Exponential trichotomy and global linearization of non-autonomous differential equations

TL;DR

This paper extends the Hartman-Grobman linearization theorem to non-autonomous differential equations with exponential trichotomy, establishing new conditions and properties for the linearization functions, including Hölder continuity and non-periodicity.

Contribution

It introduces the first linearization results for systems with exponential trichotomy, generalizing Palmer's theorem beyond dichotomy and analyzing the properties of the linearization functions.

Findings

Established linearization theorems for systems with exponential trichotomy.

Proved Hölder continuity of the linearization functions and their inverses.

Showed that for periodic systems, the linearization functions lack periodicity or asymptotic periodicity.

Abstract

Hartman-Grobman theorem was initially extended to the non-autonomous cases by Palmer. Usually, dichotomy is an essential condition of Palmer's linearization theorem. Is Palmer's linearization theorem valid for the systems with trichotomy? In this paper, we obtain new versions of the linearization theorem if linear system admits exponential trichotomy on $\mathbb{R}$. { Furthermore, the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems are H\"{o}lder continuous}. In addition, if a system is periodic, we find the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems do not have periodicity or asymptotical periodicity. To the best of our knowledge, this is the first paper studying the linearization with exponential trichotomy.

Equations

$$y^{\prime }=A(t)y+f(t,y).$$

$$x^{\prime }(t)=A(t)x(t)$$

$$x^{\prime }=A(t)x+f(t,x),$$

$$x^{\prime }=A(t)x.$$

$$\begin{array}{r}\{\begin{array}{l}PQ=QP,P+Q-PQ=I,\\ |U(t)P{U}^{-1}(s)|\le \beta e^{-\alpha (t-s)}(0\le s\le t),\\ |U(t)(I-P)U^{-1}(s)|\le \beta e^{-\alpha (s-t)}(t\le s,s\ge 0),\\ |U(t)Q{U}^{-1}(s)|\le \beta e^{-\alpha (s-t)}(t\le s\le 0),\\ |U(t)(I-Q)U^{-1}(s)|\le \beta e^{-\alpha (t-s)}(s\le t,s\le 0),\end{array}\end{array}$$

$$\begin{array}{r}\{\begin{array}{l}|U(t)P{U}^{-1}(s)|\le \beta e^{-\alpha (t-s)}(s\le t),\\ |U(t)Q{U}^{-1}(s)|\le \beta e^{-\alpha (s-t)}(t\le s).\end{array}\end{array}$$

$$x^{\prime }=(-\frac{e^t-e^{-t}}{e^t+e^{-t}})x.$$

$$|x(t)P{x}^{-1}(s)|=\frac{e^s+e^{-s}}{e^t+e^{-t}}\le \frac{e^{t+s}+e^{t-s}}{e^{2t}}\le e^{s-t}+e^{-s-t}\le 2e^{-(t-s)},(t\ge s\ge 0),$$

$$|x(t)Q{x}^{-1}(s)|=\frac{e^s+e^{-s}}{e^t+e^{-t}}\le \frac{e^{s-t}+e^{-s-t}}{e^{-2t}}\le e^{s+t}+e^{-(s-t)}\le 2e^{-(s-t)},(t\le s\le 0).$$

$$\beta =2,\alpha =1.$$

$$\begin{array}{r}G(t,s)=\{\begin{array}{l}U(t)P{U}^{-1}(s),(0\le s\le t),\\ -U(t)(I-P)U^{-1}(s),(t\le s,s\ge 0),\\ -U(t)Q{U}^{-1}(s),(t\le s\le 0),\\ U(t)(I-Q)U^{-1}(s),(s\le t,s\le 0).\end{array}\end{array}$$

$$PQ=P_2+P_3=QP,$$

$$P+Q-PQ=I.$$

$$|U(t)P_1{U}^{-1}(s)|$$

$$|U(t)P_4{U}^{-1}(s)|$$

$$\begin{array}{r}\tilde{G}(t,s)=\{\begin{array}{l}U(t)P_1{U}^{-1}(s),(s\le t),\\ U(t)(P_2+P_3)U^{-1}(s),(0\le s\le t),\\ -U(t)P_4{U}^{-1}(s),(t\le s),\end{array}\end{array}$$

$$\begin{array}{r}\{\begin{array}{l}|f(t,x)|\le \mu ,\\ |f(t,x_1)-f(t,x_2)|\le \gamma |x_1-x_2|,\\ 3\beta \gamma {\alpha }^{-1}<1.\end{array}\end{array}$$

$$m\underset{t\in \mathbb{R}}{\sup }{\int }_{-\infty }^{+\infty }|G(t,s)|{\Delta }_1^q(t,s)ds\le \frac{p}{1+p},$$

$$m\underset{t\in \mathbb{R}}{\sup }{\int }_{-\infty }^{+\infty }|G(t,s)|{\Delta }_2^q(t,s)ds\le \frac{p}{1+p},$$

$$\begin{array}{rl}|\mathcal{H}(t,x)-\mathcal{H}(t,x^{\prime })|& \le (1+p)|x-x^{\prime }{|}^q,\mathrm{if}|x-x^{\prime }|<1,\\ |\mathcal{L}(t,y)-\mathcal{L}(t,y^{\prime })|& \le (1+p)|y-y^{\prime }{|}^q,\mathrm{if}|y-y^{\prime }|<1,\end{array}$$

