Localization of the Chain Recurrent set using Shape theory and Symbolical Dynamics

Abstract The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [1] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [2] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [3].

Several authors have pointed out the connections between shape theory and topological dynamics based on the approach to shape developed by Borsuk. When dealing with spaces with complicated local behaviour, which often occurs in dynamical framework, shape theory turns out to be an appropriate tool.The use of shape theory in the study of dynamical systems was initiated by Hastings in [4].Other authors have shown how to apply shape theory to obtain global properties of attractors in the papers [5], [6], [7], [8]. Recently, using the intrinsic shape (the approach to shape without use of external spaces) [1], [9], several results are obtained showing the advantages of intrinsic approach to shape in some situations. Applying intrinsic shape in dynamical systems seems very natural. The most natural application of intrinsic shape is for investigation of objects that appear in dynamical systems, like attractors, limit sets or various types of recurrent sets.The papers [2] and [10] are good examples of that.
In the paper [11] a weak form of recurrence called chain recurrence is introduced by Conley. When we make computer simulation, calculating orbit of a point in each step we have rounding errors, so each time we obtain pseudo-orbit(ϵ -chain) instead of true orbit of observed point. So we will rather detect chain recurrent points than periodic orbits in computer experiments.
Chain recurrent set of continuous dynamical systems on compact metric spaces has many interesting properties. It is closed, ow invariant and restriction of the ow to chain recurrent set does not change chain recurrent points.It is also well known that ow restricted to limit set of any point in X is chain recurrent.
In this paper we give a theorem which shows that under certain conditions the shape of the chain recurrent set is locally invariant. An investigation of the symbolic image gives an opportunity to get a neighborhood of the chain recurrent set and using the algorithm for localizing neighborhoods provided by Osipenko in [3] we obtained a programming code which was successfully tested on a couple of systems of di erential equations. In what follows we assume that the phase space X in which all dynamics take place is always a compact metric space.

Intrinsic Shape
The rst paper about intrinsic shape is [12] where a shape morphism between compact metric spaces has been de ned for the rst time using the notion of V-continuity.
We will give a brief construction of the shape category using the intrinsic approach from [1]. The approach follows the main steps of construction of homotopy theory.
Instead of f , g : X → Y continuous functions we take (fn), (gn) : X → Y proximate sequences of functions and de ne a relation of homotopy (fn) (gn).
Like in homotopy theory where X and Y have the same homotopy type if there exist a continuous function f : X → Y with a homotopy inverse, X and Y have the same (intrinsic) shape if there exist a proximate sequence (fn) : X → Y with a homotopy inverse.
Let's start with some of the basic de nitions: For collections U and V of subsets of X, U ≺ V means that U re nes V, i.e., each U ∈ U is contained in some V ∈ V. By covering we understand a covering consisting of open sets.
De nition 1.1. Let X, Y be compact metric spaces, and V be a nite covering of Y. The function is V -continuous, if for any x ∈ X, there exists a neighborhood U of x, such that f (U) ⊆ V, for some member V ∈ V.
(The family of all U, form a covering of X, and since X is compact there is a nite subcovering. Shortly, we say The relation of homotopy is an equivalence relation and is denoted by f V g The main notion for intrinsic de nition of shape for compact metric spaces is the notion of proximate sequence. De nition 1.3. The sequence (fn) of functions fn : X → Y is a proximate sequence from X to Y, if for some sequence V V ... co nal in the set of nite coverings, for all indices m ≥ n, fn and fm are homotopic as Vn -continuous functions (co nal means that for any nite covering V there exists Vn such that Vn ≺ V). In this case we say that (fn) is a proximate sequence over (Vn).
