Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautomous Henon Map

TL;DR

This paper extends the theory of chaotic dynamics to nonautonomous two-dimensional maps, providing new conditions for chaos and hyperbolicity, exemplified through the nonautonomous Hénon map.

Contribution

It introduces a nonautonomous version of Conley-Moser conditions and derives a new sufficient condition for hyperbolic chaotic invariant sets in nonautonomous maps.

Findings

Established a precise definition of chaotic invariant sets for nonautonomous maps.

Derived new sufficient conditions for hyperbolic chaos in nonautonomous systems.

Applied the theory to the nonautonomous Hénon map to identify parameter regimes with chaotic behavior.

Abstract

In this paper we analyze chaotic dynamics for two dimensional nonautonomous maps through the use of a nonautonomous version of the Conley-Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley-Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous H\'enon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous H\'enon map to have a hyperbolic chaotic invariant set.

Figures

Equations

$$\begin{array}[]{ccccc}H&:&\quad\mathbb{R}^{2}&\quad\longrightarrow&\mathbb{R}^{2},\\ &&(x,y)&\quad\longmapsto&(A+By-x^{2},x),\end{array}$$

$$\begin{array}[]{ccccc}H^{-1}&:&\quad\mathbb{R}^{2}&\quad\longrightarrow&\mathbb{R}^{2},\\ &&(x,y)&\quad\longmapsto&(y,(x-A+y^{2})/B).\end{array}$$

$$f_n:D_n⟶D_{n+1}\forall n\in \mathbb{Z}\text{and}{f}_n^{-1}:D_{n+1}⟶D_n,$$

$$H_i^{n+1}\subset D_{n+1}\text{with}{f}_n(V_i^n)=H_i^{n+1}\text{ ,}\forall n\in \mathbb{Z}\text{ ,}i\in I.$$

$$A\equiv \{A^n{\}}_{n=-\infty }^{+\infty }\text{ is a sequence of matrices of dimension }N\times N\text{ such that}$$

$$A_{ij}^n=\{\begin{array}{l}1\text{if }{f}_n(V_i^n)\cap V_j^{n+1}\ne \varnothing \\ 0\text{otherwise}\end{array}\text{or equivalently}$$

$$A_{ij}^n=\{\begin{array}{l}1\text{if }{H}_i^{n+1}\cap V_j^{n+1}\ne \varnothing \\ 0\text{otherwise}\end{array}\forall i,j\in I.$$

$$D_x=\{x\in \mathbb{R}|\text{there exists a }y\in \mathbb{R}\text{ with }(x,y)\in D\}$$

$$D_y=\{y\in \mathbb{R}|\text{there exists an }x\in \mathbb{R}\text{ with }(x,y)\in D\}$$

$$|h(x_1)-h(x_2)|\le {\mu }_h|x_1-x_2|$$

$$|v(y_1)-v(y_2)|\le {\mu }_v|y_1-y_2|$$

$$V=\{(x,y)\in {\mathbb{R}}^2|x\in [v_1(y),v_2(y)],y\in I_y\}$$

$$H=\{(x,y)\in {\mathbb{R}}^2|y\in [h_1(x),h_2(x)],x\in I_x\}$$

$$d(H)=\underset{x\in I_x}{\max }|h_2(x)-h_1(x)|,d(V)=\underset{y\in I_y}{\max }|v_2(y)-v_1(y)|$$

$${\partial }_{v}H\equiv \{(x,y)\in H|x\in \partial I_x\}$$

$${\partial }_{h}H\equiv \partial H\setminus {\partial }_{v}H$$

$$f_n(D_n)\cap D_{n+1}\ne \varnothing \text{ ,}\forall n\in \mathbb{Z}$$

$$H_{ij}^{n+1}\equiv f_n(V_i^n)\cap V_j^{n+1}=H_i^{n+1}\cap V_j^{n+1},\text{and}{V}_{ji}^n\equiv f_n^{-1}(V_j^{n+1})\cap V_i^n$$

$$A_{ij}^n=\{\begin{array}{l}1\text{if }{H}_{ij}^{n+1}=H_i^{n+1}\cap V_j^{n+1}\ne \varnothing \\ 0\text{otherwise}\end{array}\forall i,j\in I.$$

$$f_n\left(f_n^{-1}({\partial }_h{H}_{ij}^{n+1})\right)={\partial }_h{H}_{ij}^{n+1}\subset f_n({\partial }_h{V}_i^n).$$

$$d({\tilde{V}}_i^n)\le {\nu }_{v}d(V^{n+1})\text{for some }0<{\nu }_v<1$$

$$d({\tilde{H}}_k^{n+1})\le {\nu }_{h}d(H^n)\text{for some }0<{\nu }_h<1$$

$$s=(\cdots s_{n-k}\cdots s_{n-2}{s}_{n-1}.s_n{s}_{n+1}\cdots s_{n+k}\cdots )$$

$$\sigma (s)=\sigma (\cdots s_{n-2}{s}_{n-1}.s_n{s}_{n+1}\cdots )=(\cdots s_{n-2}{s}_{n-1}{s}_n.s_{n+1}\cdots )$$

$$\tilde{\sigma }(s,n)=(\sigma (s),n+1).\text{It is also defined }f(x,y;n)=(f_n(x,y),n+1).$$

$$\begin{array}[]{ccccc}&&f&&\\ &\Lambda_{n}\times\mathbb{Z}&\longrightarrow&\Lambda_{n+1}\times\mathbb{Z}&\\ \\ \phi&\downarrow&&\downarrow&\phi\\ &&\tilde{\sigma}&&\\ &\Sigma_{\{A^{n}\}}^{N}\times\mathbb{Z}&\longrightarrow&\Sigma_{\{A^{n}\}}^{N}\times\mathbb{Z}&\end{array}$$

