Covering Relations and Non-autonomous Perturbations of ODEs

Covering relations are a topological tool for detecting periodic orbits, symbolic dynamics and chaotic behavior for autonomous ODE. We extend the method of the covering relations onto systems with a time dependent perturbation. As an example we apply the method to non-autonomous perturbations of the R\"{o}ssler equations to show that for small perturbation they posses symbolic dynamics.


Introduction
The goal of this paper is to answer the following QUESTION: Assume that the equation has a symbolic dynamics (is semiconjugated with some Bernoulli shift). Consider now small non-autonomous perturbation of (1.1) x ′ (t) = v(x(t)) + ǫ(t, x(t)). (

1.2)
Will equation (1.2) also have the symbolic dynamics if ǫ is sufficiently small? We prove that the answer to above question is positive if the symbolic dynamics is defined in terms of covering relations for Poincaré maps. This is made precise in Section 3 see Theorem 4. This result is applied to non-autonomous perturbations Rössler equations [R] to show that for small perturbation they posses symbolic dynamics.
The content of the paper can be described as follows. In Section 2 we recall from paper [GiZ] the notion of covering relations for maps. This is the basic technical tool used in this paper. In Section 2.2 we prove Theorem 2 -the basic theorem about continuation of covering relations for Poincaré maps for the non-autonomous perturbations of ODEs.
In the following sections we apply Theorem 2 to answer positively our question and present above mentioned applications. Definition 2 [GiZ] Assume N,M are h-sets, such that u(N ) = u(M ) = u and With above definition we have the following theorem (see also [MM,Z0,Z1] for its precursors).
Theorem 1 [GiZ] Let N i , i = 0, . . . , n be an h-set and N n = N 0 . Assume that for each i = 1, . . . , n we have