$$\begin{array}{r}\{\begin{array}{l}|f(t,x)|\le \varphi (t),\\ |f(t,x_1)-f(t,x_2)|\le \psi (t)|x_1-x_2|,\\ {\int }_{-\infty }^{+\infty }\varphi (t)dt<c_1,\\ {\int }_{-\infty }^{+\infty }\psi (t)dt<c_2,\end{array}\end{array}$$

$$\underset{t\in \mathbb{R}}{\sup }{\int }_{-\infty }^{+\infty }|\tilde{G}(t,s)|M(s){\Delta }_1^q(t,s)ds\le \frac{p}{1+p},$$

$$\underset{t\in \mathbb{R}}{\sup }{\int }_{-\infty }^{+\infty }|\tilde{G}(t,s)|M(s){\Delta }_2^q(t,s)ds\le \frac{p}{1+p},$$

$$\begin{array}{r}|\mathcal{H}(t,x)-\mathcal{H}(t,x^{\prime })|\le (1+p)|x-x^{\prime }{|}^q,\mathrm{if}|x-x^{\prime }|<1,\\ |\mathcal{L}(t,y)-\mathcal{L}(t,y^{\prime })|\le (1+p)|y-y^{\prime }{|}^{q^{\prime }},\mathrm{if}|y-y^{\prime }|<1,\end{array}$$

$$\Omega :=\{h(t)|h:\mathbb{R}\to X,|h(t)|\le 3\beta \mu {\alpha }^{-1}\}.$$

$$\begin{array}{rl}& \mathcal{T}{h}_1(t,\eta )={\tilde{h}}_1(t,\eta )\\ =& {\int }_{-\infty }^{0}U(t)(I-Q)U^{-1}(s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta )))ds\\ & +{\int }_0^{t}U(t)P{U}^{-1}(s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta )))ds\\ & -{\int }_t^{+\infty }U(t)(I-P)U^{-1}(s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta )))ds\\ =& {\int }_{-\infty }^{+\infty }G(t,s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta ))ds,\end{array}$$

$$\begin{array}{rl}& |{\tilde{h}}_1(t,\eta )|\\ \le & {\int }_{-\infty }^0\beta e^{-\alpha (t-s)}\mu ds+{\int }_0^t\beta e^{-\alpha (t-s)}\mu ds+{\int }_t^{+\infty }\beta e^{-\alpha (s-t)}\mu ds\\ \le & \beta \mu {\alpha }^{-1}+\beta \mu {\alpha }^{-1}+\beta \mu {\alpha }^{-1}\\ \le & 3\beta \mu {\alpha }^{-1}.\end{array}$$

$$\begin{array}{r}{\tilde{h}}_1^{{}^{\prime }}(t,\eta )=A(t){\tilde{h}}_1(t,\eta )+f(t,\eta +h_1(t,\eta )).\end{array}$$

$$\begin{array}{rl}& |{\tilde{h}}_2(t)-{\tilde{h}}_3(t)|\\ \le & {\int }_{-\infty }^0\beta e^{-\alpha (t-s)}\gamma |h_2(s)-h_3(s)|ds+{\int }_0^t\beta e^{-\alpha (t-s)}\gamma |h_2(s)-h_3(s)|ds\\ & +{\int }_t^{+\infty }\beta e^{-\alpha (t-s)}\gamma |h_2(s)-h_3(s)|ds\\ \le & 3\beta \gamma {\alpha }^{-1}\Vert h_2-h_3\Vert .\end{array}$$

$$\begin{array}{rl}& h_1(t,\eta )\\ =& {\int }_{-\infty }^{0}U(t)(I-Q)U^{-1}(s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta )))ds\\ & +{\int }_0^{t}U(t)P{U}^{-1}(s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta )))ds\\ & -{\int }_t^{+\infty }U(t)(I-P)U^{-1}(s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta )))ds\\ =& {\int }_{-\infty }^{+\infty }G(t,s)f(s,y(s,t,\eta )+h_1(s,y(s,t,\eta ))ds,\end{array}$$

$$\begin{array}{r}x(t,s,x(s,\tau ,\xi ))=x(t,\tau ,\xi ),\\ y(t,s,y(s,\tau ,\eta ))=y(t,\tau ,\eta ).\end{array}$$

$$\begin{array}{r}{h}_1(t,y(t,\tau ,\eta ))={\int }_{-\infty }^{+\infty }G(t,s)f(s,y(s,\tau ,\eta )+h_1(s,y(s,\tau ,\eta ))ds.\end{array}$$

$${\mathcal{H}}_1(t,y(t))=y+h_1(t,y(t)),t\ge 0.$$

Full text

Keywords: Exponential trichotomy; Linearization; Periodic;

MSC 2022: 34D09;34D10

1 Introduction

1.1 Brief history on trichotomy

In 1930, Perron [1] proposed the notion of (uniform or classical) exponential dichotomy. Later, many dichotomies were introduced, such as nonuniform exponential dichotomy (see Barreira and Valls [2, 3]), $(h,k)$-dichotomy (see Naulin and Pinto [4], Fenner and Pinto [5]), algebraic dichotomy (see Lin [6]), $(h,k,\mu,\nu)$-dichotomy (see Zhang et al. [7]) and so on. In 1975, Sacker and Sell [8] proposed the concept of trichotomy for linear differential systems, decomposing $\mathbb{R}^{n}$ into stable, unstable and neutral subspaces. Later, Elaydi and Hajek [9] introduced a stronger notion of trichotomy. Hong, Obaya and Gilet [10] considered the existence of a class of ergodic solutions for some differential equations by using exponential trichotomy. Barreira and Valls [11, 12] showed that the existence of a nonuniform exponential trichotomy under sufficiently small $C^{1}$ perturbations. Popa, Ceausu and Bagdaser [13] considered linear discrete-time systems by generalized exponential trichotomy. Adina and Bogdan [14] study the uniform exponential trichotomy of variational difference equations. In Banach spaces, Kovacs [15] considered three concepts of uniform exponential trichotomy on the half-line in the general framework of evolution operators.