$${\mathcal{V}}^n\equiv \bigcup _{i,j\in I}{V}_{ji}^n=\bigcup _{i,j\in I}{f}_n^{-1}(V_j^{n+1})\cap V_i^n,$$

$${\mathcal{H}}^{n+1}\equiv \bigcup _{i,j\in I}{H}_{ij}^{n+1}=\bigcup _{i,j\in I}{H}_i^{n+1}\cap V_j^n,f_n({\mathcal{V}}^n)={\mathcal{H}}^{n+1}$$

$$S_{\mathcal{K}}^u\equiv \{({\xi }_z,{\eta }_z)\in {\mathbb{R}}^2||{\eta }_z|\le {\mu }_h|{\xi }_z|\text{, }z\in \mathcal{K}\}\text{(unstable sector bundle)}$$

$$S_{\mathcal{K}}^s\equiv \{({\xi }_z,{\eta }_z)\in {\mathbb{R}}^2||{\xi }_z|\le {\mu }_v|{\eta }_z|\text{, }z\in \mathcal{K}\}\text{(stable sector bundle)}$$

Full text

Keywords: chaotic dynamics, invariant set, hyperbolic.

1 Introduction

Studies of the Hénon map (Hénon [1976]) have played a seminal role in the development of our understanding of chaotic dynamics and strange attractors. The map depends on two parameters, $A$ and $B$, and has the following form:

[TABLE]

where we will require $B\not=0$ in order to endure the existence of the inverse map,

[TABLE]

The “heart” of chaotic dynamics is exemplified by the so-called “Smale horseshoe map” (see Smale [1980] for a general description, with background). The essential feature of the Smale horseshoe map for chaos is that the map contains an invariant Cantor set on which the dynamics are topologically conjugate to a shift map defined on a finite number of symbols (a “chaotic invariant set”, sometimes also referred to as a “chaotic saddle”). Devaney and Nitecki [1979] gave sufficient conditions, in terms of the parameters $A$ and $B$, for the Hénon map to have an invariant Cantor set on which it is topologically conjugate to a shift map of two symbols. The proof uses a technique due to Conley and Moser (see Moser [1973]) that is referred to as the “Conley-Moser conditions” (but for earlier work in a similar spirit see Alekseev [1968a, b, 1969]). Holmes [1982] used these conditions to show the existence of a chaotic invariant set in the so-called “bouncing ball map”. The Conley-Moser conditions were given a more detailed exposition, along with a slight weakening of the hypotheses, in Wiggins [2003]. More recently, the Conley-Moser conditions were used to show the existence of a chaotic invariant set in the Lozi map (Lopesino et al. [2015]).

The purpose of this paper is to carry out a similar analysis for a nonautonomous version of the Hénon map. The generalization of the Conley-Moser conditions for nonautonomous systems, i.e. in the discrete time setting with dynamics defined by an infinite sequence of maps, was given in Wiggins [1999]. We extend the nonautonomous Conley-Moser conditions further by providing an additional condition which is sufficient for the nonautonomous chaotic invariant set to be hyperbolic. Hyperbolicity of nonautonomous invariant sets is discussed in general in Katok and Hasselblatt [1995]. Earlier work on chaos in nonautonomous systems is described in Lerman and Silnikov [1992]; Stoffer [1988a, b]. Recent interesting work is described in Lu and Wang [2010, 2011].

While the development of the “dynamical systems approach to nonautonomous dynamics” is currently a topic of much interest, it is not a topic that is widely known in the applied dynamical systems community (especially the fundamental work that was done in the 1960’s). An applied motivation for such work is an understanding of fluid transport from the dynamical systems point of view for aperiodically time dependent flows. Wiggins and Mancho [2014] have given a survey of the history of nonautonomous dynamics as well as its application to fluid transport.

This paper is outlined as follows. In Section 2 we develop the required concepts for “building” chaotic invariant sets for two-dimensional nonautonomous maps. In Section 3 we prove the “main theorem” generalizing the Conley-Moser conditions that provide necessary conditions for two-dimensional nonautonomous maps to have a chaotic invariant set. In the course of the proof of the theorem the nature of chaotic invariant sets, and chaos, for nonautonomous maps is developed. This theorem was first given in Wiggins [1999], but in Section 3.1 we develop the theory further by providing a more analytical, rather than topological, construction for one of the Conley-Moser conditions that allows us to conclude that the nonautonomous chaotic invariant set is hyperbolic. In Section 4 we develop a version of the nonautonomous Hénon map and use the previously developed results to give sufficient conditions for the map to possess a nonautonomous chaotic invariant set. In Section 5 we discuss directions for future work along these lines.

2 Preliminary concepts

In this section we describe the basic setting and concepts that we will use throughout the remainder of the paper.

Nonautonomous dynamics will be defined by a sequence of maps and domains, $\{f_{n},D_{n}\}_{n=-\infty}^{+\infty}$, acting as follows:

[TABLE]

where, for our purposes, $D_{n}$ will be an appropriately chosen domain in $\mathbb{R}^{2}$, for all $n$.

Similar to the Smale horseshoe construction (Wiggins [2003]), on each domain $D_{n}$ we must construct a finite collection of vertical strips $V_{i}^{n}\subset D_{n}$ ($\forall n\in\mathbb{Z}$ and $\forall i\in I=\{1,2,...,N\}$) which map to a finite collection of horizontal strips $H_{i}^{n+1}$ located in $D_{n+1}$:

[TABLE]

Associated with these mappings we will need to define a transition matrix as follows:

[TABLE]

[TABLE]

[TABLE]

However, first we must precisely define the notion of the domains that we will use, horizontal and vertical strips in those domains, and provide a characterization of the intersection of horizontal and vertical strips in the domain appropriate for our purposes.

