Compact Global Chaotic Attractors of Discrete Control Systems

Abstract The paper is dedicated to the study of the problem of existence of compact global chaotic attractors of discrete control systems and to the description of its structure. We consider so called switched systems with discrete time xn+1 = fv(n)(xn), where v: Z+ → {1; 2; : : : ;m}. If m≥2 we give sufficient conditions (the family M := {f1; f2; : : : ; fm} of functions is contracting in the extended sense) for the existence of a compact global chaotic attractor. We study this problem in the framework of non-autonomous dynamical systems (cocycles)


Introduction
The aim of this paper is the study of the problem of existence of compact global chaotic attractors of discrete control systems (see, for example, Bobylev, Emel'yanov and Korovin [3], Cheban [7,8] and the references therein). Let W be a metric space, M := {f i : i ∈ I} be a family of continuous mappings of W into itself and (W, f i ) i∈I be the family of discrete dynamical systems, where (W, f ) is a discrete dynamical system generated by positive powers of continuous map f : W → W . On the space W we consider a discrete inclusion A solution of the discrete inclusion DI(M) is called (see, for example, [3,11]) a sequence {{x j } | j ≥ 0} ⊂ W such that (1) x j = f i j x j−1 for some f ij ∈ M (trajectory of DI(M)), i.e.
We can consider that it is a discrete control problem, where at each moment of the time j we can apply a control from the set M, and DI(M) is the set of possible trajectories of the system.
The problem of existence of compact global attractors for a discrete inclusion arise in a number of different areas of mathematics (see, for example, [8,9] and the references therein).
It is well known the following result. In the book [8] (Chapter VI) it was generalized this theorem for the finite family M = {f 1 , f 2 , . . . , f m } when it is contracting in the generalized sense, i.e., there are two positive numbers N and q ∈ (0, 1) such that for all x 1 , x 2 ∈ W and n ∈ N, where i 1 , i 2 , . . . , i n ∈ {1, 2, . . . , m}.
In this paper we consider an arbitrary family of discrete dynamical systems (W, f ) (f ∈ M, M contains, generally speaking, an infinite number of mappings f ) on the complete metric space W and we give the conditions which guarantee the existence of compact global attractor for M. If M consists of a finite number of maps, then we prove that M admits a compact global chaotic attractor. We study this problem in the framework of non-autonomous dynamical systems (cocyles). This paper is organized as follows.
In Section 2 we give some notions and facts (set-valued dynamical systems, compact global attractors, an ensemble (coolage) of dynamical systems, cocycles) from the theory of set-valued dynamical systems which we use in our paper. Section 3 is dedicated to the study of compact global chaotic attractors of discrete control systems. We give also the description of the dynamics of global attractors for this type of control systems. The main result of Section 3 (Theorem 3.2) contains the conditions of existence of chaotic attractor for discrete control systems.
In Section 4 we study the problem of existence of compact global attractor for discrete control system in the case when M contains an infinite number of mappings f and they are not (in general) invertible. The main result (Theorem 4.9 and Theorem 4.12) of Section 4 give the conditions of existence of compact global attractors and describes its dynamics.
Section 5 contains some applications of general results obtained in Sections 3 and 4 for certain classes of control systems with continuous time.

Some Notions and Facts from Dynamical and Control Systems
In this Section we collect some notions and facts from the theory of set-valued dynamical systems which we use in our paper.