1.2 $C^{0}$ linearization of the differential equations

On the other hand, we are interested in the linearization of the ordinary differential equations. Hartman and Grobman [16, 17] made a basic contribution to the linearization problem for autonomous differential equations (called Hartman-Grobman theorem). Later, Hartman-Grobman theorem are generalized in scalar reaction-diffusion equations, Cahn-Hilliard equation, phase field equations and random dynamical systems (see Lu [18], Bates and Lu [19], Barreira and Valls [20]). Pugh [21] used certain powerful functional analytic skills to obtain another proof way of Hartman-Grobman theorem. In Banach spaces, Hein and Prüss [22] extended Hartman-Grobman theorem to abstract semilinear hyperbolic evolution equations. Reinfelds [23] proved that some specific differential equations are strictly dynamically equivalent. Reinfelds and Sermone [24] gave a linearization result in nonlinear differential equations with impulse effect. For the dynamical equivalence of quasilinear impulsive equations, one can refer to Reinfelds [25, 26], Sermone [28, 27], Reinfelds and $\breve{S}$teinberga [29, 30].

In 1973, Palmer [31] successfully generalized the Hartman-Grobman theorem to non-autonomous differential equations

[TABLE]

In order to weaken the conditions of Palmer’s linearization theorem, some improvements were reported: without exponential dichotomy (see Backes, Dragičević and Palmer [32]), for nonuniform dichotomy (see Barreira and Valls [33, 34, 35, 36]), for generalized dichotomy and ordinary dichotomy (Jiang [37, 38]), for nonuniform contraction (see Castañeda and Huerta [39, 40]), for differential equations with piecewise constant argument (see Zou, Xia and Pinto [41]), for dynamic systems on time scales (Xia et al. [42], Pötzche [43]), for the instantaneous impulsive system (see Fenner and Pinto [44], Xia and Chen [45]), Papaschinopoulos [46], Castañeda, Gonzálze and Robledo [47], Pinto and Robledo [48], for nonuniform $(h,k,\mu,\nu)$-dichotomy with ordinary differential equations (see Zhang, Fan and Zhu [7]), for nonuniform $(h,k,\mu,\nu)$-dichotomy with nonautonomous impulsive differential equations (see Zhang, Chang and Wang [50]), for non-instantaneous impulsive nonautonomous (see Li, Wang and O’Regan [51, 52, 53, 54]), for admissibility and roughness of nonuniform exponential dichotomies (see Zhou and Zhang [55, 56]), Dragičević, Zhang and Zhou [58] (admissibility and nonuniform exponential dichotomies), for generalized exponential dichotomies with invariant manifolds (see Zhang [57]). Above mentioned works are for the $C^{0}$ linearization. Recently, there are some interesting advance in the $C^{1}$ linearization for hyperbolic diffeomorphisms (see e.g Backes and Dragičević [59]; Dragičević, Zhang and Zhang [61, 62]; Zhang, Zhang and Jarczyk [63]; Zhang and Zhang [64, 65]; Zhang, Lu and Zhang [66]).

1.3 Motivation and novelty

Palmer’s linearization theorem [31] needs two essential conditions: (i) the nonlinear term $f$ is bounded and Lipschitzian; (ii) the linear system

[TABLE]

admits exponential dichotomy. In this paper, we pay particular attention to the effect of the exponential trichotomy imposing on the linearization of the non-autonomous ordinary differential equations. Motivated by the works of Palmer [31], Backes, Dragičević and Palmer [32], Elaydi and Hajek’s exponential trichotomy (see Elaydi and Hajek [9]), we give new versions of the linearization theorems based on exponential trichotomy. The main contributions of the present paper is to improve Palmer’s linearization theorem in four aspects:

(I): The linear system admits exponential trichotomy, which is weaker than exponential dichotomy.

(II): We prove that the equivalent functions $\mathscr{H}(t,x)$ and its inverse $\mathscr{L}(t,y)$ are Hölder continuous.

(III): The periodicity of the equivalent function is investigated. We prove that periodicity is not an invariant property under exponential trichotomy. More specially, for the periodic systems, if the linear system admits exponential trichotomy, the equivalent function $\mathscr{H}(t,x)$ and its inverse $\mathscr{L}(t,y)$ are not periodic, asymptotically periodic or almost periodic. While, if the linear system admits exponential dichotomy, its equivalent functions are periodic (see [42]).

(IV): The nonlinear term $f$ could be unbounded or non-Lipschitzian in our second linearization theorem.

1.4 Outline of this paper

The structure of our paper as follows. In Section 2, we give some basic definitions. In Section 3, we give the our theorems. In Section 4, we prove our results. Finally, we give some examples to show our linearization theorems.