To begin, let $D\subset\mathbb{R}^{2}$ denote a closed and bounded set. We consider two associated subsets of $\mathbb{R}$:

[TABLE]

[TABLE]

Therefore $D_{x}$ and $D_{y}$ represent the projections of $D$ onto the $x$-axis and the $y$-axis respectively. From this it is easy to see that $D\subset D_{x}\times D_{y}$. We consider two closed intervals $I_{x}\subset D_{x}$ and $I_{y}\subset D_{y}$. We next define $\mu_{h}$-horizontal and $\mu_{v}$-vertical curves on these domains.

Definition 2.1**.**

*Let $0\leq\mu_{h}<+\infty$. A $\mu_{h}$-horizontal curve $\overline{H}$ is defined to be the graph of a function $h:I_{x}\rightarrow\mathbb{R}$ where $h$ satisfies the following two conditions:

1. The set $\overline{H}=\{(x,h(x))\in\mathbb{R}^{2}\text{ }|\text{ }x\in I_{x}\}$ is contained in $D$.

2. For every $x_{1},x_{2}\in I_{x}$ we have the Lipschitz condition*

[TABLE]

*Similarly, let $0\leq\mu_{v}<+\infty$. A $\mu_{v}$-vertical curve $\overline{V}$ is defined to be the graph of a function $v:I_{y}\rightarrow\mathbb{R}$ where $v$ satisfies the following two conditions:

1. The set $\overline{V}=\{(v(y),y)\in\mathbb{R}^{2}\text{ }|\text{ }y\in I_{y}\}$ is contained in $D$.

2. For every $y_{1},y_{2}\in I_{y}$ we have the Lipschitz condition*

[TABLE]

Next we “fatten” these curves into strips.

Definition 2.2**.**

Given two nonintersecting $\mu_{v}$-vertical curves $v_{1}(y)<v_{2}(y)$, $y\in I_{y}$, we define a $\mu_{v}$-vertical strip as

[TABLE]

Similarly, given two nonintersecting $\mu_{h}$-horizontal curves $h_{1}(x)<h_{2}(x)$, $x\in I_{x}$, we define a $\mu_{h}$-horizontal strip as

[TABLE]

The width of horizontal and vertical strips is defined as

[TABLE]

We will need to consider different parts of the boundary of the strips in relation to the domain on which they are defined. The following three definitions provide the necessary concepts.

Definition 2.3**.**

The vertical boundary of a $\mu_{h}$-horizontal strip $H$ is denoted

[TABLE]

The horizontal boundary of a $\mu_{h}$-horizontal strip $H$ is denoted

[TABLE]

Definition 2.4**.**

We say that $H$ is a $\mu_{h}$-horizontal strip contained in a $\mu_{v}$-vertical strip $V$ if the two $\mu_{h}$ horizontal curves defining the horizontal boundaries of $H$ (denoted by $\partial_{h}H$) are contained in $V$, with the remaining boundary components of $H$ (denoted by $\partial_{v}H$) contained in $\partial_{v}V$. These two last subsets, $\partial_{h}H$ and $\partial_{v}H$ are referred to as the horizontal and vertical boundaries of $H$, respectively. See Figure 1.

Definition 2.5**.**

Let $V$ and $\widetilde{V}$ be $\mu_{v}$-vertical strips. $\widetilde{V}$ is said to intersect $V$ fully if $\widetilde{V}\subset V$ and $\partial_{h}\widetilde{V}\subset\partial_{h}V$. See Figure 2.

3 The main theorem

In this section we prove the main general theorem which provides sufficient conditions for the existence of a chaotic invariant set for nonautonomous maps. In the course of the proof our meaning of “chaos” for nonautonomous dynamics will be made precise.

Following the original development of the Conley-Moser conditions (Moser [1973]), there are three geometrical and analytical conditions that, if satisfied, provide sufficient conditions for an autonomous map (in the original formulation) to have a chaotic invariant set. These are referred to as A1, A2, and A3. The conditions A1 and A2 provide sufficient conditions for the existence of a topological chaotic invariant set. The conditions A1 and A3 provide sufficient conditions for a hyperbolic chaotic invariant set. Conditions A1 and A2 were developed for nonautonomous dynamics in Wiggins [1999]. In this section we recall A1 and A2, but we also give a new construction of A3 for nonautonomous dynamics555We point out a minor technical point. In previous development of the Conley-Moser conditions (e.g. Moser [1973]; Wiggins [2003]) the set-up considers the mapping of horizontal strips to vertical strips. However, for the Hénon map it is more natural to consider vertical strips mapping to horizontal strips. Of course, the choice of what we refer to as “horizontal” and “vertical” is arbitrary. However, the same choice of coordinate labeling as is used in the previous literature can be used for the Hénon map if we impose a rotation P=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right) on the sequence of maps $\{f_{n}\}_{n=-\infty}^{+\infty}$ or, alternatively, take each map $f_{n}$ as $f_{-n}^{-1}$ for every $n\in\mathbb{Z}$.. In particular, we will show that A1 and A3 imply that A1 and A2 also hold.

The following two lemmas will play an important role in the proof of the main theorem.

Lemma 3.1

*i) If $V_{1}\supset V_{2}\supset\cdots\supset V_{k}\supset\cdots$ is a nested sequence of $\mu_{v}$-vertical strips with $d(V_{k})\rightarrow 0$ as $k\rightarrow\infty$, then $\cap_{k=1}^{\infty}V_{k}\equiv V_{\infty}$ is a $\mu_{v}$-vertical curve.

ii) If $H_{1}\supset H_{2}\supset\cdots\supset H_{k}\supset\cdots$ is a nested sequence of $\mu_{h}$-horizontal strips with $d(H_{k})\rightarrow 0$ as $k\rightarrow\infty$, then $\cap_{k=1}^{\infty}H_{k}\equiv H_{\infty}$ is a $\mu_{h}$-horizontal curve.*

Lemma 3.2

Suppose $0\leq\mu_{v}\mu_{h}<1$. Then a $\mu_{v}$-vertical curve and a $\mu_{h}$-horizontal curve intersect in a unique point.