Set-valued dynamical systems and their compact global attractors.
Let (X, ρ) be a complete metric space, S be a group of real (R) or integer (Z) numbers, T (S + ⊆ T) be a semi-group of additive group S. If A ⊆ X and x ∈ X, then we denote by ρ(x, A) the distance from the point x to the set A, i.e. ρ(x, A) = inf{ρ(x, a) : a ∈ A}. We denote by B(A, ε) an ε-neighborhood of the set A, i.e. B(A, ε) = {x ∈ X : ρ(x, A) < ε}, by C(X) we denote the family of all non-empty compact subsets of X. For every point x ∈ X and number t ∈ T we put in correspondence a closed compact subset β(π(t, x), π(t 0 , x 0 )) = 0 for all x 0 ∈ X and t 0 ∈ T, where β(A, B) = sup{ρ(a, B) : a ∈ A} is a semi-deviation of the set A ⊆ X from the set B ⊆ X.
In this case it is said [17] that there is defined a set-valued semi-group dynamical system.
Let T ⊂ T ⊂ S. A continuous mapping γ x : T → X is called a motion of the set-valued dynamical system (X, T, π) issuing from the point x ∈ X at the initial moment t = 0 and defined on T , if The set of all motions of (X, T, π), passing through the point x at the initial moment t = 0 is denoted by F x (π) and F(π) := {F x (π) | x ∈ X} (or simply F).
The trajectory γ ∈ F(π) defined on S is called a full (entire) trajectory of the dynamical system (X, T, π).
Let (X, T, π) be compactly dissipative and K be a compact set attracting every compact subset of X. Let us set It can be shown [6], [8] that the set J defined by equality (2) does not depend on the choice of the attractor K, but it is characterized only by the properties of the dynamical system (X, T, π) itself. The set J is called Levinson center of the compact dissipative dynamical system (X, T, π). A discrete inclusion DI(M) is called (see, for example, [3,11]) a set of all sequences for some f ij ∈ M (trajectory of DI(M)), i.e.
Denote by F x 0 the set of all trajectories of discrete inclusion (4) (or DI(M)) issuing from the point x 0 ∈ W and F : Below we will give a new approach concerning the study of discrete inclusions DI(M) (or difference inclusion (4)). Denote by C(Z + , W ) the space of all continuous mappings f : Z + → W equipped with the compact-open topology. Denote by (C(Z + , X), Z + , σ) a dynamical system of translations (shifts dynamical system or dynamical system of Bebutov [15,16]) on C(Z + , W ), i.e. σ(k, f ) := f k and f k is a k ∈ Z + shift of f (i.e. f k (n) := f (n + k) for all n ∈ Z + ).
We may now rewrite equation (3) in the following way: where ω ∈ Ω is the operator-function defined by the equality ω(j) := f i j+1 for all j ∈ Z + . We denote by ϕ(n, x 0 , ω) the solution of equation (5) issuing from the point x 0 ∈ E at the initial moment n = 0. Note that (4)) is equivalent to the family of non-autonomous equations (5) (ω ∈ Ω).
From the general properties of difference equations it follows that the mapping ϕ : Z + × W × Ω → W satisfies the following conditions: ω)) for all n, τ ∈ Z + and (x 0 , ω) ∈ W × Ω; (iii) the mapping ϕ is continuous; (iv) for any n, τ ∈ Z + and ω 1 , ω 2 ∈ Ω there exists ω 3 ∈ Ω such that Let T 1 ⊆ T 2 be two sub-semigroups of group S, X, Y be two metric (or topological) spaces and (X, Let W, Ω be two topological spaces and (Ω, T 2 , σ) be a semi-group dynamical system on Ω.
From the presented above it follows that every DI(M) (respectively, inclusion (4)) in a natural way generates a cocycle W, ϕ, (Ω, Z + , σ) , where Ω = C(Z + , M), (Ω, Z + , σ) is a dynamical system of shifts on Ω and ϕ(n, x, ω) is the solution of equation (5) issuing from the point x ∈ W at the initial moment n = 0. Thus, we can study inclusion (4) (respectively, DI(M)) in the framework of the theory of cocycles with discrete time.
Below we need the following result.

Chaotic attractors of discrete control systems
In Section 3 we give the conditions of existence of chaotic attractor for discrete control systems.
Denote by A the set of all mapping ψ : Z + × R + → R + possessing the following properties: (G1) ψ is continuous; (G2) there exists a positive number t 0 such that: (a) ψ(t 0 , r) < r for all r > 0; (b) the mapping ψ(t 0 , ·) : R + → R + is monotone increasing. (G3) ψ(t + τ, r) ≤ ψ(t, ψ(τ, r)) for all t, τ ∈ Z + and r ∈ R + . Remark 3.1. 1. Note that the functions ψ(t, r) = N q t r (N > 0 and q ∈ (0, 1)) and ψ(t, r) = r 1+rt belong to A, where (t, r) ∈ Z + × R + . 2. Let f : R + be a continuous function satisfying the conditions: Then the mapping ψ : The set S ⊂ W is (i) nowhere dense, provided the interior of the closure of S is empty set, int(cl(S)) = ∅; (ii) totally disconnected, provided the connected components are single points; (iii) perfect, provided it is closed and every point p ∈ S is the limit of points q n ∈ S with q n = p.
The set S ⊂ W is called a Cantor set, provided it is totally disconnected, perfect and compact.
The subset M of (X, T, π) is called (see, for example, [14]) chaotic, if the following conditions hold: where P er(π) is the set of all periodic points of (X, T, π).
Recall that a point x ∈ X of the dynamical system (X, T, π) is called Poisson's stable, if x belongs to its ω-limit set ω x := t≥0 τ ≥t π(τ, x).
Theorem 3.2. Suppose that the following conditions are fulfilled: Then the following statement hold: Proof. Let Y = Ω := C(Z + , Q) and (Y, Z + , σ) be a semi-group dynamical system of shifts on Y . Then Y is compact. By Theorem 2.1, P er(σ) = Ω and Ω is compact and invariant.
The set I is said to be a chaotic attractor of DI(M), if (i) the set J is chaotic, i.e. J is transitive and J = P er(σ), where J is the Levinson center of the skew-product dynamical system (X, Z + , π) generated by DI(M); (ii) I = pr 1 (J). [9] for the special case, when N > 0 and q ∈ (0, 1)). [2,3,4] (see also the bibliography therein). In [2,3,4] the statement close to Theorem 3.2 was proved. Namely:

