{l}})_{1},\phi(kT,x^{n_{l}})_{2})\to(\phi(kT,x^{*})_{1},\phi(kT,x^{*})_{2})\\\
&=\tilde{\varphi}(k,y^{*}),\end{split}$

which is a contradiction to $y^{\prime}\neq y^{*}$.

(2) $\tilde{\varphi}(0,y)=y.$

We have $\tilde{\varphi}(0,y)=\tilde{\varphi}(0,h(x))=h(\phi(0,x))=h(x)=y.$

(3)
$\tilde{\varphi}(n_{1}+n_{2},y)=\tilde{\varphi}(n_{1},\tilde{\varphi}(n_{2},y)).$

Let $\widetilde{x}=\phi(n_{2}T,x)$, then
$\tilde{\varphi}(n_{2},y)=\tilde{\varphi}(n_{2},h(x))=h(\phi(n_{2}T,x))=h(\widetilde{x})$.
Therefore,

$\displaystyle\tilde{\varphi}(n_{1},\tilde{\varphi}(n_{2},y))$
$\displaystyle=\tilde{\varphi}(n_{1},h(\widetilde{x}))$
$\displaystyle=h(\phi(n_{1}T,\widetilde{x}))$
$\displaystyle=(\phi(n_{1}T,\widetilde{x})_{1},\phi(n_{1}T,\widetilde{x})_{2})$
$\displaystyle=(\phi(n_{1}T,\phi(n_{2}T,x))_{1},\phi(n_{1}T,\phi(n_{2}T,x))_{2})$
$\displaystyle=(\phi((n_{1}+n_{2})T,x)_{1},\phi((n_{1}+n_{2})T,x)_{2}),$
$\displaystyle\tilde{\varphi}(n_{1}+n_{2},y)=$ $\displaystyle
h(\phi((n_{1}+n_{2})T,x))$ $\displaystyle=$
$\displaystyle(\phi((n_{1}+n_{2})T,x)_{1},\phi((n_{1}+n_{2})T,x)_{2}),$

that is,
$\tilde{\varphi}(n_{1}+n_{2},y)=\tilde{\varphi}(n_{1},\tilde{\varphi}(n_{2},y))$.

We have checked that $(h(M),\mathbb{Z},\tilde{\varphi})$ is a dynamical
system. By Definition 2.4, $(M,\mathbb{Z},\phi)$ is embedded into
$(h(M),\mathbb{Z},\tilde{\varphi})$.

Thus, Theorem 4.2 is proved. ∎

######  Remark 4.1.

It is easy to see that $h:\omega^{p}(x^{0})\to h(\omega^{p}(x^{0}))$ is a
Lipschitz mapping, by using Floquet theory and constructing nested invariant
cones, the inverse mapping $h^{-1}$ restricted to $h(\omega^{p}(x^{0}))$ is
expected to be also Lipschitz(which means that the embedding is Lipschitz), we
will prove it in the following work.

##  5 Discussions

###  5.1 Transform (1.3)+(1.4) into (1.3)+(1.8)

Consider (1.3)+(1.4) and let

$\begin{split}&\delta_{i}=\left\\{\begin{aligned} 1,\quad\text{ if
}\frac{\partial f_{i}}{\partial x_{i-1}}\geq 0\\\ -1,\quad\text{ if
}\frac{\partial f_{i}}{\partial x_{i-1}}<0\end{aligned}\right.,\quad
i=1,\cdots,n\\\ &\Delta=\delta_{1}\cdots\delta_{n}.\end{split}$ (5.1)

Let $y_{i}=\mu_{i}x_{i}$($i=1,\cdots,n$), where

$\mu_{1}=\delta_{1},\mu_{2}=\delta_{1}\delta_{2},\cdots,\mu_{n}=\delta_{1}\cdots\delta_{n}=\Delta.$

Then,

$\dot{y}_{i}=\mu_{i}f_{i}(\mu_{i-1}x_{i-1},\mu_{i}x_{i},\mu_{i+1}x_{i+1})\triangleq
g_{i}(y_{i-1},y_{i},y_{i+1}).$ (5.2)