The proof of both these two lemmas can be found in Wiggins [2003].

We assume that for each $D_{n}\subset\mathbb{R}^{2}$ we have:

[TABLE]

Furthermore, we assume that on each $D_{n}$ we can find a set of disjoint $\mu_{v}$ vertical strips, $D_{V}^{n}\equiv\cup_{i=1}^{N}V_{i}^{n}$, such that each $f_{n}$ is one-to-one on $D_{V}^{n}\equiv\cup_{i=1}^{N}V_{i}^{n}$. We then define

[TABLE]

with inverse function $f_{n}^{-1}$ defined on $D_{H}^{n+1}\equiv\cup_{i=1}^{N}H_{i}^{n+1}=f_{n}\left(\cup_{i=1}^{N}V_{i}^{n}\right)$ for every $n\in\mathbb{Z}$, (see Figure 3).

The transition matrix $\{A^{n}\}_{n=-\infty}^{+\infty}$ is defined as follows:

[TABLE]

Now we can state the first two Conley-Moser conditions for a sequence of maps that are sufficient to prove the existence of a chaotic invariant set for nonautonomous systems.

Assumption 1** **(A1)

For all $i,j\in I$ such that $A_{ij}^{n}=1$, $H_{ij}^{n+1}$ is a $\mu_{h}$-horizontal strip contained in $V_{j}^{n+1}$ with $0\leq\mu_{v}\mu_{h}<1$. Moreover, $f_{n}$ maps $V_{ji}^{n}$ homeomorphically onto $H_{ij}^{n+1}$ with $f_{n}^{-1}(\partial_{h}H_{ij}^{n+1})\subset\partial_{h}V_{i}^{n}$.

Also, since $f_{n}$ maps $V_{ji}^{n}$ homeomorphically onto $H_{ij}^{n+1}$ with $f_{n}^{-1}(\partial_{h}H_{ij}^{n+1})\subset\partial_{h}V_{i}^{n}$ then $f_{n}^{-1}$ maps $H_{ij}^{n+1}$ homeomorphically onto $V_{ji}^{n}$ ($\forall i,j\in I$) with

[TABLE]

Assumption 2** **(A2)

Let $V^{n+1}$ be a $\mu_{v}$-vertical strip which intersects $V_{j}^{n+1}$ fully. Then $f_{n}^{-1}(V^{n+1})\cap V_{i}^{n}\equiv\widetilde{V}_{i}^{n}$ is a $\mu_{v}$-vertical strip intersecting $V_{i}^{n}$ fully for all $i\in I$ such that $A_{ij}^{n}=1$. Moreover,

[TABLE]

Similarly, let $H^{n}$ be a $\mu_{h}$-horizontal strip contained in $V_{j}^{n}$ such that also $H^{n}\subset H_{ij}^{n}$ for some $i,j\in I$ with $A_{ij}^{n-1}=1$. Then $f_{n}(H^{n})\cap V_{k}^{n+1}\equiv\widetilde{H}_{k}^{n+1}$ is a $\mu_{h}$-horizontal strip contained in $V_{k}^{n+1}$ for all $k\in I$ such that $A_{jk}^{n}=1$. Moreover,

[TABLE]

Now we develop symbolic dynamics in a form appropriate for nonautonomous dynamics. Let

[TABLE]

denote a bi-infinite sequence with $s_{l}\in I$ ($\forall l\in\mathbb{Z}$) where adjacent elements of the sequence satisfy the rule $A_{s_{n}s_{n+1}}^{n}=1$, $\forall n\in\mathbb{Z}$.

Similarly to the symbolic dynamics implemented for the Smale horseshoe (see page 575 of Wiggins [2003]), here we denote the set of all such symbol sequences by $\Sigma_{\{A^{n}\}}^{N}$. If $\sigma$ denotes the shift map

[TABLE]

on $\Sigma_{\{A^{n}\}}^{N}$, we define the “extended shift map” $\tilde{\sigma}$ on $\widetilde{\Sigma}\equiv\Sigma_{\{A^{n}\}}^{N}\times\mathbb{Z}$ by

[TABLE]

Now we can state the main theorem.

Theorem 3.3** (Main theorem).**

Suppose $\{f_{n},D_{n}\}_{n=-\infty}^{+\infty}$ satisfies A1 and A2. There exists a sequence of sets $\Lambda_{n}\subset D_{n}$, with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$, such that the following diagram commutes

[TABLE]

where $\phi(x,y;n)\equiv(\phi_{n}(x,y),n)$ with $\phi_{n}(x,y)$ a homeomorphism mapping $\Lambda_{n}$ onto $\Sigma_{\{A^{n}\}}^{N}$.

Remark 3.4**.**

The sequence of sets $\{\Lambda_{n}\}_{n=-\infty}^{+\infty}$ is what we mean by a chaotic set for nonautonomous dynamics. Consequently our “main theorem” is a theoretical result which gives sufficient conditions for the existence of such a sequence of sets. The original proof can be found in Wiggins [1999], keeping in mind the geometrical considerations mentioned before.

Next, we will generalize the third Conley-Moser condition to the nonautonomous case. This will provide an alternative, and more analytical (as opposed to topological) method for proving that the second Conley-Moser condition holds, and it will also provide the additional information that the chaotic invariant set is hyperbolic.