2 Statement of main results

Let $(X,|\cdot|)$ be an arbitrary Banach space. $A(t)$ is a $n\times n$ continuous and bounded matrice defined on $\mathbb{R}$ respectively. $f:\mathbb{R}\times X\to X$ is a continuous map.

Consider the systems

[TABLE]

and

[TABLE]

Definition 2.1**.**

[49*]** Suppose that there exists a function $\mathscr{H}:\mathbb{R}\times X\to X$ such that

$(i)$ for each fixed $t$, $\mathscr{H}(t,\cdot)$ is a homeomorphism of $X$ into $X$;

$(ii)$ $|\mathscr{H}(t,x)-x|$ is uniformly bounded with respect to $t$;

$(iii)$ assume that $\mathscr{L}(t,\cdot)=\mathscr{H}^{-1}(t,\cdot)$ also has property $(ii)$;

$(iv)$ if $x(t)$ is a solution of system (3), then $\mathscr{H}(t,x(t))$ is a solution of system (4); and if $y(t)$ is a solution of system (4), then $\mathscr{L}(t,y(t))$ is a solution of system (3).

If such a map $\mathscr{H}_{t}(\mathscr{H}_{t}:=\mathscr{H}(t,\cdot))$ exists, then system (3) is topologically conjugated to system (4) and the transformation $\mathscr{H}(t,x)$ is called an equivalent function.*

Definition 2.2**.**

[9]** Linear system (4) is said to possess an exponential trichotomy, if there exists projections $P$, $Q$ and constants $\beta\geq 1$, $\alpha>0$ such that

[TABLE]

hold; here $U(t)$ is a fundamental matrix of the linear system (4).

Remark 2.3**.**

If we take $P=I-Q$ in the Definition 2.2, then (5) becomes

[TABLE]

We obtain an exponential dichotomy on $\mathbb{R}$.

Remark 2.4**.**

The first inequality of (5) can be divided into the first and the fourth inequalities of (5). The second inequality of (5) can be divided into the second and the third inequalities of (5). Thus, it is always true that exponential dichotomy on $\mathbb{R}$ implies exponential trichotomy. However, the converse is clearly false as it may be shown by simple example.

Next example shows that the linear system admits an exponential trichotomy, but it does not admit an exponential dichotomy.

Example 2.5**.**

[9*]** *Consider the scalar equation

[TABLE]

Then $x(t)=\frac{2}{e^{t}+e^{-t}}x_{0}$ is the solution of equation (7) with $x(0)=x_{0}$. Now we take $P=I$, $Q=I$. Obviously, $I-P=0$, $I-Q=0$ , $PQ=QP$, $P+Q-QP=I$.

[TABLE]

The last two inequalities of definition 2.2 obviously hold in this case. This implies that equation (7) admits an exponential trichotomy with

[TABLE]

However, equation (7) doesn’t satisfy exponential dichotomy.

Exponential trichotomy in Definition 2.2 has the Green function:

[TABLE]

Now we consider exponential trichotomy by decomposition of fundamental matrix $U(t)$. We assume that $U(t)=(\varphi_{1}(t),\varphi_{2}(t),...,\varphi_{n}(t))$ is a fundamental matrix of the linear system $x^{\prime}=A(t)x$. $\varphi_{i}(t)$ is bounded on $\mathbb{R}$ for $i=1,...,r-k,r+l,...,n$, $\varphi_{j}(t)$ is unbounded on $t\leq k$ (and bounded on $t\geq k$) for $j=r-k+1,...,r+l-1$. Then, we chose projection $P=P_{1}+P_{2}+P_{3}=diag\{I_{r-k},0\}+diag\{0_{r-k},I_{r},0\}+diag\{0_{r},I_{r+l-1},0\}$ and $Q=P_{2}+P_{3}+P_{4}=diag\{0_{r-k},I_{r},0\}+diag\{0_{r},I_{r+l-1},0\}+diag\{0_{r-l+1},I_{n}\}.$ Furthermore,

[TABLE]

We verify projections $P,Q$ satisfy the first inequality of Definition 2.2. Then $|U(t)P_{1}U^{-1}(s)|$ and $|U(t)P_{4}U^{-1}(s)|$ are bounded on $\mathbb{R}$. $|U(t)(P_{2}+P_{3})U^{-1}(s)|$ is unbounded on $t\leq s$ (and bounded on $t\geq s$). We denote that

[TABLE]

Then we obtain another Green function $\tilde{G}(t,s)$,

[TABLE]

which appears in the statement of Theorem 3.6. We will study linearization based on these two Green functions.

3 Main results

Theorem 3.1**.**

Suppose that linear system (4) admits an exponential trichotomy (8) (that is, fundamental matrix $U(t)$ satisfying (5)) and $f(t,x)$ satisfies

[TABLE]

Then, nonlinear system (3) is topologically conjugated to its linear system (4).

Remark 3.2**.**

In Theorem 3.1, if exponential trichotomy reduce to exponential dichotomy, Theorem 3.1 still holds. Indeed, that is Palmer’s linearization theorem, (see Palmer [31]).

In what follows, we introduce an assumption motivated by Backes, Dragičević and Palmer [32].

Condition (I): suppose that there is a continuous function $\Delta_{1}(t,s)>0$ such that if $z_{1}(t)$ and $z_{2}(t)$ are the solution of system (4), then $|z_{1}(t)-z_{2}(t)|\leq\Delta_{1}(t,s)|z_{1}(s)-z_{2}(s)|$; there is another continuous function $\Delta_{2}(t,s)>0$ such that if $z_{3}(t)$ and $z_{4}(t)$ are the solution of system (3), then $|z_{3}(t)-z_{4}(t)|\leq\Delta_{2}(t,s)|z_{3}(s)-z_{4}(s)|$ ($\Delta_{1}(t,s)>0$ and $\Delta_{2}(t,s)>0$ are continuous functions).