The problem of the existence of compact global attractors for DI(M) with finite M (collage or iterated function system (IFS)) was studied before in works
(i) in [2] it was announced the first and proved the second statement of Theorem 3.2, if ψ(t, r) = q t r (t ∈ Z + and q ∈ (0, 1)); (ii) in [3,4] Then the following statements hold: (i) there exists a unique fix point p ∈ W of f , i.e., f (p) = p; (ii) the point p is uniformly attracting, i.e., for all x 1 , x 2 ∈ X and t ∈ Z + , where α : C(X) × C(X) → R + is the Hausdorff distance and F t : for all A ∈ C(X). Note that K = F (K), i.e., K ⊆ X is a compact invariant set of dynamical system (X, F ) and by (8) we get for all A ∈ C(X). Therefore, (X, F ) is compactly dissipative and from (9) and the invariance of K it follows that the set K is Levinson center of (X, F ). Theorem 4.4. Let (X, T, π) be a set-valued dynamical system and let exist ψ ∈ A such that for all x 1 , x 2 ∈ X and t ∈ T. Then (X, T, π) is compactly dissipative.
Proof. Let (X, T, π) be a set-valued dynamical system, ψ ∈ A and the condition (10) be fulfilled. Let us choose t 0 > 0 (t 0 ∈ T) so that ψ(t 0 , r) < r for all r > 0 and ψ(t 0 , ·) be monotone increasing, and consider the cascade (X, P ) generated by positive powers of the set-valued mapping P (x) := π(t 0 , x) (x ∈ X). Denote byψ : Z + × R + → R + the mapping defined by equalityψ(t, r) := ψ(t 0 t, r) for all (t, r) ∈ Z + × R + . It easy to check thatψ ∈ A and that the mapping P is aψ-contraction and by Theorem 4.2 the dynamical system (X, P ) is compactly dissipative. From Lemma 4.3 the compact dissipativity of the dynamical system (X, Z + , π) follows. The theorem is proved. Then the following statements hold: Proof. It is easy to check that Let ψ ∈ A from the definition of ψ-contraction of the family M. We will prove that (11) α

Thus, for an arbitrary point
Similarly we have the inequality α(A, B)). (12) and (13). Now to finish the proof of theorem it is enough to cite Theorem 4.4.