3.1 Nonautonomous third Conley-Moser condition

We begin by giving a natural definition of stable and unstable sector bundles for the nonautonomous situation:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

with ${\cal K}$ being either ${\cal V}^{n}$ or ${\cal H}^{n+1}$. Then we can state the following assumption: the third Conley-Moser condition for the nonautonomous setting.

Assumption 3** **(A3)

$Df_{n}(S^{u}_{{\cal V}^{n}})\subset S^{u}_{{\cal H}^{n+1}}$*, $Df^{-1}_{n}(S^{s}_{{\cal H}^{n+1}})\subset S^{s}_{{\cal V}^{n}}$.

Moreover, if $(\xi_{f_{n}(z_{0}^{n})},\eta_{f_{n}(z_{0}^{n})})\equiv Df_{n}(z_{0}^{n})\cdot(\xi_{z_{0}^{n}},\eta_{z_{0}^{n}})\in S^{u}_{{\cal H}^{n+1}}$ then*

[TABLE]

If $(\xi_{f_{n}^{-1}(z_{0}^{n+1})},\eta_{f_{n}^{-1}(z_{0}^{n+1})})\equiv Df_{n}^{-1}(z_{0}^{n+1})\cdot(\xi_{z_{0}^{n+1}},\eta_{z_{0}^{n+1}})\in S^{s}_{{\cal V}^{n}}$ then

[TABLE]

Obviously we need to impose an additional condition in order to guarantee the existence of the Jacobian matrices $Df_{n}$ and $Df_{n}^{-1}$. From now we will consider that $f_{n},f_{n}^{-1}\in C^{1}$ for every $n\in\mathbb{Z}$ on their respective domains. Now we establish an important relationship between assumptions A2 and A3.

Theorem 3.5**.**

If nonautonomous A1 and A3 are satisfied for $0<\mu<1-\mu_{h}\mu_{v}$ then A2 is satisfied.

Part of the proof of this theorem is based on the following result.

Lemma 3.6

*Let $\{f_{n},D_{n}\}_{n=-\infty}^{+\infty}$ be a sequence of maps satisfying A1 and A3.

For every $n\in\mathbb{Z}$ and every pair of indices $i,j\in I$ we have that

i) if $\overline{V}^{n+1}\subset V_{j}^{n+1}$ is a $\mu_{v}$-vertical curve, then $f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ is a $\mu_{v}$-vertical curve in case $\overline{V}^{n+1}\cap H_{i}^{n+1}\not=\emptyset$.

ii) if $\overline{H}^{n}\subset V_{ji}^{n}$ is a $\mu_{h}$-horizontal curve, then $f_{n}(\overline{H}^{n})\cap H_{i}^{n+1}$ is a $\mu_{h}$-horizontal curve in case $\overline{H}^{n}\cap V_{i}^{n}\not=\emptyset$.*

Proof 3.7**.**

*We omit the proof of ii) as it follows the same line of reasoning as i).

We consider a $\mu_{v}$-vertical curve $\overline{V}^{n+1}\subset V_{j}^{n+1}$. By definition there exist an interval $T\subset\mathbb{R}$ and a function $v:T\rightarrow\mathbb{R}$ such that $\overline{V}^{n+1}$ is the graph of $v$ and also the Lipschitz condition $|v(t_{1})-v(t_{2})|\leq\mu_{v}|t_{1}-t_{2}|$ holds for a constant $\mu_{v}>0$ and every pair of points $t_{1},t_{2}\in T$.

It follows from Assumption 1 that $(f_{n}^{-1})$ is a homeomorphism over $H_{ij}^{n+1}=H_{i}^{n+1}\cap V_{j}^{n+1}$. In particular a homeomorphism over $\overline{V}^{n+1}\cap H_{i}^{n+1}\not=\emptyset$. This implies that*

[TABLE]

Since the curve $\overline{V}^{n+1}$ can be parametrized by $(v(t),t)|_{t\in T}$ (take also $v\in C^{1}$) then this last subset $f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ can also have a parametrization but over a smaller domain $T^{*}\subset T$,

[TABLE]

The image of ant tangent vector of $\overline{V}^{n+1}$ under $Df_{n}^{-1}$ has the form

[TABLE]

This last relation follows directly from the Lipschitz condition:

[TABLE]

By applying Assumption 3 we obtain that the tangent vectors belong to $S^{s}_{{\cal V}^{n}}$,

[TABLE]

Moreover, as we also assume that $(f_{n}^{-1})\in C^{1}$, any tangent vector

[TABLE]

From these two relations it follows that $\dot{y}(t)$ cannot change its sign in the entire domain $T^{*}$. Consequently for every pair of points $(x_{1},y_{1}),(x_{2},y_{2})\in f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ there exist $t_{1},t_{2}\in T^{*}$ such that $(x_{k},y_{k})=(x(t_{k},y(t_{k}))$, $(k=1,2)$ and we have the inequality

[TABLE]

[TABLE]

and this result implies that $f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ is a $\mu_{v}$-vertical curve.

Proof 3.8** (Proof of Theorem 3.5).**

*The theorem will be proved by verifying the following steps.

Step 1: Let $\overline{V}^{n+1}\subset V_{j}^{n+1}$ be a $\mu_{v}$-vertical curve. Then $f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ is a $\mu_{v}$-vertical curve for every $i\in I$ such that $\overline{V}^{n+1}\cap H_{i}^{n+1}\not=\emptyset$.

Step 2: Let $V^{n+1}$ be a $\mu_{v}$-vertical strip which intersects $V_{j}^{n+1}$ fully. Then $f_{n}^{-1}(V^{n+1})\cap V_{i}^{n}$ is a $\mu_{v}$-vertical strip that intersects $V_{i}^{n}$ fully for every $i\in I$ such that $V^{n+1}\cap H_{i}^{n+1}\not=\emptyset$.