We mention that if (fn) and (f n ) are proximate sequences from X to Y, then there exists a sequence (Vn) of nite coverings, such that (fn) and (f n ) are proximate sequences over (Vn). Two proximate sequences (fn), (f n ) are homotopic if for some sequence V V ... co nal in the set of nite coverings, (fn) and (f n ) are proximate sequences over (Vn), and for all integers n, fn and f n are homotopic as Vn -continuous functions. This is an equivalence relation and we denote (fn) (f n ). The homotopy class is denoted by (fn) . Let (fn) : X → Y be proximate sequence over (Vn), and let (gn) : Y → Z be a proximate sequence over (Wn).For a covering Wn of Z, there exist a covering V kn of Y such that g(V kn ) ≺ Wn . Then the composition of these two proximate sequences is the proximate sequence (hn) = (gn f kn ) : X → Z. This proximate sequence is unique up to homotopy.
Compact metric spaces and homotopy classes of proximate sequences form the category whose isomorphisms induce classi cation which coincide with the standard shape classi cation, i.e., isomorphic spaces in this category have the same shape.

Basic notions and theorems about dynamical systems
One approach to dynamical systems introduced by Poincare and followed by Morse and Conley can be described as follows: First of all a nite family of invariant sets {M i } n i= is located such that the ow in the complement set X\ n j= M j is fairly simple. Then some kind of local analysis is performed around those sets( local means that only the ow in arbitrary small neighborhoods is involved). Finally this local information is put together in some way involving the global topology of the phase space. Trivial examples of such sets are nitely many critical points p , p , ..., pn which contain all limit sets on a compact manifold. This approach leads to introduction of a Morse decomposition.
Before introducing the concept of these sets we shall give a brief review of the basic notions from dynamical system theory.
A ow in a metric space (X, d) is a continuous map φ : X × R → X satisfying the following two condition: for all x ∈ X and t, s ∈ R If we replace the set R with R + we get the corresponding notion of semi-dynamical system. The map φ is called a phase map, and the corresponding space X a phase space.
One of the important problems in dynamical systems concerns the asymptotic behavior of trajectories as time goes to plus or minus in nity. Limit sets are fundamental tools for this problem.
Positive limit set for arbitrary subset N ⊆ X is the set: Analogously, negative limit set for arbitrary subset N ⊆ X is the set: If we replace the set R with R + or R − , we get the corresponding notion of positive and negative invariance. If φ(x, t) = x, for all t ∈ R, the point x is called rest point.
A set M is stable if every neighborhood U of M admits a positively invariant neighborhood V of M such that V ⊆ U.
A compact invariant set Y ⊆ X is called an attractor if it admits a neighborhood U such that ω(U) = Y . Analogously, a compact invariant set Y ⊆ X is called an repellor if it admits a neighborhood U such that It is known that every attractor is a stable set (for example [13]).
De nition 2.1. Let K = {K j | j ∈ J} be a family of disjoint compact invariant subsets of the phase space X. A Lyapunov function for K is a continuous function τ : X → R such that: .

Example 2.2.
Consider the complete metric space S , the -dimensional sphere, which we identify here with R π. On S the di erential equationẋ = sin (x) de nes a dynamical system. In this case we have CR(φ) = S which is easily proved. The following lemma is proved in [2]  Theorem 2.1. Every global attractor A of a semi-dynamical system de ned on a compact metric space X has the same shape as the phase space, i.e. Sh(A) = Sh(X).
Using the results previously stated we proved in [2] also the following theorem: In the next section we shall prove a more general form of this theorem using the following result also known as Keesling's reformulation of Beck's theorem, [16].

Shape of the Chain Recurrent Set
In the paper [2] we introduced the notion of a proper covering V and de ned a retraction r V adjoint to V. Also we invoked previous results from theory of retracts and Non-continuous Topology in order to answer the question that is naturally imposed by theorem 2.2 about the shape of members of a Morse decomposition with more than two elements. Here we give a new proof only using the notion of a Lebesque number for a covering which enables us to state the claim about the local shape properties of the chain recurrent set CR(φ). We use the following notation: T(x, ε)-open ball centered at x and radius ε, S(x, ε) sphere centered at x and radius ε.