Remark 3.3**.**

Condition (I) is valid, one can refer to Appendix A in [32] for the detail of functions $\Delta_{1}(t,s)$ and $\Delta_{2}(t,s)$.

Theorem 3.4**.**

Suppose that the conditions in Theorem 3.1 and condition ($I$) are satisfied. Let $p>0$ and $0<q<1$ such that

[TABLE]

where $m=\max\{\gamma,2\mu^{1-q}\gamma^{q}\}$. Then

[TABLE]

where $\mathscr{L}(t,\cdot)=\mathscr{H}^{-1}(t,\cdot)$ and $p$ is a positive constant.

Theorem 3.5**.**

Suppose that linear system (4) admits an exponential trichotomy (11) (that is, fundamental matrix $U(t)$ satisfying (5)) and $f(t,x)$ satisfies

[TABLE]

where $\phi,\psi:\mathbb{R}\to[0,\infty)$ are integrable functions; $c_{1},c_{2}$ are positive constant. Then, nonlinear system (3) is topologically conjugated to its linear system (4).

Theorem 3.6**.**

Suppose that the conditions in Theorem 3.5 and condition (I) are satisfied.

[TABLE]

where $M(t)=\max\{\psi(t),2\phi^{1-q}(t)\psi^{q}(t)\}$ and $q$ is in (0,1). Then

[TABLE]

where $\mathscr{L}(t,\cdot)=\mathscr{H}^{-1}(t,\cdot)$ and $p$ is a positive constant.

Theorem 3.7**.**

In system (3), assuming that $A(t)$ and $f(t,x)$ has $T$ period with respect to $t$. Then the equivalent function $\mathscr{H}(t,x)$ and its inverse $\mathscr{L}(t,y)$ in Theorem 3.1 and Theorem 3.5 do not have periodicity, asymptotical periodicity or almost periodicity with respect to t.

4 Proofs of main results

Let $x(t,t_{0},x_{0})$ be the solution of system (3) satisfies the initial value condition $x(t_{0})=x_{0}$, $y(t,t_{0},y_{0})$ be the solution of system (4) satisfies the initial value condition $y(t_{0})=y_{0}$.

Proof of Theorem 3.1.

Step 1. We prove the case of $t\geq 0$. Let $\Omega$ denote the space of all continuous maps $h$ satisfies

[TABLE]

Then, $(\Omega,\|\cdot\|)$ is a Banach space. For $t\geq 0$, $\xi,\eta\in X$ and $h_{1}\in\Omega$, we define the following mapping:

[TABLE]

where $G(t,s)$ is defined in (8). It follows from (5), we obtain

[TABLE]

Moreover, by differentiating (16), we get

[TABLE]

Hence, from (17) and (18), we know $\tilde{h}_{1}(t,\eta)$ is continuous and $\tilde{h}_{1}(t,\eta)\in\Omega$. For any $h_{2},h_{3}\in\Omega$, from (12) and (16), we have

[TABLE]

Note that $3\beta\gamma\alpha^{-1}<1$. Thus, $\mathscr{T}$ : $\Omega\to\Omega$ is a contraction map. Therefore, there exists a unique fixed point $h_{1}\in\Omega$ such that

[TABLE]

where $G(t,s)$ is defined in (8). Using the identities, we have

[TABLE]

Then, if $y(t)$ is a solution of system (4), we have

[TABLE]

Taking

[TABLE]

By direct differentiation (23), we conclude that

[TABLE]

The above proof implies that if $y(t)$ is a solution of (4), then $\mathscr{H}_{1}(t,y(t))$ is a solution of (3). Next, we show that the existence of $\mathscr{L}_{1}$. Set

[TABLE]

Similarly to $\tilde{h}_{1}$, we can easily prove that $l_{1}\in\Omega$. From (21), if $x(t)$ is a solution of system (3), we have

[TABLE]

Taking

[TABLE]

By direct differentiation (27), we have

[TABLE]

The above proof implies that if $x(t)$ is a solution of (3), then $\mathscr{L}_{1}(t,x(t))$ is a solution of (4).

Next we prove $\mathscr{H}_{1}(t,\mathscr{L}_{1}(t,x))=x$ and $\mathscr{L}_{1}(t,\mathscr{H}_{1}(t,y))=y$, for $t\geq 0$. Let $x(t)$ be any solution of system (3). Then we know that $\mathscr{L}_{1}(t,x(t))\triangleq w_{1}(t)$ is the solution of system (4) and $\mathscr{H}_{1}(t,\mathscr{L}_{1}(t,x(t)))\triangleq g_{1}(t)$ is the solution of system (3). It follows from (23) and (27),

[TABLE]

Then,

[TABLE]

Therefore, $\mathscr{H}_{1}(t,\mathscr{L}_{1}(t,x))=x$. Let $y(t)$ be any solution of system (4). Then we know that $\mathscr{H}_{1}(t,y(t))\triangleq\hat{g}_{1}(t)$ is the solution of system (3) and $\mathscr{L}_{1}(t,\mathscr{H}_{1}(t,y(t)))\triangleq\hat{w}_{1}(t)$ is the solution of system (4). It follows from (23) and (27),

[TABLE]

Then, $\hat{w}_{1}(t)=y(t)$. Therefore, $\mathscr{L}_{1}(t,\mathscr{H}_{1}(t,y))=y$.

Step 2. We prove if $t<0$, Theorem 3.1 still holds. For $t<0$, $\xi,\eta\in X$ and $v\in\Omega$, we define the following mapping:

[TABLE]

where $G(t,s)$ is defined in (8). It follows from (5), we obtain

[TABLE]

Moreover, by differentiating (32), we get

[TABLE]

Hence, from (32) and (33), we know $\tilde{v}(t,\eta)$ is continuous and $\tilde{v}(t,\eta)\in\Omega$. For any $v_{1},v_{2}\in\Omega$, from (12), we have

[TABLE]

Note that $3\beta\gamma\alpha^{-1}<1$. Thus, $\mathscr{F}$ : $\Omega\to\Omega$ is a contraction map. Therefore, there exists a unique fixed point $v\in\Omega$ such that

[TABLE]

Then, if $y(t)$ is a solution of system (4), we have

[TABLE]

Taking

[TABLE]

By direct differentiation (38), we conclude that

[TABLE]

The above proof implies that if $y(t)$ is a solution of (4), then $\mathscr{H}_{2}(t,y(t))$ is a solution of (3). Next, we construct the function $\mathscr{L}_{2}$. Set

[TABLE]

Similarly to $\tilde{v}$, we can prove that $l_{2}\in\Omega$. From (21), if $x(t)$ is a solution of system (3), we have

[TABLE]

Taking

[TABLE]

By direct differentiation, we get

[TABLE]

The above proof implies that if $x(t)$ is a solution of (3), then $\mathscr{L}_{2}(t,x(t))$ is a solution of (4).

Next we prove $\mathscr{H}_{2}(t,\mathscr{L}_{2}(t,x))=x$ and $\mathscr{L}_{2}(t,\mathscr{H}_{2}(t,y))=y$, for $t<0$. Let $x(t)$ be any solution of system (3). Then we know that $\mathscr{L}_{2}(t,x(t))\triangleq w_{2}(t)$ is the solution of system (4) and $\mathscr{H}_{2}(t,\mathscr{L}_{2}(t,x(t)))\triangleq g_{2}(t)$ is the solution of system (3). From (38) and (42),

[TABLE]

Then,

[TABLE]

Therefore, $\mathscr{H}_{2}(t,\mathscr{L}_{2}(t,x))=x$. Let $y(t)$ be any solution of system (4). Then we know that $\mathscr{H}_{2}(t,y(t))\triangleq\hat{g}_{2}(t)$ is the solution of system (3) and $\mathscr{L}_{2}(t,\mathscr{H}_{2}(t,y(t)))\triangleq\hat{w}_{2}(t)$ is the solution of system (4). From (23) and (27),

[TABLE]

Then, $\hat{w}_{2}(t)=y(t)$. Therefore, $\mathscr{L}_{2}(t,\mathscr{H}_{2}(t,y))=y$. We have proved that the Theorem 3.1 holds for $t<0$.

Step 3. At last, we prove that if $t\to 0$, we have $H_{2}(t,y(t))\to H_{1}(0,y(0))$. Recall that if $t=0$, we obtain

[TABLE]

Now note that $2\beta\gamma\alpha^{-1}<1$, we have $h_{1}=v$. Thus, if $t=0$,

[TABLE]

if $t\to 0$,

[TABLE]

Hence, if $t\to 0$, we have $H_{2}(t,y(t))\to H_{1}(0,y(0))$. Therefore, the proof of Theorem 3.1 is completed.

**Proof of Theorem 3.4.

Step 1.** For $t\geq 0$, we prove the equivalent function $\mathscr{H}_{1}$ is Hölder continuous. Suppose that constants $p>0$ and $0<q<1$. From (12), we can obtain

[TABLE]

where $x_{1},x_{2}\in X$. Let $\hat{\Omega}$ denote the space of all continuous maps $\varpi$ satisfies

[TABLE]

for $t\geq 0$, $x_{1},x_{2}\in X$. It follows from (16), we get

[TABLE]

where $\varpi\in\hat{\Omega}$. By using (12) and (47)

[TABLE]

where $m=\max\{\gamma,2\mu^{1-q}\gamma^{q}\}$. Furthermore, by using (12) and(49), we obtain

[TABLE]

for $t\geq 0$, $\eta_{1},\eta_{2}\in X$. Therefore, ${\mathscr{T}}\varpi\in\hat{\Omega}$. Thus, the unique fixed point $h_{1}$ of ${\mathscr{T}}$ belongs to $\hat{\Omega}$. From (23) and (50), we get

[TABLE]

where $\tilde{m}_{1}=1+p$, $0<|y_{1}-y_{2}|<1$. Therefore, $\mathscr{H}_{1}$ is Hölder continuous. Next, we prove $\mathscr{L}_{1}$ is also Hölder continuous. From (25), we get

[TABLE]

Then,

[TABLE]

It follows from (25) and (47),

[TABLE]

Then,

[TABLE]

where $\tilde{m}_{2}=1+p$, $0<|x_{1}-x_{2}|<1$. Hence, for $t\geq 0$, we prove $\mathscr{H}_{1}$ and $\mathscr{L}_{1}$ are Hölder continuous.