Step 3: Show that $d(\widetilde{V}_{i}^{n})\leq(\mu/(1-\mu_{h}\mu_{v}))\cdot d(V^{n+1})$ for $\widetilde{V}_{i}^{n}=f_{n}^{-1}(V^{n+1})\cap V_{i}^{n}$.

We omit the part of the proof dealing with horizontal strips since it follows from the same reasoning used to prove the part concerning vertical strips.

We begin with Step 1. Let $\overline{V}^{n+1}\subset V_{j}^{n+1}$ be a $\mu_{v}$-vertical curve. For each $i\in I$ such that $\overline{V}^{n+1}\cap H_{i}^{n+1}\not=\emptyset$, by applying A1 we obtain that $H_{ij}^{n+1}=V_{j}^{n+1}\cap H_{i}^{n+1}\not=\emptyset$ is a $\mu_{h}$-horizontal strip contained in $V_{j}^{n+1}$. Since implicitly we are taking $\overline{V}^{n+1}$ as one of the two components of the vertical boundary of a vertical strip $V^{n+1}$ intersecting $V_{j}^{n+1}$ fully, the curve $\overline{V}^{n+1}$ intersects $\partial_{h}H_{i}^{n+1}$ in exactly two points.

Also because $f_{n}^{-1}$ maps the horizontal boundaries of each subset $H_{ij}^{n+1}(=H_{i}^{n+1}\cap V_{j}^{n+1})$ onto the horizontal boundaries of $V_{i}^{n}$, we have that $f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ is a curve linking the two horizontal boundaries of $V_{i}^{n}$. Finally if we apply Lemma 3.6 to this curve it follows that $f_{n}^{-1}(\overline{V}^{n+1})\cap V_{i}^{n}$ is also a $\mu_{v}$-vertical curve.

*To prove Step 2 we apply Step 1 to the $\mu_{v}$-vertical boundaries of the $\mu_{v}$-vertical strip $V^{n+1}$ which intersects $V_{j}^{n+1}$ fully. It then follows that $f_{n}^{-1}(V^{n+1})\cap V_{i}^{n}$ is also a $\mu_{v}$-vertical strip for every $i\in I$ such that $V^{n+1}\cap H_{i}^{n+1}\not=\emptyset$. Moreover this last strip intersects each $V_{i}^{n}$ fully because of the geometric considerations in Step 1.

For proving Step 3 first we need to fix an iteration $n\in\mathbb{Z}$ and an index $i\in I$. The width of each $\mu_{v}$-vertical strip $\widetilde{V}_{i}^{n}$ will be the distance between two points $p_{0},p_{1}\in\widetilde{V}_{i}^{n}$ with the same $y$-component and located in separate vertical boundaries, $d(\widetilde{V}_{i}^{n})=|p_{1}-p_{0}|$.*

By taking segment $p(t)$ considered in Figure 4, $\dot{p}(t)=p_{1}-p_{0}$ is a vector with its $y$-component equal to zero. Therefore $\dot{p}(t)\in S^{u}_{{\cal V}^{n}}$, $\forall t\in[0,1]$. Now we have that the curve $f_{n}(p(t))\equiv z(t)=(x(t),y(t))$ located in $D_{n+1}$ is a $\mu_{h}$-horizontal curve because of the second part of Lemma 3.6,

[TABLE]

Furthermore since the graph of $z(t)=(x(t),y(t))$ is a $\mu_{h}$-horizontal curve, we obtain that

[TABLE]

by denoting $(x_{i},y_{i})\equiv(x(i),y(i))=z(i)=f_{n}(p(i))=f_{n}(p_{i})$ for $i=0,1$.

Using this last fact and also the geometric considerations in Figure 5, it follows that

[TABLE]

[TABLE]

[TABLE]

Also as a result of the last part of Assumption 3 there exists a positive constant $\mu$, which we impose to be $\mu<1-\mu_{h}\mu_{v}$, such that

[TABLE]

[TABLE]

*Note that the two expressions containing the integrals are equal since $\dot{x}(t)$ does not change its sign at any point. This is due to the fact that the graph of $z(t)=(x(t),y(t))$ is a $\mu_{h}$-horizontal curve.

Finally we arrive at the result,*

[TABLE]

[TABLE]

4 Nonautonomous Hénon map

We have now developed the necessary tools for proving the existence of a chaotic invariant set for the nonautonomous Hénon map. Recall our general notation for nonautonomous dynamics (a sequence of maps defined on a sequence of domains), $\{f_{n},D_{n}\}_{n=-\infty}^{+\infty}$.

We will construct domains $D_{n}$ for the nonautonomous Hénon map, each of them containing an associated pair of horizontal strips and another of vertical strips. Moreover, the transition matrices will be shown to be identical for each map $f_{n}$, with A=\left(\begin{array}[]{cc}1&1\\ 1&1\end{array}\right) for each iteration $n$.

Recall that the autonomous Hénon map has the form:

[TABLE]

Following Devaney and Nitecki [1979], sufficient conditions for the existence of a chaotic invariant set in the autonomous context can be proven when the parameters satisfy the following inequalities:

[TABLE]

Note that when $B=-1$ the map is orientation-preserving and area-preserving. For our version of the nonautonomous Hénon map, we will take $B=-1$ for the sequence of maps $\{f_{n},D_{n}\}_{n=-\infty}^{+\infty}$ in order to retain these properties, but we will allow $A$ to vary for each iteration $n$. Therefore, we will take:

[TABLE]

where

[TABLE]

This choice is motivated by the fact that $A_{2}=5+2\sqrt{5}\approx 9.47$ is the minimum threshold for parameter $A$ for which the autonomous Hénon map satisfies the autonomous versions of Assumptions 1 and 3 of the Conley-Moser conditions.