Moreover,

  * (i)

if $2\leq i\leq n$,

$\begin{split}&\operatorname{sgn}(\frac{\partial g_{i}}{\partial
y_{i-1}})=\operatorname{sgn}(\mu_{i}\frac{\partial f_{i}}{\partial
f_{i-1}}\mu_{i-1})=\mu^{2}_{i}=1>0,\\\ &\operatorname{sgn}(\frac{\partial
g_{i-1}}{\partial y_{i}})=\operatorname{sgn}(\mu_{i-1}\frac{\partial
f_{i-1}}{\partial f_{i}}\mu_{i})=\mu^{2}_{i}=1>0;\end{split}$

  * (ii)

$i=1$,

$\begin{split}&\operatorname{sgn}(\frac{\partial g_{1}}{\partial
y_{n}})=\operatorname{sgn}(\mu_{1}\delta_{1}\mu_{n})=\Delta=\delta_{1}\cdots\delta_{n},\\\
&\operatorname{sgn}(\frac{\partial g_{n}}{\partial
y_{1}})=\operatorname{sgn}(\mu_{n}\delta_{1}\mu_{1})=\Delta=\delta_{1}\cdots\delta_{n}.\end{split}$

Therefore, then the Jacobi matrix of (5.2) is one of the following forms:

  * (i)

$\Delta=1$,

$J(g)=\left(\begin{array}[]{cccc}*&\geq 0&&>0\\\ >0&*&\geq 0&\\\
&\ddots&\ddots&\geq 0\\\ \geq 0&&>0&*\end{array}\right)$

and it is a positive feedback system;

  * (ii)

$\Delta=-1$,

$J(g)=\left(\begin{array}[]{cccc}*&\geq 0&&<0\\\ >0&*&\geq 0&\\\
&\ddots&\ddots&\geq 0\\\ \leq 0&&>0&*\end{array}\right)$

and it is a negative feedback system.

###  5.2 Dissipative conditions

In this subsection, we impose a dissipative condition on system (1.3)+(1.4) as
the following:

$\noindent{\bf(H)}\quad f_{i}(t,x_{i-1},x_{i},x_{i+1})x_{i}<0,\quad\text{for
any}\ |x_{i}|\geq C,|x_{i\pm 1}|\leq|x_{i}|,\ i=1,\cdots,n,\ \text{and all}\
t\in[0,T],$

where $C>0$ is constant.

Fix $x\in\Omega$ and let $\phi(t,x)$ be the solution of (1.3)+(1.4) with
$\phi(0,x)=x$. Then, similar as the arguments in [42, Lemma 4.3], $\phi(t,x)$
will enter $A=\\{x\in\mathbb{R}^{n}:|x_{i}|\leq C\ \text{for all }i\\}$ at
some finite time and then remains there. And hence, we have

######  Theorem 5.1.

Assume that (H) holds true. Then, for any $x\in\Omega$, the $\omega$-limit set
$\omega^{p}(x)$ of the Poincaré map $P$ of (1.3)+(1.4) at $x$ can be
continuous embedded into a compact subset of $\mathbb{R}^{2}$.

###  5.3 $1$-cooperative and $2$-cooperative systems

As we have known that, the system (1.1) is a $2$-cooperative system
equivalents to that the linear variation system of (1.1) is of the following
form:

  * •

$(-1)a_{1n}(t),(-1)a_{n1}(t)\geq 0$;

  * •

$a_{ij}(t)\geq 0$ for all $i,j$ with $|i-j|=1$;

  * •

$a_{ij}(t)=0$ for all $i,j$ with $1<|i-j|<n-1$.

Moreover, if there exist $i,j\in\\{1,\cdots,n\\}$ such that
$a_{i-1,i}(t)\equiv a_{j,j+1}(t)\equiv 0$, then by adjusting the order of
$x_{1},\cdots,x_{n}$ appropriately, and use a transformation like in Section
5.1, a $2$-cooperative system can be converted into a tridiagonal structure
with its linear variation system in the following form:

  * •

$a_{ij}(t)\geq 0$ for all $i,j$ with $|i-j|=1$;

  * •

$a_{ij}(t)=0$ for all $i,j$ with $1<|i-j|<n-1$.

Particularly, if $a_{ij}(t)>0$ for all $i,j$ with $|i-j|=1$, (1.1) is a
tridiagonal cooperative system as in [32, 35], and then a $1$-cooperative
system.

For instance, consider the following four-dimensional system of Lotka-Volterra
type:

$\displaystyle\dot{x_{1}}$
$\displaystyle=x_{1}[\gamma_{1}-a_{11}x_{1}-a_{14}x_{4}],$ (5.3)
$\displaystyle\dot{x_{2}}$
$\displaystyle=x_{2}[\gamma_{2}+a_{21}x_{1}-a_{22}x_{2}+a_{23}x_{3}],$
$\displaystyle\dot{x_{3}}$
$\displaystyle=x_{3}[\gamma_{3}+a_{32}x_{2}-a_{33}x_{3}+a_{34}x_{4}],$
$\displaystyle\dot{x_{4}}$
$\displaystyle=x_{4}[\gamma_{4}+a_{43}x_{3}-a_{44}x_{4}],$

where $\gamma_{i}>0,\ a_{ij}>0,\ x_{i}(0)\geq 0,\ 1\leq i\leq 4$. Here $x_{i}$
denotes the density of species $i$ at time $t$, $r_{i}$ is the intrinsic
growth rate of species $i$, $a_{ij},i\neq j$ measures the action of species
$j$ on the growth of species $i$, and $a_{ii}$ represents the crowding effect
within species $i$ (see [2]).

If $a_{14}=0$, then (5.3) is a tridiagonal cooperative system. Therefore, any
bounded solutions of the system converge to an equilibrium point (see [32]).
If $a_{14}<0$, then the system is cooperative, which is discussed in the
literatures [12, 13, 18, 29, 33] and Poincaré-Bendixson theorem holds true. If
$a_{14}>0$, the system is neither cooperative nor competitive but a strongly
$2$-cooperative system (see [33, 40]), and Poincaré-Bendixson theorem also
established(see [40]). In [2, Lemma 3.1], Elkhader used Hofbauer’s conclusion
in [18] to show that the solution of system (5.3) is bounded under the
condition $a_{22}(a_{33}a_{44}-a_{43}a_{34})-a_{23}a_{32}a_{44}>0$; moreover,
under further assumptions, boundary equilibria and interior equilibrium can
exist.

Now consider the time period Lotka-Volterra system:

$\displaystyle\dot{x_{1}}$
$\displaystyle=x_{1}[\gamma_{1}(t)-a_{11}(t)x_{1}-a_{14}(t)x_{4}],$ (5.4)
$\displaystyle\dot{x_{2}}$
$\displaystyle=x_{2}[\gamma_{2}(t)+a_{21}(t)x_{1}-a_{22}(t)x_{2}+a_{23}(t)x_{3}],$
$\displaystyle\dot{x_{3}}$
$\displaystyle=x_{3}[\gamma_{3}(t)+a_{32}(t)x_{2}-a_{33}(t)x_{3}+a_{34}(t)x_{4}],$
$\displaystyle\dot{x_{4}}$
$\displaystyle=x_{4}[\gamma_{4}(t)+a_{43}(t)x_{3}-a_{44}(t)x_{4}],$

where $(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{4}_{+}$, $a_{ij}(t)$ and
$\gamma_{i}(t)$ are continuous periodic functions with a minimum positive
period of $T$, and all other assumptions for (5.3) are also hold for (5.4). If
$a_{14}(t)\equiv 0$, then (5.4) is a time $T$-periodic tridiagonal cooperative
system; and hence, any bounded solutions of the system converge to a
$T$-peridoic orbit(see [35]). If $a_{14}(t)>0$, then it is strongly
$2$-cooperative system, and our conclusion Theorem 4.2 can be used here.

There are some interesting questions for (5.4). For instance, whether
conditions similar to $a_{22}(a_{33}a_{44}-a_{43}a_{34})-a_{23}a_{32}a_{44}>0$
can also imply dissipative of (5.4)? Whether there exist boundary or interior
periodic orbits under further assumptions? We plan to answer these questions
in the following work.

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