Step 2. Similarly to the above proof. For $t<0$, we can easily prove the equivalent function $\mathscr{H}_{2}$ and $\mathscr{L}_{2}$ are Hölder continuous.

**Proof of Theorem 3.5.

**Let $\Lambda$ denote the space of all continuous maps $\varrho$ satisfies

[TABLE]

Then, $(\Lambda,\|\cdot\|)$ is a Banach space. For $t\in\mathbb{R}$, $\xi,\eta\in X$ and $\varrho_{1}\in\Lambda$, we define the following mapping:

[TABLE]

where $\tilde{G}(t,s)$ is defined in (11). From (9) and (10), we obtain

[TABLE]

From (5) and (9), we obtain

[TABLE]

It follows from (55) and (56), we get

[TABLE]

Moreover, by differentiating (54), we get

[TABLE]

Hence, from (57) and (58), we know $\tilde{\varrho}_{1}(t,\eta)$ is continuous and $\tilde{\varrho}_{1}(t,\eta)\in\Lambda$. For any $\varrho_{2},\varrho_{3}\in\Lambda$, from (14) and (54), we have

[TABLE]

Note that $(\beta+2\kappa_{1}+\kappa_{2})c_{2}<1$. Thus, $\mathscr{R}$ : $\Lambda\to\Lambda$ is a contraction map. Therefore, there exists a unique fixed point $\varrho_{1}\in\Lambda$ such that

[TABLE]

where $\tilde{G}(t,s)$ is defined in (11). By using identities (21), if $y(t)$ is a solution of system (4), we have

[TABLE]

where $\tilde{G}(t,s)$ is defined in (11). Taking

[TABLE]

By direct differentiation (61), we conclude that

[TABLE]

The above proof implies that if $y(t)$ is a solution of (4), then $\mathscr{H}_{3}(t,y(t))$ is a solution of (3). Next, we show that the existence of $\mathscr{L}_{3}$. Set

[TABLE]

Similarly to $\tilde{\varrho}_{1}$, we can easily prove that $\rho_{1}\in\Lambda$. From (21), if $x(t)$ is a solution of system (3), we have

[TABLE]

Taking

[TABLE]

By direct differentiation (27), we have

[TABLE]

The above proof implies that if $x(t)$ is a solution of (3), then $\mathscr{L}_{3}(t,x(t))$ is a solution of (4).

Next we prove $\mathscr{H}_{3}(t,\mathscr{L}_{3}(t,x))=x$ and $\mathscr{L}_{3}(t,\mathscr{H}_{3}(t,y))=y$, for $t\in\mathbb{R}$. Let $x(t)$ be any solution of system (3). Then we know that $\mathscr{L}_{3}(t,x(t))\triangleq\omega_{1}(t)$ is the solution of system (4) and $\mathscr{H}_{3}(t,\mathscr{L}_{3}(t,x(t)))\triangleq\omega_{2}(t)$ is the solution of system (3).

[TABLE]

Then,

[TABLE]

Therefore, $\omega_{2}(t)=x(t)$. Let $y(t)$ be any solution of system (4). Then we know that $\mathscr{H}_{3}(t,y(t))\triangleq\hat{\omega}_{1}(t)$ is the solution of system (3) and $\mathscr{L}_{3}(t,\mathscr{H}_{3}(t,y(t)))\triangleq\hat{\omega}_{2}(t)$ is the solution of system (4).

[TABLE]

Then, $\hat{\omega}_{2}(t)=y(t)$. Therefore, $\mathscr{L}_{3}(t,\mathscr{H}_{3}(t,y))=y$. Hence, the proof of Theorem 3.5 is completed.

**Proof of Theorem 3.6.

Step 1.**We prove the equivalent function $\mathscr{H}_{1}$ is Hölder continuous. Suppose that constants $c\geq 1$, $p>0$ and $0<q<1$. From (14), we can obtain

[TABLE]

where $x_{1},x_{2}\in X$. Let $\hat{\Lambda}$ denote the space of all continuous maps $\chi$ satisfies

[TABLE]

for $t\in\mathbb{R}$, $x_{1},x_{2}\in X$. It follows from (54), we get

[TABLE]

where $\tilde{G}(t,s)$ is defined in (11). By using (14) and (70)

[TABLE]

Furthermore, by using (14) and (72), we obtain

[TABLE]

for $t\in\mathbb{R}$, $\eta_{1},\eta_{2}\in X$, where $\tilde{G}(t,s)$ is defined in (11). Therefore, ${\mathscr{R}}\chi\in\hat{\Lambda}$. Thus, the unique fixed point $\varrho_{1}$ of ${\mathscr{R}}$ belongs to $\hat{\Lambda}$. From (23) and (50), we get

[TABLE]

where $\tilde{c}=1+p$, $0<|y_{1}-y_{2}|<1$. Therefore, $\mathscr{H}_{3}$ is Hölder continuous. Next, we prove $\mathscr{L}_{3}$ is also Hölder continuous. From (63), we get

[TABLE]

where $\tilde{G}(t,s)$ is defined in (11). Then,

[TABLE]

It follows from (63) and (70),

[TABLE]

Then,

[TABLE]

where $\tilde{c}=1+p$, $0<|x_{1}-x_{2}|<1$. Hence, we prove $\mathscr{H}_{3}$ and $\mathscr{L}_{3}$ are Hölder continuous.