In the following we will prove that the nonautonomous Hénon map satisfies the conditions described in Theorem 3.3. In particular, we will prove the following theorem.

Theorem 4.1**.**

If $A^{*}\geq 9.5$ then the nonautonomous Hénon map $f_{n}=(A(n)-y-x^{2},x)$ with $A(n)=A^{*}+\epsilon\cdot\cos(n)$, $\epsilon=0.1$ has a nonautonomous chaotic invariant set in $\mathbb{R}^{2}$.

Proof 4.2**.**

*We carry out the proof for the specific case where $A_{0}=9.5$. The case for $A_{0}>9.5$ follows similar reasoning as for the case $A_{0}=9.5$, with the main difference being that some values in the inequalities appearing when checking Assumption 3 must be changed. We begin with the first Conley-Moser condition.

Assumption 1. The domain $D_{n}$ on which each function $f_{n}$ will be defined is the square*

[TABLE]

*analogously to the domain considered for the autonomous Hénon map.

The horizontal strips and the vertical strips associated to any iteration $n\in\mathbb{Z}$ will be taken as*

[TABLE]

and since $f_{n}$ is a homeomorphism we also note that vertical strips “move” to horizontal strips in forward iteration,

[TABLE]

*Moreover the index $I$ indicating the number of strips in either $D_{H}^{n}$ or $D_{V}^{n}$ is $I=\{1,2\}$ and the strips are defined by

[TABLE]

[TABLE]

*These are determined by the images of $D$ with respect to $f_{n}$ and $f_{n}^{-1}$ for every $n\in\mathbb{Z}$. They result easy to compute. Let

[TABLE]

[TABLE]

the segments which conform the boundary of $D$. Their images with respect $f_{n}$ and $f_{n}^{-1}$ are either another segment or a parabola, and as $f_{n}$ is a homeomorphism, both $f_{n}(D)$ and $f_{n}^{-1}(D)$ are two strips with a parabolic form.

The key points of $f_{n}(D)$ and $f_{n}^{-1}(D)$ shown in Figure 6 have the following coordinates,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The coordinates of the these points satisfy $A(n)>2R$, $\forall n\in\mathbb{Z}$ and

[TABLE]

with strict inequality when $n\not=0$. Only in case $n=0$, the points $p_{6},p_{3}=q_{6},q_{3}$ are inside the domain $D$ and actually these are three vertices of the square $D$. In any case, it follows that the points $p_{1},p_{2},p_{4},p_{5}$ and $q_{1},q_{2},q_{4},q_{5}$ do not belong to $D$ for any $n$.

We denote the arguments of the parabolic curves by $X$ and $Y$, so that their equations take the form:

[TABLE]

[TABLE]

With this notation the absolute value of the derivatives of these functions take the form:

[TABLE]

and their maximum values are assumed when $X=R$ and $Y=R$, respectively. Therefore we have:

[TABLE]

*Due to reasons explained later, for convenience we will choose the thresholds $\mu_{h}=\mu_{v}=0.615$ for the maximum values that the slopes of the horizontal and vertical boundaries can assume, respectively. Using this fact, one can conclude that $H_{i}^{n+1}$ is a $\mu_{h}$-horizontal strip and $V_{i}^{n}$ a $\mu_{v}$-vertical strip for every $i\in I$ and $n\in\mathbb{Z}$. Moreover, $\mu_{h}\cdot\mu_{v}=(0.615)^{2}=0.378225<1$. This proves part of Assumption 1.

Furthermore, for any $i,j\in I$ and $n\in\mathbb{Z}$ we have that the horizontal boundaries of $f_{n}(V_{i}^{n})=H_{i}^{n+1}$ are two $\mu_{h}$-horizontal curves which link the left and right sides of the square $D$. Since the two $\mu_{v}$-vertical curves bounding $\partial_{v}V_{j}^{n+1}$ link the upper and the bottom sides of $D_{n+1}$, both boundaries $\partial_{h}f_{n}(V_{i}^{n})$ and $\partial_{v}V_{j}^{n+1}$ intersect in four different points. From this fact it follows that $H_{ij}^{n+1}=f_{n}(V_{i}^{n})\cap V_{j}^{n+1}$ is a $\mu_{h}$-horizontal strip contained in $V_{j}^{n+1}$.

Since $f_{n}$ is a homeomorphism for every $n\in\mathbb{Z}$,*

[TABLE]

[TABLE]

*This can be checked by an easy computation.

Therefore the nonautonomous Hénon map satisfies Assumption 1.

Assumption 3. To begin our verification that A3 is also satisfied, we need to recall the notation for several concepts developed earlier:*

[TABLE]

[TABLE]

[TABLE]

*with ${\cal K}$ being either ${\cal V}^{n}$ or ${\cal H}^{n+1}$.

Now given any point $z_{0}=(x_{0},y_{0})\in{\cal H}^{n+1}$ and any $(\xi_{z_{0}},\eta_{z_{0}})\in S^{s}_{{\cal H}^{n+1}}$ (which by definition, $|\xi_{z_{0}}|\leq\mu_{v}|\eta_{z_{0}}|$) we have that*

[TABLE]

belongs to $S^{s}_{{\cal V}^{n}}$ if and only if the inequality

[TABLE]

Since it is also true that

[TABLE]

[TABLE]

*In case $\left(2|y_{0}|\mu_{v}-\mu_{v}^{2}\right)\geq 1$, the previous inequality (55) will hold.

To verify this we need to check if*

[TABLE]

*At this point we give a geometrical argument.