be a compact metric space with a ow φ. For arbitrary Morse decomposition for X, which admits Lyapunov function τ
Proof. First let us note that we can suppose that the ow on the set n i= M i is stationary. Namely, if this is not the case then using Theorem 2.3 with the choice S = n i= M i we will get a new ow in which this is the case. Also let us emphasize that M = {M , M , M , ..., Mn} is also a Morse decomposition for the new ow. It is enough to consider sets M i such that ≤ i ≤ (n − ), since the result about attractor and repelor is already proved. So let us pick arbitrary set M i , ≤ i ≤ (n− ). We denote τ(M i ) = c i , and form the sets where ϵ > is su ciently small such that the following holds: Now let's consider the semi-dynamical systems (U + , φ |U + ) and (U − , φ |U − ). This is possible because the sets are positively and negatively invariant, respectively. We will show this assumption. Namely , so for arbitrary t > using the property of Lyapunov function, we have τ(φ(x, t)) < τ(x) ≤ c i + ϵ. Of course the possibility x ∈ M i is trivial so we assume that x / ∈ M i . Let us assume that τ(φ(x, t)) < c i . Then using continuity of φ and τ from τ(φ(x, t)) < c i < τ(x) we get that a real number t * > exists such that Similarly, U − is negatively invariant. Now we shell prove that for the rst semidynamical system (U + , φ |U + ) M i is an attractor, i.e. ω(U + ) = M i , and for the second one (U − , φ |U − ) a repelor, i.e. α(U − ) = M i . Namely, for the rst one it is clear that ω(U + ) ⊇ M i . If we assume that z ∈ ω(U + ) then from the invariance of the limit set ω(U + ) as well as from ω(U + ) ⊆ U + , we have that φ(z, R) ⊆ U + which means that α(z) ⊆ U + , ω(z) ⊆ U + , but from the fact that all limit sets of points are contained in n i= M i , we get that α(z) ∪ ω(z) ⊆ M i from where we have that z ∈ M i . The proof for the second semi-dynamical system is similar. Now it easily follows from de nitions of limit sets that for arbitrary ϵ > , ∃t ∈ R + such that ∀t ≥ t , We are ready to de ne a map a : (U + ∪ U − ) × R + → M i in the following way: For points (x, t) ∈ U + × R + , we go with the ow until the point φ(x, t), then we measure the distance d(φ(x, t), M i ) which by compactness of the set M i is achieved in some point m t . Of course this point may not be unique but never the less we can pick any such point. So we de ne a(x, t) = m t x , for (x, t) ∈ U + × R + . For points (x, t) ∈ U − × R + : we go with the ow until the point φ(x, −t), then we measure . Of course this point may not be unique but never the less we can pick any such point. So we de ne a(x, t) = m t x , for (x, t) ∈ U − × R + . Let us note that this map is well de ned on M i × R + because the following holds: a(x, t) = φ(x, t) = x = φ(x, −t), ∀(x, t) ∈ M i × R + , of course having in mind that the ow on M i is stationary.
Let (Vn) be a co nal sequence of coverings for M i . We shall de ne a sequence of real numbers tn ∈ R + , such that the maps fn : U →M i de ned by fn(x) = a(x, tn) are Vn-continuous.