Proof of Theorem 3.7. Firstly, we prove some lemmas, consider periodic system

[TABLE]

where $\sigma(t+T,x)=\sigma(t,x)$. Systems (77) satisfy the existence and uniqueness of the solution. Suppose that $\bar{X}(t,(t_{0},x_{0}))$ is the solution of system (77) satisfying

[TABLE]

Lemma 4.1**.**

For any $t,s\in\mathbb{R}$, $x\in\mathbb{R}^{n}$, we have

[TABLE]

Proof. From variation formula, we have

[TABLE]

Then,

[TABLE]

Denote $F_{1}(t)=\bar{X}(t+T,(s+T,x))$. By (79), we know $F_{1}(t)$ is the solution of system (77). Since $F_{1}(s)=x$, $\bar{X}(s,(s,x))=x$, according to the existence and uniqueness of the solution, we get $\bar{X}(t+T,(s+T,x))=\bar{X}(t,(s,x))$.

Lemma 4.2**.**

Suppose that periodic system $x^{\prime}=A(t)x$ $(A(t+T)=A(t))$ have an exponential trichotomy ($U(t)$ is the fundamental matrix of system (4) satisfying (5)). For any $t,s\in\mathbb{R}$, we have

[TABLE]

Proof. $U(t)$ is the fundamental matrix of linear system $x^{\prime}=A(t)x$. It’s easy to get $U(t+T)$ is also the fundamental matrix of linear system $x^{\prime}=A(t)x$. Thus, there exists an invertible matrix $D$ such that $U(t+T)=U(t)D$. Taking $B=\frac{1}{T}\ln D$ and $M(t)=U(t)e^{-Bt}$. Then,

[TABLE]

From (80), we get

[TABLE]

Similar to the above proof, we can easily get $U(t+T)(P_{2}+P_{3})U^{-1}(s+T)=U(t)(P_{2}+P_{3})U^{-1}(s)$ and $U(t+T)P_{4}U^{-1}(s+T)=U(t)P_{4}U^{-1}(s)$. Thus, $\tilde{G}(t+T,s+T)=\tilde{G}(t,s)$. Similar to $\tilde{G}$, we get $G(t+T,s+T)=G(t,s)$.

Now we start to prove the periodicity or asymptotically periodic of $\mathscr{H}(t,x)$ and $\mathscr{L}(t,y)$. From (61), (65), Lemma 4.1 and Lemma 4.2, we obtain,

[TABLE]

In addition,

[TABLE]

Thus, the equivalent function $\mathscr{H}_{3}$ and its inverse $\mathscr{L}_{3}$ in theorem 3.5 do not have periodicity.

Next we prove the equivalent function $\mathscr{H}_{3}$ and its inverse $\mathscr{L}_{3}$ in theorem 3.5 do not have asymptotical periodicity. Since

[TABLE]

and

[TABLE]

Thus, $\mathscr{H}_{3}$ do not have asymptotical periodicity. The proof of $\mathscr{L}_{3}$ is similar to $\mathscr{H}_{3}$, we omit.

Next, we prove equivalent function $\mathscr{H}_{1}$, $\mathscr{H}_{2}$, $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ in theorem 3.1 do not have periodicity or asymptotical periodicity.

[TABLE]

and

[TABLE]

and

[TABLE]

Hence, $\mathscr{H}_{1}$ do not have periodicity or asymptotically periodicity. The almost periodicity is similar to the above proof, we omit. The proof of $\mathscr{H}_{2}$, $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ are similar to $\mathscr{H}_{1}$, we omit.

5 Some example

Example 5.1**.**

Consider the scalar equations

[TABLE]

and

[TABLE]

where $\delta$ is a positive constant.

From example 2.5, we know that equation (83) admits an exponential trichotomy with $\beta=2,\alpha=1$. Let, $f(t,x)=\delta\sin t\sin x(t)$, then

[TABLE]

Hence, equation (82) satisfies the condition of Theorem 3.1 if $\delta<\frac{1}{6}$. Therefore, equation (81) is topologically conjugated to equation (82).

Example 5.2**.**

Consider the scalar equations

[TABLE]

and

[TABLE]

where $\epsilon$ is a sufficiently small positive constant.

From example 2.5, we know that equation (83) admits an exponential trichotomy with $\beta=2,\alpha=1$. Then $U(t)=\frac{2}{e^{t}+e^{-t}}$ is a solution of (83). Taking $P=1$, we get $|U(t)PU^{-1}(s)|<1$ $(t\geq s)$. Apart from this, we have $|\displaystyle\int_{-\infty}^{+\infty}\frac{\epsilon}{1+t^{2}}\sin xdt|<\displaystyle\int_{-\infty}^{+\infty}\frac{\epsilon}{1+t^{2}}dt<\epsilon\pi$. Furthmore,

[TABLE]

Hence, equation (84) satisfies the condition of Theorem 3.5 if $\epsilon<\frac{1}{4\pi}$. Therefore, equation (83) is topologically conjugated to equation (84).

Data Availability Statement

No data was used for the research in this article. It is pure mathematics.

Conflict of Interest

The authors declare that they have no conflict of interest.

Contributions

We declare that all the authors have same contributions to this paper.

Ethical Approval

Not applicable.

Funding

This paper was jointly supported from the Natural Science Foundation of Zhejiang Province (No. LZ23A010001), National Natural Science Foundation of China under Grant (No. 11671176, 11931016), Grant Fondecyt (No. 1170466), Fondecyt 038-2021-Perú.