Continuation of covering relations for Poincaré maps for non-autonomous perturbations
Let v : R k → R k be a C 1 function. Let us consider an autonomous differential equation Let V 0 , V 2 , . . . , V n−1 , V n be Poincaré sections of the system generated by the equation (we do not require that they are different). Let 1 ≤ i ≤ n, and let x be the solution of the problem . . , n, by σ i (x 0 ) we will denote the first time for which the solution x reaches the section V i , When it will be evident from the context which sections we wish to consider, we will sometimes omit the index i.
We will also define functions (Poincaré maps) Now in the context of the question asked in the introduction we assume that we have a finite set of covering relations for sets N i for Poincaré maps defined by an ODE, which leads via Theorem 1 to symbolic dynamics. For example assume that we have the following covering relations N i P =⇒ N j for i, j = 0, 1 (compare topological horseshoes in [Z3]). Then from Theorem 1 it follows that we have a semiconjugacy onto the Bernoulli shift on two symbols.
Let us now consider the equation (2.4) with a time dependent perturbation We will try to show a similar result (for example topological horseshoe) for this perturbed equation. It seams very likely that for small perturbation the above result should hold. Let us start with the fact that for small perturbations of the equation the covering relations (2.3) for the solution still hold. Let us clarify what we will exactly understand by the functions f ji in the setting of the perturbed equation (2.5). Let us consider the equation (2.5) with the following initial conditions Let x be the solution of problem (2.6). We will define functions f T ji which will be analogous to the functions f ji . As before Let us note that for |ǫ| sufficiently small the functions above are well defined [GiZ]. What is more if there exists a covering relation N j fji =⇒ N i then for small |ǫ| the term in the sense that if we change the ǫ, then the lower and the upper bound of the expression does not change for all x 0 ∈ N j . The goal of this section is to establish the following . . , V n be the Poincaré sections for the solution of the equation . . , n be h-sets, we denote this family by H Assume that we have a set Γ of covering relations N i fij =⇒ N j for some N i , N j ∈ H, where f ji are Poincaré maps for (2.9).
Then there exits δ = δ(Γ) such that for all continuous ǫ : R k+1 → R k such that |ǫ| < δ we have: For any t 0 ∈ R and for any infinite chain of covering relations from Γ . . , such that for the solution x of the equation Before we move on to the proof of this theorem we shall need some preliminary results.
Lemma 1 Assume that f i is a Poincaré map for (2.9). If (2.14) where x = (p, q) and A i is the linear map from the definition of the covering relation for the covering This Lemma states that for all the Poincaré maps f T i , for any T , there exists a homotopy H i T = H i (·, T, ·) which transports the function f T i,c into the linear function (A i , 0), which is independent from T . What is more, the family of functions H i T is continuous with respect to T . Proof: The first part of the lemma regarding the fact that is a consequence of the Theorem 13 from [GiZ]. To prove the second part of the lemma, let us consider the following differential equation where λ ∈ [0, 1 2 ]. We can define Poincaré maps f λ,T i in the same manner as we have defined the functions f T i in (2.7). From the first part of the lemma we know that which satisfies the conditions 1, 2.1 and 2.2, from the definition of the covering relation.
We can now define our homotopy as We need to show that this homotopy satisfies the conditions (2.11), (2.12), (2.13), (2.14). The first two conditions are evident from the definition of H. From the fact that The fact that follows from the conditions 2.1 and 2.2 for the covering Hence all the conditions (2.11), (2.12), (2.13), (2.14) hold.
The following lemma will be the main tool for the proof of the Theorem 2.
Lemma 2 Let v : R k → R k be C 1 -function, let V 1 , . . . , V n be the Poincaré sections for the solution of the equation then there exists a δ > 0, δ depends only on the set of covering relations in the chain (2.16) and not on the length of the chain, such that for all continuous and for any t 0 ∈ R there exists a point x 0 ∈ N 0 and a sequence t 0 < t 1 < . . . < t n , such that for the solution x of the equation Proof: From Lemma 1 we know that the first part of the lemma is true.
Without any loss of generality we will will give the proof for t 0 = 0. We will also assume that c Ni = Id for i = 0, . . . , n Let us define a function This artificial function will be needed to close the loop of covering relations (compare Thm. 1), so that it is possible to define the function (2.20) later on. Let us define functions T )).
If we start from the set N 0 , then from (2.8) we know that there exists s 1 , s 2 , . . . s n and r 1 , r 2 , . . . , r n such that Let us define and We will show that there exists an x = ((x 0 , t 0 ), . . . , (x n , t n )) ∈ int XN such that F (x) = 0. Once we find the x, we will have our x 0 , t 0 , . . . , t n , because from the definition we know that for i = 1, . . . , n and from the fact that F (x) = 0 we shall have = 0 which will mean that What is more, from our construction and the fact that F (x) = 0 we will know that t 0 = 0 and that t i − σ(x i−1 , t i−1 ) = 0 (2.21) which means that 0 = t 0 < t 1 < . . . < t n .
Our goal is therefore to find the x ∈ int XN for which F (x) = 0. Let us define a homotopy H : where for i = 1, . . . , n, H i is the homotopy from the Lemma 1 and G(λ, ·, ·) = (A n+1 , 0) . Let us note that H(0, x) = F (x) and that Let us assume that we have the following two lemmas which we will prove after completing this proof Let us now complete our proof using the two lemmas. From Lemmas 3 and 4 we know that deg(F, intXN, 0) = deg(H(0, ·), intXN, 0) = deg(H(1, ·), intXN, 0) = ±1 which means that there exists an x ∈ intXN such that F (x) = 0, x = ((x 0 , t 0 ), . . . , (x n , t n )) hence we have found our x i ∈ int N i and t i . where B is linear. From the degree for affine maps (4.2) we have deg (H(1, ·), intXN, 0) = sgn(detB) which means that to prove the lemma it is sufficient to show that B is an isomorphism. Let us recall the definition of B.
Proof of Lemma 4: From the homotopy property, it is sufficient to show that H(λ, x) = 0, for all x ∈ ∂XN and λ ∈ [0, 1]. (2.24) We will consider an x from the boundary of XN x = ((x 0 , t 0 ), . . . , (x n , t n )). If x ∈ ∂XN then there exists an i such that one of the following conditions holds First let us consider the case (2.25). For i = 1, . . . n if x i ∈ N + i and H(λ, x) = 0 then in particular From the statement of Lemma 1, condition (2.14) we know that This and the fact that x i−1 ∈ N i−1 contradicts (2.28). We therefore know that (2.25) does not hold for i = 1, . . . , n. For i = 0 if x 0 ∈ N + 0 and H(λ, x) = 0 then which means that q 0 = 0 which contradicts the fact that x 0 = (p 0 , q 0 ) ∈ N + 0 = B u (0, 1) × ∂B s (0, 1).
Let us now consider the case (2.26).
We are now left with the case (2.27). For i = 1, . . . , n if (2.27) holds and H(λ, x) = 0 then in particular (2.30) Our construction of XN (2.18) which guarantees that gives us and therefore from (2.27) This clearly contradicts (2.30). For i = 0 from the definition of H(λ, ·) and the fact that H(λ, x) = 0 we get straight away the fact that t 0 = 0 which means that it is not possible for t 0 ∈ {−1, 1}.
We have shown that for any λ ∈ [0, 1] and x ∈ ∂XN , H(λ, x) = 0, this fact and the homotopy property of the index concludes our proof.
Proof of Theorem 2: Let us consider the sequence of covering relations from Γ N 0 Let us consider a finite subsequence of the sequence (2.31) From Lemma 2 we know that for |ǫ| < δ there exists x m ∈ N 0 and a sequence t 0 = t m 0 < t m 1 < . . . < t m m , such that for the solution x of the equation (2.10) we have x(t i ) ∈ intN i for i = 1, . . . , m and that δ depends only on the family Γ and not on the length of the sequence. We therefore have a sequence {x m } ∞ m=1 ⊂ N 0 . Since N 0 is compact there exists a subsequence x m k which converges to a certain x 0 ∈ N 0 . In the curse of the proof of Lemma 2 we have shown that (2.21) From this fact and from the continuity of the solution of the problem with respect to the initial conditions, it follows that the solution x(t), of the problem (2.32) x(0) = x 0 passes through the sets N 0 , N 1 , . . . and therefore there exists a sequence t 0 < t 1 < . . . such that x(t i ) ∈ intN i for i = 1, 2, . . .