Construction of the sequence (tn): We choose ε = λn where λn ≤ is the Lebesque number of the covering Vn. There exist strictly monotonically increasing sequences t + n , t − n −→ ∞ such that the following conditions are satis ed: We will choose the time sequences in such a way to satisfy the following inequality, namely t − n ≥ c i − βn αn − c i ·t + n . Finally we make the choice Proof. We de ne the following map Hn : U × I −→ M i by: Let us note that the connecting relations are obvious. We shall prove that Hn is Vn-continuous homotopy.  (Hn(x , t ), λn / ), so for the same Vn ∈ Vn is valid Hn W ∩ U − \M i , W ⊆ Vn. Moreover, let us note that there exists neighborhoods W of x and W of t such that the following inclusion is valid: Now, we shall discuss the points x ∈ U + \M i , t ∈ I. First let us note that Hn(x , t ) = a(x , ( − t )tn + t tn). Now, for arbitrary ε > there exists neighborhood W of x and W of t such that d φ(x, hn,m(t)), φ(x , hn,m(t )) < ε, for all x ∈ W , t ∈ W , where hn,m(t) = ( − t)tn + ttn. Again, having in mind that hn,m(t) ≥ tn, we obtain the following estimates:
According to the previous lemma, we can con rm that there exists a shape morphism [(fn)] : U −→ M i . where for the subsequence kn we will impose additional condition, namely to satisfy the following inequality:

Now we are ready to de ne a map
Hn which will be a homotopy candidate between (i • f kn ) and ( U ) by: The connection relation is obvious. Let us check how close this map is to continuous: i) For points (x , t ), x ∈ U, t > / the map is continuous because the phase map and the Lyapunov function are actually such. ii) Let x ∈ intM i , t < / . There exists a neighborhood W of x and W of t such that Hn(x, t) = φ x, t kn ( − t) , ∀x ∈ W , ∀t ∈ W , so the continuity follows. iii) Let us discuss the points (x , t ), x ∈ intM i , t = / . We shall de ne the following map hn : I −→ R by W of t such that Hn(x, t) = φ x, hn(t) = x, ∀x ∈ W , ∀t ∈ W , so we obtain continuity in this case as well. iv) Consider the points (x , t ), x ∈ ∂M i , ≤ t < / . Hn(x , t ) = φ x , t kn ( − t ) . We choose arbitrary neighborhood Q of this point (Hn(x , t )). There exist neighborhoods W of x and W ⊆ [ , / ) of t such that Hn(W ∩ M i , W ) ⊆ Q. There exists ε > such that T (Hn(x , t ), ε/ ) ⊆ T (Hn(x , t ), ε) ⊆ Q. Let us consider the following map hn(x, t) = tt kn (τ(x) − c i ) + t kn ( − t) which is continuous. There exist neighborhoods W of x and W of t such that ∀x ∈ (U + \M i ) ∩ W , ∀t ∈ W : hn(x, t)), a(x, hn(x, t)) < ε , φ x, hn(x, t) ∈ T (Hn(x , t ), ε/ ).
We choose W ∩ W ∩ W = W * as a neighborhood of x and W ∩ W ∩ W = W * as a neighborhood of t . Hence Hn(W * , W * ) ⊆ Q, so the continuity is con rmed in this case as well.
v) Let's consider the case for the points (x , t ) such that x ∈ ∂M i , t = / .
We choose arbitrary neighborhood Q of Hn(x , / ). There exist neighborhoods W of x and W of t = / such that Hn(W , W ∩ ( / , ]) ⊆ Q. From the previous case there exist neighborhoods W of x and W of t = / such that Hn(W , W ∩ [ , / ]) ⊆ Q. We choose W * = W ∩ W as a neighborhood for x and W * = W ∩ W as a neighborhood for t = / . From the previous discussion we obtain the following inclusion Hn(W * , W * ) ⊆ Q and the continuity in these points follows. vi) Let us consider the points (x , t ), t < / and such that |τ(x ) − c i | > γn. We suppose that x ∈ U + . We consider the function hn : X × R −→ R de ned by hn(x, t) = tt kn (τ(x) − c i ) + t kn ( − t). Note that hn(x, t) ≥ max{t + n , t − n }. Namely, hn(x, t) > tt kn γn + t kn ( − t) (in a neighborhood of the point x ) and the right side of the inequality if we treat it as a linear map on the segment [ , / ] it obtains its minimum at t = / so: There exist neighborhoods W of x and W of t such that Specially d φ(x, hn(x, t)), a(x , hn(x , t )) ≤ λn / . Now, from the continuity of the phase map φ and a choice for ε < λn / there exist neighborhoods W of x and W of t such that Let W * = W ∩W and W * = W ∩W are the neighborhoods for x and t correspondingly. Note the following inequality: hn(x , t )), a(x , hn(x , t )) < λn + λn + λn = λn , ∀x ∈ W * , ∀t ∈ W * . Now, for arbitrary x , x ∈ W * and t , t ∈ W * , we obtain that: which is exactly Vn-continuity in the point (x , t ). Similarly for the points (x , t ), t < / , |τ(x ) − c i | > γn and x ∈ U − . vii) Now let us consider the points ( . From the assumption x ∈ Un. We shall prove that the set τ − (c i , c i + γn + δ) is positively invariant. Namely, if p ∈ τ − (c i , c i + γn + δ) then for arbitrary t > we have We shell explain only the rst inequality c i < τ φ(p, t) . In the contrary c i ≥ τ φ(p, t) then p ∈ M i which is a contradiction. Now, from hn(x, t)), a(x, hn(x, t)) < λn , ∀x ∈ W , ∀t ∈ W .