Inequality (11) follows from inequalities
Lemma 4.6. Let M be a compact subset of C(W ), M be ψ-contracting and W, φ, (Ω, Z + , σ) be a cocycle generated by DI(M). For each ω ∈ Ω, n ∈ N, and Let K denote a nonempty compact subset from W. Then there exists a real number for all ω ∈ Ω, all m, n ∈ Z + , and all x 1 , x 2 ∈ K, where m ∧ n := min (n, m).
Proof. Let ω, m, n, x 1 , and x 2 be as stated in the lemma. By Theorem 4.5 the set- Then the setK := ∪ ∞ n=0 F n (K) is compact and positively invariant. Without any loss of generality we can suppose that m < n. Then observe that . Then x 3 belongs toK. Hence we can write  α(A, B)) for all A, B ∈ C(W ) and n ∈ Z + . Thus the mapping F : C(W ) → C(W ) is a ψ-contraction on the metric space (C(W ), α); and we have where F t := F t−1 • F (t ≥ 2). In particular {F t (K)} is a Cauchy sequence in (C(W ), α). Notice that φ(ω, n, x) ∈ F n (K). It follows from Theorem 4.5, that if lim n→+∞ φ(ω, t, x) exists, then it belongs to I.
That the later limit does exist follows from the fact that, for fixed ω ∈ Ω and x ∈ W , {φ(ω, n, x)} n∈Z + is a Cauchy sequence: by Lemma 4.6 for all x ∈ K and ω ∈ Ω. Notice that the right hand site here tends to zero as m and n tend to infinity. The uniformity (with respect to x ∈ K and ω ∈ Ω) of the convergence follows from the fact that the constant C is independent of (ω, x) ∈ Ω × K. Since the map φ(·, n, ·) : Ω × W → W (n ∈ Z + ) is continuous then φ : Ω → W is continuous too.
Finally, we prove that φ is onto. Let a ∈ I. Then, since by Theorem 4.5 I = lim n→+∞ F n ({x}) it follows that there exists a sequence {ω n } ⊆ Ω such that lim n→∞ φ(ω n , n, x) = a.
To prove the second statement we note that φ(σ(m, ω)) = φ(ω) and ϕ(m, a,ω) = a, if the point ω is m-periodic (i.e., σ(m, ω) = ω). Proof. The space Ω is the closure of the set of periodic points. Lift this statement to I using the map φ : Ω → I. From Lemma 4.8 it follows that if ω ∈ Ω is m-periodic point of the dynamical system (Ω, Z + , σ) then the point φ(ω) will be m-periodic point of DI(M). Now it is sufficient to note that if S ⊂ Ω is such that its closure equals Ω, then the closure of φ(S) equals I.
for all t ∈ Z + and x ∈ W.

Some applications
Consider a control dynamical system governed by the differential equation (14) x where E and B are some Banach spaces. Then the equation admits a unique solution ψ(t, r) with initial condition ψ(0, r) = r and the mapping ω : R 2 + → R + possesses the following properties: (i) the mapping ψ : R 2 + → R + is continuous; (ii) ψ(t, r) < r for all r > 0 and t > 0; (iii) for all t ∈ R + the mapping ψ(t, ·) : R + → R + is increasing;  (15) (θ(u) = u α for all u ∈ R + and α > 1) with initial condition ψ(0, r) = r.
Let H be a Hilbert space with the scalar product ·, · and f ∈ C(H, H) be a function satisfying (16) Re for all u 1 , u 2 ∈ H, where θ : R + × R + → R + is some function with properties (H1)-(H4).

Theorem 5.3. [10]
If the function f verifies the condition (16), then (i) the equation generates a semigroup dynamical system (H, R + , π), where π(t, u) is a unique solution of equation (17) defined on R + with the initial condition π(0, u) = u; (ii) the following inequality holds for all u 1 , u 2 ∈ H and t ∈ R + , where ψ(t, r) is a unique solution of equation (15) with initial data ψ(0, r) = r for all r ∈ R + , | · | is the norm generated by the scalar product ·, · in the space H.
A point x 0 ∈ H is said to be m-periodic (m ∈ N) for control system (14), if there exist an m-periodic control ω ∈ S(R + , P) (ω(t + m) = ω(t) for all t ∈ R + ) such that the unique solution x(t) of equation with initial data Proof. This statement directly it follows from Theorems 3.2 and 4.9.

Monotone evolution equations.
Let H be a real Hilbert space with the inner product , , | · | := , and E be a reflexive Banach space contained in H algebraically and topologically. Furthermore, let E be dense in H, and here H can be identified with a subspace of the dual E of E and , can be extended by continuity to E × E. Remind [5,6,12] that an operator A is called: -monotone, if for every u 1 , u 2 ∈ D(A) : Au 1 − Au 2 , u 1 − u 2 ≥ 0; -semi-continuous, if the function ϕ : R → R defined by the equality ϕ(λ) := A(u + λv, w) is continuous.
Note that the family of monotone operators can be partially ordered by including graphics.
A monotone operator is called maximal, if it is maximal among the monotone operators.
A nonlinear "elliptic" operator The following result is established in [13] (Ch.II and Ch.IV). If x ∈ H and p = p p−1 , then there exists a unique solution ϕ(t, u) ∈ C(R + , H) of (20).
Theorem 5.5. Suppose that the operator A satisfies the conditions above. Then equation(20) generates on the space H a semi-group dynamical system (H, R + , π) satisfying the following condition: for all t ∈ R + and u 1 , u 2 ∈ H, where ψ(t, r) is a unique solution of equation y = −2αy β/2 with initial data ψ(0, r) = r.
Proof. This statement directly it follows from Theorem 7.10 [10].
Consider a finite set of differential equations Proof. This statement it follows from Theorems 3.2 and 4.9.