$\bullet$ The horizontal lines $\{Y=\pm 1.1205\}$ cut the parabola $\{X=A(n)-R-Y^{2}\}$ at two points:*

[TABLE]

[TABLE]

$\bullet$* The horizontal lines $\{Y=\pm 1.1205\}$ cut the parabola $\{Y=A(n+1)+R-X^{2}\}$ at two points with a positive $x$-component:*

[TABLE]

[TABLE]

From here we have that

[TABLE]

[TABLE]

[TABLE]

The inequalities $\bar{x}_{1}<\bar{x}_{2}<x_{2}=x_{1}$ (note that $\bar{x}_{1}<\bar{x}_{2}$ is trivial due to the definitions) also hold for every parameter $A(n)=A^{*}+\epsilon\cos(n)$ satisfying $A^{*}\geq 9.5$ and $\epsilon=0.1$. The reason comes from comparing the derivatives of $\bar{x}_{2}$ and $x_{2}$ with respect to $A^{*}$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Clearly $\frac{d\bar{x}_{2}}{dA^{*}}<\frac{dx_{2}}{dA^{*}}$ for every $A^{*}\geq 9.5$. It follows that $\bar{x}_{2}<x_{2}$ for $A^{*}\geq 9.5$.

*This setup shows that every point $z_{0}=(x_{0},y_{0})\in{\cal H}^{n+1}$ satisfies that $|y_{0}|>1.1205=\frac{1}{2}\left(\mu_{v}+\frac{1}{\mu_{v}}\right)$, since the four areas composing ${\cal H}^{n+1}$ are either beneath the line $\{Y=-1.1205\}$ or above the line $\{Y=1.1205\}$, as can be observed in Fig. 7.

*Since $z_{0}\in{\cal H}^{n+1}$ is an arbitrary point, the inclusion $Df^{-1}_{n}(S^{s}_{{\cal H}^{n+1}})\subset S^{s}_{{\cal V}^{n}}$ is proven. *

*For the second inclusion $Df_{n}(S^{u}_{{\cal V}^{n}})\subset S^{u}_{{\cal H}^{n+1}}$ we focus on the fact that ${\cal V}^{n}=f_{n}^{-1}({\cal H}^{n+1})$ and since $f_{n}^{-1}(x,y)=(y,A(n)-x-y^{2})$ transforms the $y$-components of the points of ${\cal H}^{n+1}$ into the $x$-components of the points of ${\cal V}^{n}$, it immediately follows that *

[TABLE]

[TABLE]

As in the previous case this inequality allows us to prove the inclusion,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

*and then the inclusion $Df_{n}(S^{u}_{{\cal V}^{n}})\subset S^{u}_{{\cal H}^{n+1}}$ is proved.

Finally for the last part of Assumption 3 we will only prove the inequality*

[TABLE]

since the inequality

[TABLE]

*is proved by using the same argument.

[TABLE]

[TABLE]

[TABLE]

This last inequality is true if we require that $\mu_{v}<\mu<1-\mu_{h}\mu_{v}$. This interval exists since $\mu_{v}=0.615$ is less than $(1-\mu_{h}\mu_{v})=0.621775$. Then we have that

[TABLE]

Analogously for any $z_{0}=(x_{0},y_{0})\in{\cal V}^{n}$,

[TABLE]

*and the proof that the nonautonomous Hénon map satisfies A1 and A3 with $0<\mu<1-\mu_{h}\mu_{v}$ is complete. Consequently it also satisfies A2 by using Theorem 3.5.

By applying the main theorem it follows that there exists a chaotic invariant set $\{\Lambda_{n}\}_{n=-\infty}^{+\infty}$ (with respect to the nonautonomous Hénon map $\{f_{n}\}$) contained in $\{D_{n}\}_{n=-\infty}^{+\infty}$ (let say $\Lambda_{n}\subset D_{n}=D$ and $f_{n}(\Lambda_{n})=\Lambda_{n+1}$) which is conjugate to a shift map of two symbols.*

Remark 4.3**.**

*Comparing this result to what happens for the autonomous Hénon map, it is curious that for some $n\in\mathbb{Z}$ the quantity $A(n)=9.5+\epsilon\cos(n)$ is less than $A_{2}=5+2\sqrt{5}\approx 9.47$, which is the minimum threshold for parameter $A$ for which the autonomous Hénon map satisfies the autonomous Assumption 3.

In other words, this given example shows that although for some $n\in\mathbb{Z}$ the values that parameter $A$ takes imply that $f_{n}$ does not satisfy the autonomous Assumption 3 separately, this fact does not necessarily mean that the nonautonomous Assumption 3 is not satisfied for the sequence $\{f_{n}\}_{n=-\infty}^{+\infty}$.*

5 Summary and Conclusions

In this paper we have considered a nonautonomous version of the Hénon map and have provided necessary conditions for the map to possess a nonautonomous chaotic invariant set. This is accomplished by using a nonautonomous version of the Conley-Moser conditions given in Wiggins [1999]. We sharpen these conditions by providing a more analytical condition that, as a consequence, enables us to show that the chaotic invariant set is hyperbolic. In the course of the proof we provide a precise characterization of what is mean by the phrase “hyperbolic chaotic invariant set” for nonautomous dynamical systems. Currently there is much interest in nonautomous dynamics and a thorough analysis of a specific example might provide a benchmark for further studies, just as the work in Devaney and Nitecki [1979] provided a benchmark for studies of chaotic dynamics for autonomous maps. Indeed, our generalization of the Hénon map to the nonautonomous setting provides an approach to generalizing the map to even more general nonautonomous settings, such as a consideration of “noise”. This would be an interesting topic for future studies.

Acknowledgments

The research of FB-I, CL and AMM is supported by the MINECO under grant MTM2014-56392-R. The research of SW is supported by ONR Grant No. N00014-01-1-0769. We acknowledge support from MINECO: ICMAT Severo Ochoa project SEV-2011-0087.