Application to Rössler equations.
In this section we combine Theorem 2 and results from [Z1] to show that small non-autonomous perturbations of Rössler [R] posses symbolic dynamics.
First we need to recall some definitions.
Let k be a positive integer. Let Σ k := {0, 1, . . . , k − 1} Z , Σ + k := {0, 1, . . . , k − 1} N . Σ k , Σ + k are topological spaces with the Tichonov topology. On Σ k , Σ + k we have the shift map σ given by Obviously Σ + A , Σ A are invariant under σ. Let F : X → X be any continuous map and N ⊂ X. By F |N we will denote the map obtained by restricting the domain of F to the set N . The maximal invariant part of N (with respect to F ) is defined by The Rössler equations are given by [R] x = −(y + z) y = x + by (3.3) where a = 5.7, b = 0.2. These are parameters values originally considered by Rössler. The flow generated by Eq. (3.3) exhibits a so-called strange attractor. We will investigate the Poincaré map P generated by (3.3) on the section Θ := {(x, y, z)| x = 0, y < 0,ẋ > 0}.
The following result was proved in [Z1] (see also [Z3]) Theorem 3 For all parameter values in sufficiently small neighborhood of (a, b) = (5.7, 0.2) there exists Poincaré section N ⊂ Θ such that the Poincaré map P induced by Eq. ( 3.3) is well defined and continuous. There exists continuous map π : Inv(N, P ) → Σ 3 , such that The preimage of any periodic sequence from Σ A contains periodic points of P .
Above theorem is a consequence of Theorem 1 and the following Lemma, which was established in [Z1] with computer assistance (computer assisted proof) Lemma 5 There are h-sets N 0 , N 1 , N 2 ⊂ Θ such that for all parameter values in sufficiently small neighborhood of (a, b) = (5.7, 0.2) N ⊂ Dom(P ) and the following conditions hold Let us denote by R a,b : R 3 → R 3 the vector field on the right-hand side of (3.3). By applying Theorem 2 to Lemma 5 we immediately obtain the following Theorem 4 Let us fix (a, b) = (5.7, 0.2). Let A be as in Theorem 3. Consider a non-autonomous perturbation of (3.3) (3.5) There exists δ > 0, such that for any t 0 ∈ R and any sequence c = (c i ) ∈ Σ + A there exists a solution of (3.5), x c : [t 0 , ∞) → R 3 and a sequence t 0 < t 1 < t 2 < · · · < t n < t n+1 < . . . , such that Above theorem says nothing about the size of δ. To obtain a numerical value for δ one can take one of two approaches analytical from the computer assisted proof in [Z1] one can obtain global bounds Z ⊂ R 3 , Z compact, such that all trajectories linking |N i | with its Poincaré image are in Z. For ǫ sufficiently small the same will be true for (3.5). Now using bounds for the Poincaré return times on |N i | we can compute an upper bound of the distance between the solution of (3.3) and (3.5). Then we compute ǫ for which the covering relations listed in Lemma 5 survive.
computational we can replace (3.5) by a differential inclusion Now for various values of δ we can perform an rigorous integration of (3.6) looking for the largest possible δ for which the covering relations listed Lemma 5 are still satisfied (for any continuous selector). For an algorithm for rigorous integration of differential inclusions see [Z4].

Other examples.
Other based on Theorem 1 computer assisted proofs of the existence of nontrival symbolic dynamics give rise to theorems analogous to Theorem 4. This applies to the following systems • Lorenz equations [GaZ], • Chua circuit [G], • Kuramoto-Shivashinsky ODE [W].
The precise formulation of these results is left to the reader. Degree property for affine maps. [L] Suppose that f (x) = B(x−x 0 )+c, where B is a linear map and x 0 ∈ R n . If the equation B(x) = 0 has no nontrivial solutions (i.e if Bx = 0, then x = 0) and x 0 ∈ D, then deg(f, D, C) = sgn(detB). (4.2)