Specially, for x = x and t = t we have that hn(x , t )), a(x , hn(x , t )) < λn . Now, from the continuity of the phase map for a choice of ε < λn / there exist neighborhoods W of x and W of t such that d φ(x, hn(x, t)), φ(x , hn(x , t )) < ε, ∀x ∈ W , ∀t ∈ W .
There exists an element Vn of Vn such that Hn(W * , W * ∩ [ , / ]) ⊆ Vn is valid. For gn(x, t) = t kn (τ(x) − c i )( − t) and ε = λn / there exist neighborhoods W of x and W of / such that d φ(x, gn(x, t)), φ(x, gn(x, / )) < λn / . Analogously there exists neighborhood W of x such that: d φ(x, gn(x, / )), φ(x , gn(x , / )) < λn . Let W ** = W ∩ W ∩ W is a neighborhood of x and W ** = W a neighborhood of / . We obtain the following estimates: This means that there exists V * n ∈ Vn such that Hn(W ** , W ** ∩ ( / , ]) ⊆ V * n . Also, let us note that the following inclusion Hn(W ** , W ** ∩ [ / , ]) ⊆ V * n . Namely, the neighborhood W ** can be chosen su ciently small such that: Having this in mind for the points (x , / ), (x , / ), x , x ∈ W ** we have that the following holds: Similarly, for the points (x , / ), (x , t ), x , x ∈ W ** , t > / we have : Hence, V * n ∩ Vn ≠ ∅. But this implies that we have st(Vn)-continuity in this points. Now, we shall discuss the points τ(x ) ≥ αn. From the inequality τ(x ) − c i ≥ αn − c i we obtain that . Now, from the inequality tn ≥ t + n we obtain that: , which implies that φ x, tt kn (τ(x) − c i ) + t kn ( − t) ∈ T(M i , λn / ). So we have st(Vn)-continuity in this points as well. Now we shall assume that x ∈ U − \M i , which is the remaining case. We make additional assumption τ(x ) > βn. We shall prove that τ(x) > βn, x ∈ U − implies that x ∈ T(M i , λn / ). On the contrary, there exists t > such that τ φ(x, −t) ∈ S(M i , λn / ). But, this would imply that which is a contradiction. Now, from the inequality: we have that φ(x , t kn (τ(x ) − c i )) ∈ T(M i , λn / ). According to the previous discussion, similarly we obtain st(Vn)-continuity in this points. It remains the case when τ(x ) ≤ βn. Note the inequality τ(x ) − c i ≤ βn − c i in which all the values are negative. This implies that , which implies that we have st(Vn)-continuity in this points as well. The proof is complete.
Let us note that the condition imposed on the Lyapunov function is essential in theorem 3.1 as shown in the following example: The Hawaiian earring is a Morse set which is not a shape retract of any of its neighborhoods. This shows that theorem 3.1 does not hold for general Lyapunov functions. Now, we are ready to consider the shape of chain recurrent set CR(φ). We shall use the following theorem of Conley [14]. Proof. From the previous theorem of Conley, the connected components of CR(φ), M = {CR j (φ)|j ∈ J}, form a Morse decomposition. According to the previously proven theorem, each of this components admits a compact neighborhood U i with the same shape as CR i (φ), i.e. Sh(U i ) = Sh CR i (φ) and such that is disjoint from the others. Now, if we choose U = ∪ n i= U i , we get that In the section that follows we shall verify this result on a couple of systems of di erential equations by investigating the symbolic image (an oriented graph) with respect to a given covering. We used the algorithm suggested by Osipenko in [3] to get a sequence of embedded neighborhoods which converges to the chain recurrent set by writing a programming code in Mathematica.

Symbolical analysis of the Chain Recurrent Set
The theoretical background of the symbolical images and the constructive methods which are applied on them were described in detail by Osipenko in [17]. The main idea is construction of a directed graph which represents the structure of the state space for the investigated dynamical system. This graph is called the symbolical image of the focused system and can be seen as approximation of the system ow. Valuable information about the global structure of the system may come from the analysis of this symbolic image. From the computation point of view the usage of such a graph has the advantage that once it is constructed, all investigations are matters of graph analysis which provides an opportunity among other things in it's pallet an algorithmic way of thinking. Hence basic operations on the symbolical image gives an opportunity to obtain a sequence of neighborhoods which localize invariant sets, or in our case of interest, the chain recurrent set. Numerical computations are performed for several dynamical systems in order to verify our theoretical result.
Let us consider a discrete dynamical system governed by a homeomorphism f de ned on a compact manifold M.
We use the following algorithm from [3] based on which a programming code in "Mathematica" is successfully obtained and tested on couple of systems of di erential equations: Step1. Starting with an initial covering C, the symbolical image G = (T, R) of the map f is found. The cells of the initial covering may have arbitrarily large diameter d .
Step2. The recurrent vertices i k of the graph G are recognized. Using the recurrent vertices, a closed neighborhood P = i k M(i k ) of the chain recurrent set CR(f ) is obtained.
Step3. The cells corresponding to the recurrent vertices i.e. {M(i k )| i k is recurrent} are partitioned. For example, the largest diameter of the cells may be divided with 2. Thus the new covering is de ned.
Step4. The symbolic image G = (T, R) is constructed for the new covering. It should be noted that the new symbolic image may be constructed on the set P = i k M(i k ). In other words, the cells corresponding to non recurrent vertices do not participate in the construction of the new covering and the new symbolic image.
Step5. Go back to step2. Repeating this partitioning process we obtain a sequence of neighborhoods P , P , P , ... of the chain recurrent set CR(f ).
Before giving some examples of system of di erential equations we shall discuss the transition from discrete to continuous phase maps. Namely, the idea is quite simple, discretization of systems continuous in time. Let a system of di erential equations be given byẋ(t) = F(x(t)). What we need is some kind of mapping which transforms an orbit continuous in time into one discrete in time. A shift operator along trajectories is needed. Such a mapping has the form f (x) = φ(x, t), with φ(x, ) = x. It can be calculated by solving the equation. If we x a value for t = t we obtain the required discretization. Consider the following system of di erential equations: In polar coordinates we have:ṙ = r( − r ) θ = .
with initial values r( ) = r and θ( ) = θ . The rst equation can be solved either as a Bernoulli equation or as a separable di erential equation. The solution is given by: r(t, r ) = ( + ( r − )e − t ) − and θ(t, θ ) = t + θ Now we easily obtain the required discretization by choosing a value for t = for example. (The value of t is important for practical purposes above all because it determines the speed with which the non recurrent cells are being erased but only user experience and heuristic testing can lead to the most proper setting of t). We input this map in the programming code and obtain the following neighborhoods of CR(f ):  Consider the following system of di erential equations: x = ẏ y = −x + y cos(x).
In this system we cannot obtain our discretization map like in the previous example by integrating the system, but using the built in function NDSOLVE in the programming package "Mathematica" we obtain approximation of it.
These are the results:  Here are the results: