uniform dissipativeness, robust synchronization

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International Journal of Bifurcation and Chaos, Vol. 21, No. 2 (2011) 513–526 c World Scientific Publishing Company  DOI: 10.1142/S0218127411028568

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS HILDEBRANDO M. RODRIGUES Departamento de Matem´ atica Aplicada e Estat´ıstica, Instituto de Ciˆencias Matem´ aticas e de Computa¸c˜ ao, Universidade de S˜ ao Paulo, Caixa Postal 668, 13560-970, S˜ ao Carlos, SP, Brazil [email protected] JIANHONG WU Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Canad´ a M3J1P3 [email protected] LU´IS R. A. GABRIEL Departamento de Matem´ atica Aplicada e Estat´ıstica, Instituto de Ciˆencias Matem´ aticas e de Computa¸c˜ ao, Universidade de S˜ ao Paulo, Caixa Postal 668, 13560-970, S˜ ao Carlos, SP, Brazil Received March 13, 2010 In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation. Keywords: Discrete system; uniform dissipativeness; attractor; synchronization; constructing a Liapunov function.

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1. Introduction In this paper we study the uniform dissipativeness, with respect to parameter variation, in order to obtain uniform estimates of global attractors for general discrete systems. The central idea is based on the classical Liapunov direct method. The novelty is the use of a Liapunov function, whose derivative along solutions of the considered systems is allowed to be positive in a bounded set, to obtain a uniform estimation of the attractors for those discrete systems arising from the discretization of some well-known chaotic continuous dynamical systems. As will be shown, to adopt the Liapunov function of a continuous system to its discrete analogue we will need to impose some restrictions on the step size of the discretization, and one goal of this work is to use two important examples to show how to derive these restrictions by incorporating the Language multiplier method into the Liapunov direct approach. There are many application problems where estimating the global attractors is an important step towards the full understanding of the complex dynamical behaviors of given systems, and normally many parameters are involved in these systems and the parameter values vary due to measurement errors and imperfection of hardware implementation. Estimation of the attractors for a given dynamical system exhibiting chaotic behaviors is particularly important for secure communication, but obtaining such an estimation is a very challenging task since chaotic behaviors are normally associated with the coexistence of both expansion and contraction in disjoint subspaces of the high dimensional phase space. Historically, this task has been achieved by using a Liapunov function, but due to the aforementioned coexistence of expansion and contraction in different subspaces the construction of a Liapunov function with negative derivative along solutions of the considered system is difficult. Our general results will require the negative derivative of the Liapunov function along solutions of the considered system only in a restricted region. This reduces substantially the difficulty of constructing a Liapunov function from the physical background of the system, as will be illustrated by various applications presented in the paper. In 2004, Lu´ıs R. A. Gabriel, under the advice of one of the authors (H. M. Rodrigues), presented his Master Dissertation “Comportamento Assint´otico de Sistemas n˜ao Lineares Discretos”.

That work includes some results related to those of this paper, but for autonomous systems and can be found in the address www.teses.usp.br/teses/ disponiveis/55/55135/tde-12012005-230105/. Similar theoretical results were published by [Alberto et al., 2007], for autonomous systems. The general results in our paper can be regarded as the generalization of the two above papers and analogue of the uniform invariance principle for continuous dynamical systems developed previously in [Rodrigues et al., 2000, 2001] and [Gameiro & Rodrigues, 2001]. We also obtain some previous related results in [Rodrigues, 1996] for autonomous systems, in [Affraimovich & Rodrigues, 1998] for nonautonomous systems and in [Carvalho et al., 1998] for infinite dimensional parabolic systems. However, as will be shown, to use the same Liapunov function of a continuous system for its discrete analogue we will need to impose some restrictions on the step size of the discretization, and our general results, coupled with the Language multiplier method, allow us to derive these restrictions explicitly. Our work is mainly motivated by the synchronization problem of secured communication system, such as the discretization of the following masterslave Lorenz system consisting of a Master System x(t) ˙ = −ax(t) + ay(t), r y(t) ˙ = −y(t) − (x(t) + α(t)) − (x(t) + α(t))z, 4 5 z(t) ˙ = −bz(t) + (x(t) + α(t))y(t) − br, 4 and a Slave-System u(t) ˙ = −au(t) + av(t), r v(t) ˙ = −v(t) − (x(t) + α(t)) − (x(t) + α(t))w(t), 4 5 w(t) ˙ = −bw(t) + (x(t) + α(t))v(t) − br, 4 where α is the signal to be transmitted and the parameter values a, b, r are chosen so that the master system alone should exhibit chaotic behaviors in order to maintain the security of the communication. As indicated above, due to the arbitrary choice of the signal α, we will have to deal with nonautonomous systems. Roughly speaking, we say that two similar coupled systems synchronize if the

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distance between their solutions is sufficiently close to zero for sufficiently large values of t. A general approach in the continuous case was developed in [Gameiro & Rodrigues, 2001] to establish the synchronization property for the above master-slave system, that requires a uniform estimation of the attractors for both the master and the slave systems. Our ultimate goal is to develop a parallel theory for the discretized analogue and indeed a very general theory for robust synchronization of coupled discrete systems: this has obvious implications in terms of both hardware implementation and numerical simulations. This part of our work focuses on the first part of this theory — the uniform estimation of the global attractor. The remaining part of this paper is organized as follows. In Sec. 2, we state and prove our main general results of uniform dispativeness for nonautonomous systems and uniform invariance for autonomous systems. These general results are then illustrated by applications to the estimation of the global attractor for the discretized analogue of the well-known Lorenz system (Sec. 3) and a periodictime chaotic system (Sec. 4) when the parameters are in the region where chaotic behaviors are observed. Illustrated by these two examples is also the approach to extend the Liapunov function from continous systems to their discrete analogues, that involves a delicate treatment of the step size of the discretization. We will also obtain, in Sec. 4, a uniform estimation for the attractor of a nonlinear perturbation of a linear stable parametric system that requires some careful analysis of the spectral radius for a family of linear bounded operators in Banach spaces.

2. Main Results Let Λ be a subset of a Banach space E and let V : Rp × Λ → R+ and T : Rp × Z × Λ → Rn be continuous functions. Consider the discrete system

515

Therefore, if x(n) is a solution of (1) for n ∈ J, then V˙ (x(n), n, λ) = V (x(n + 1), λ) − V (x(n), λ) provided that n ∈ J and n + 1 ∈ J. Theorem 1 (Uniform Dispativeness). Let a, b: Rp →

R+ be continuous functions such that a(x) ≤ V (x, λ) ≤ b(x),

∀ (x, λ) ∈ Rp × Λ (2)

and a(x) → ∞ as |x| → ∞. For every ρ > 0, define Aρ := {x ∈ Rp : a(x) ≤ ρ} and Bρ := {x ∈ Rp : b(x) ≤ ρ}. We assume that there exists H > 0 so that for every ρ ∈ [0, H] the set Aρ is connected. Assume further that there exists a continuous function c : AH → R such that −V˙ (x, n, λ) ≥ c(x),

∀ (x, n, λ) ∈ AH × Z × Λ, (3)

and that the set {x ∈ AH : c(x) < 0} = ∅, and the set C := {x ∈ AH : c(x) ≤ 0} is bounded. Finally, assume that we can find positive constants R, µ and H so that max b(x) < R < ∞ − µ < min c(x) x∈C

x∈C

and

R + µ < H.

(4)

Then for each λ ∈ Λ the following holds: (i) If x0 ∈ BR and if x(n) is the solution of (1) with initial x0 , then x(n) ∈ AR+µ for every n ∈ Z+ . (ii) If x0 ∈ AH then there exists n0 = n0 (x0 , λ) ≥ 0 such that x(n0 ) ∈ BR and so x(n) ∈ AR+µ for every n ≥ n0 . (iii) If x0 ∈ AH and x(n) is the solution of (1) with initial value x0 , then there exists an increasing subsequence nj such that x(nj ) ∈ BR . (iv) If x(n) satisfies (1) for every n ∈ Z and if x(n) ∈ AH for every n ∈ Z, then x(n) ∈ AR+µ for every n ∈ Z.

A solution of (1) with n ∈ J for a given subset J of Z is a sequence x(n) that satisfies (1) for all n ∈ J. Unless specified otherwise, by a solution of (1), we will mean a solution defined for all n ≥ 0 (namely, J = Z+ ) and x(0) will be called the initial value. For each (x, n, λ) ∈ Rp × Z × Λ we define

Whenever the argument does not need to specify the dependence on λ, we shall write T (·, n) = T (·, n, λ) and V (·) = V (·, λ). We first note from the above assumptions that if 0 ≤ ρ ≤ H then Bρ ⊂ Vρ ⊂ Aρ , where Vρ := {x ∈ AH : V (x) ≤ ρ}. Also, C ⊂ BR and C = BR . We now claim that if R + µ ≤ ρ ≤ H then Vρ is positively invariant with respect to (1). To show this, we note that if x0 ∈ BR then −V˙ (x(0), 0) ≥ c(x0 ) ≥ −µ and so

V˙ (x, n, λ) := V (T (x, n, λ), λ) − V (x, λ).

V (x(1)) − V (x(0)) = V˙ (x(0), 0) ≤ µ.

x(n + 1) = T (x(n), n, λ).

(1)

Proof.

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Therefore, V (x(1)) ≤ V (x(0) + µ ≤ b(x0 ) + µ ≤ R + µ ≤ ρ. This shows that if x0 ∈ BR then x(1) ∈ Vρ . On the other hand, if x(0) ∈ Vρ \BR , then c(x(0)) > 0 and so −V (T (x(0), 0)) + V (x(0)) = −V˙ (T (x(0), 0)) ≥ c(x(0)) > 0, from which it follows that V (x(1)) = V (T (x(0), 0)) < V (x(0)) ≤ ρ. This, via the standard induction argument, implies that Vρ is positively invariant. In particular, if we take ρ = R + µ and if x(0) ∈ BR then x(n) ∈ VR+µ ⊂ AR+µ for all n ∈ Z+ and this completes the proof of (i). We now prove (ii). We claim that if x(0) ∈ AH \BR then there exists m ≥ 1, m = m(x(0), λ) such that x(m) ∈ BR . Let ρ = ρ(x(0), λ) := max{V (x(0)), R + µ}. Then the set Vρ is positively invariant. Let W := Vρ \BR . It is easy to see that if x ∈ W then c(x) > 0. Let β := minx∈W c(x). Since x(0) ∈ W we have c(x(0)) > 0. Then −[V (T (x(0)) − V (x(0))] = −V˙ (x(0), 0) ≥ c(x(0)) ≥ β or V (T (x(0)) − V (x(0)) ≤ −β and so, V (x(1)) = V (T (x(0), 0) ≤ V (x(0)) − β. If x(1) ∈ BR , we are done, otherwise if x(1) ∈ Vρ \BR then we can repeat the above procedure to obtain V (x(2)) ≤ V (x(1)) − β ≤ V (x(0)) − 2β. If x(2) ∈ BR , we are done, otherwise we continue the above procedure. Inductively, if after n−1 times we still have x(n − 1) ∈ / BR then V (x(n)) ≤ V (x(0)) − nβ. This shows that there exists m such that x(m) ∈ BR , otherwise we would have a contradiction because V ≥ 0. This proves the claim. Conclusions (ii) and (iii) are obvious consequence of the above claim. It remains to prove (iv). Under the hypothesis of (iv), and suppose that there exists m ∈ Z such that x(m) ∈ AH \VR+µ . Let ν := min{c(x) : x ∈ AH \VR+µ }. It is clear that ν > 0. From (ii) it follows that x(m − 1) ∈ AH − VR+µ . Then −[V (x(m)) − V (x(m − 1) = −V˙ (T (x(m − 1), m − 1) ≥ c(x(m − 1)) ≥ ν.

This implies that V (x(m)) + ν ≤ V (x(m − 1)). Since x(m − 2) ∈ AH \VR+µ , if we repeat the above procedure we obtain, V (x(m)) + 2ν ≤ V (x(m − 1)) + ν ≤ V (x(m − 2)). Inductively, after n steps we will obtain V (x(m)) + n ν ≤ V (x(m − n)) ≤ b(x(m − n)). This would give a contradiction because b is a bounded function on AH .  We now consider a special case where the system is autonomous. The next proposition can be found in [LaSalle, 1976]. Proposition 1 (Invariance Principle). Let V : G → R be continuous, where G is an open set of Rp . Let T : Rp → Rp be a continuous function. Suppose that V˙ (x) = V (T (x))−V (x) ≤ 0 for every x ∈ G. If x0 ∈ G and T n (x0 ) ∈ G for every n ∈ N and is bounded, then the ω-limit set ω(x0 ) = ∅ and there exists a real number c such that T n (x0 ) → M ∩ V −1 (c) where M is the largest positively invariant set contained in the set E := {x ∈ Rp : V˙ = 0} ∩ G. Theorem 2 (Uniform Invariance Principle). Let T : Rp × Λ → Rp , V : Rp × Λ → R+ . Suppose that there are continuous functions a, c : Rp → R such that 0 ≤ a(x) ≤ V (x, λ), a(x) → ∞ as |x| → ∞, and −V˙ (x, λ) ≥ c(x) ≥ 0 for every x ∈ Rp and every λ ∈ Λ. Let x0 ∈ Rp and λ0 ∈ Λ. Then Ec := {x ∈ Rp : c(x) = 0} = ∅ and T n (x0 , λ0 ) tends, as n → ∞, to the largest invariant set of T (·, λ0 ) contained in Ec .

We denote T (·) = T (·, λ0 ) and V (·) = V (·, λ0 ). Since c(x) ≥ 0, ∀ x ∈ Rp , −V˙ (x) ≥ c(x) ≥ 0 ⇒ −V˙ (x) ≥ 0, ∀ x ∈ Rp . Moreover, Proof.

V˙ (x) ≤ 0 ⇒ V (T (x0 )) − V (x0 ) ≤ 0 ⇒ V (T (x0 )) ≤ V (x0 ) ⇒ V (T n (x0 )) ≤ · · · ≤ V (T 2 (x0 )) ≤ V (T (x0 )) ≤ V (x0 ) ⇒ V (T n (x0 )) ≤ V (x0 ) =: ρ(x0 , l0 ) = ρ ⇒ T n (x0 ) ∈ Vρ = Vρ (λ0 ) := {x ∈ Rm : V (x) ≤ ρ}.

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Then T n (x0 ) is bounded because x ∈ Vρ ⇒ V (x, λ) ≤ ρ ⇒ a(x) ≤ V (x, λ) ≤ ρ ⇒ a(x) ≤ ρ. Since a(x) → ∞ as |x| → ∞, then Aρ := {x ∈ Rp : a(x) ≤ ρ} is bounded. We conclude that V is a Liapunov function in Rp . Since every solution x(n) = x(n, λ0 ) = T n (x0 ) is bounded in Rp , ∀ n ∈ N, then by Proposition 1, x(n) tends to the largest invariant set contained in E = Eλ0 = {x ∈ Rp : V˙ (x) = 0} as n → ∞. Furthermore, x ∈ E ⇒ V˙ (x) = 0 ⇒ 0 = −V˙ (x) ≥ c(x) ≥ 0. ⇒ c(x) = 0 ⇒ x ∈ Ec ⇒ E ⊂ Ec . By Proposition 1, ∅ = ω(x(0)) ⊂ E, and this completes the proof. 

3. Applications to Discretized Lorenz System As an application of the general results in the previous sections, we consider the map T (x, y, z, n, λ) := (T1 (x, y, z, n, λ), T2 (x, y, z, n, λ), T3 (x, y, z, n, λ)), where x, y, z ∈ R, n ∈ Z, and λ := (a, r, b, (αn ), h), where a ∈ [am , aM ], r ∈ [rm , r+ ], b ∈ [bm , bM ], h are positive real numbers and (αn ) ∈ ∞ . The map T is obtained by discretizing the Lorenz equation, and hence T1 (x, y, z, n, λ) = x + h[−ax + ay],  r T2 (x, y, z, n, λ) = y + h −y − (x + α) 4  − (x + α)z ,   5 T3 (x, y, z, n, λ) = z + h −bz + (x + αn )y − br . 4 (5) Our goal in this section is to use the above uniform dissipativeness theorem to obtain an uniform estimate of the attractor of T . Let us consider the function V (x, y, z, λ) := rx2 + 4ay 2 + 4az 2 . Then for every (x, y, z, λ) we have the following inequalities: a(x, y, z) := rm x2 + 4am y 2 + 4am z 2 ≤ V (x, y, z, λ), and V (x, y, z, λ) ≤ rM x2 + 4aM y 2 + 4aM z 2 := b(x, y, z).

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Let us compute the derivative: −V˙ (x, y, z, λ, n) = −V (T1 , T2 , T3 ) + V (x, y, z) = rx2 + 4ay 2 + 4az 2  − r[x + h(−ax + ay)]2   2 r + 4a y + h −y − (x + α) − (x + α)z 4 2

  5 +4a z + h −bz + (x + αn )y − br 4 = rx2 + 4ay 2 + 4az 2  − r[x2 + h(2x)(−ax + ay) + h2 (−ax + ay)2 ]    r + 4a y 2 + h(2y) −y − (x + α) − (x + α)z 4 2   r 2 + h −y − (x + α) − (x + α)z 4   5 2 + 4a z + h(2z) −bz + (x + αn )y − br 4

 2 5 2 + h −bz + (x + αn )y − br 4 = h[2arx2 − 2arxy + 8ay 2 + 2ary(x + αn ) + 8abz 2 + 10abrz] − h2 g(x, y, z, n, λ), where g(x, y, z, n, λ) := r(−ax + ay)2  2 r + 4a −y − (x + αn ) − (x + α)z 4 2  5 + 4a −bz + (x + αn )y − br . 4 Therefore, −V˙ (x, y, z, λ, n) h = 2arx2 + 8ay 2 + 2aryαn + 8abz 2 + 10abrz + hg(x, y, z, n, λ)

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≥ 2am rm x2 + 8am y 2 − 2aM rM |y||αn | + 8am bm z 2 − 10aM bM rM |z| − h|g(x, y, z, n, λ)| ≥ 2am rm x2 + 8am y 2 − 2aM rM |y|γ + 8am bm z 2 − 10aM bM rM |z| − h|g(x, y, z, n, λ)|,

−V˙ (x, y, z, n, λ) ≥ c(x, y, z),

where we assumed that |αn | ≤ γ, ∀ n ∈ Z+ . Let d(x, y, z) := 2am rm x2 + 8am y 2 − 2aM rM |y|γ + 8am bm z 2 − 10aM bM rM |z|. Completing the squares, we have d(x, y, z) = 2am rm x2 + 8am y 2 − 2|y|  +

aM rM γ 8am

2

 −

+ 8am bm z 2 − 2|z|  +

5aM bM rM 8am bm

2

aM rM γ 8am

aM rM γ 8am



5aM bM rM 8am bm



aM rM γ = 2am rm x + 8am |y| − 8am  5aM bM rM 2 + 8am bm |z| − 8am bm 2



2

2

(aM rM γ)2 (5aM bM rM )2 − . 8am 8am bm

The minimum of the function d is given by −((aM rM γ)2 /8am ) − ((5aM bM rM )2 /8am bm ). Let D := {(x, y, z) : d(x, y, z) ≤ 0}. Let R1 and R be given so that max{b(x, y, z) : (x, y, z) ∈ D} < R1 < R. We choose H larger than R. Given δ > 0 there exists h0 > 0, sufficiently small, so that h max{|g(x, y, z, n, λ)| : (x, y, z) ∈ AH } < δ for 0 < h < h0 . If (x, y, z) ∈ AH , we have d(x, y, z) − |hg(x, y, z)| ≥ d(x, y, z) − := d1 (x, y, z).

(aM rM γ)2 (5aM bM rM )2 − −δ 8am 8am bm

∀ (x, y, z) ∈ AH .

The minimum of c(x, y, z) in C is given by   (aM rM γ)2 (5aM bM rM )2 + +δ −h 8am 8am bm   (aM rM γ)2 (5aM bM rM )2 + +δ ≥ −h0 8am 8am bm > −µ

2

5aM bM rM 8am bm 

For C := {(x, y, z) : d1 (x, y, z) ≤ 0}, one can show that if δ is sufficiently small, then there exists h0 such that max{b(x, y, z) : (x, y, z) ∈ C} < R for every 0 < h < h0 . Now letting c(x, y, z) := hd1 (x, y, z), we have that

for an appropriate µ > 0, that can be made as small as we wish if h0 is sufficiently small. Take h0 sufficiently small so that R + µ < H. Then all the conditions of Theorem 1 are satisfied for every 0 < h < h0 and so the attractor is contained in AR+µ . It is important to determine the maximum of b(x, y, z) on D, that will occur on the boundary of D, that is an ellipsoid. Due to the geometry of D, this maximum coincides with the maximum of b(0, y, z) on the set {(y, z) : d(0, y, z) = 0}, that is a ball. To find it, we will use the Lagrange multipliers method. Let (y, z) := b(0, y, z) = 4aM y 2 + 4aM z 2 , and q(y, z) := d(0, y, z), that is q(y, z) = 8am y 2 − 2aM rM |y|γ + 8am bm z 2 − 10aM bM rM |z|. Then y = βqy if and only if 8aM y = 16βam y − 2aM rM γβ and z = βqz if and only if 8aM z = 16βam bm z − 10aM bM rM β. This implies y=

aM rM γβ , 8βam − 4aM

z=

5aM bM rM β . 8βam bm − 4aM

(6)

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Substituting these values to q(y, z) = 0, we get  8am

aM rM γ 2 aM rM γβ − 8βam − 4aM 8am  5aM bM rM 2 5aM bM rM β + 8am bm − 8βam bm − 4aM 8am bm

=

(aM rM 8am

γ)2

+

(5aM bM rM 8am bm

)2



aM rM γ 8am =−

2

 −

2

699 799 156 641 ∼ = 594.819468623618, 1 176 490 000

we have −V˙ (x, y, z, λ, n) ≥ d1 (x, y, z) h

.

> −595 − δ = −596,

Taking γ = 4, after some calculations, the above equation can be written as

and so −V˙ (x, y, z, λ, n) ≥ hd1 (x, y, z) = c(x, y, z) > −596 h > −596 h0

16bm (aM )2 (2βam bm − aM )2

for all h ∈ (0, h0 ). If we restrict h0 so that h0 < 1/596, then we have

+ (5aM bM )2 (2βam − aM )2 = [16bm + (5bM )2 ](2βam bm − aM )2 × (2βam − aM )2 .

c(x, y, z) > −1 := −µ.

(7)

Now we have a fourth order polynomial function in β, that we will solve numerically using the software Mathematica. Consider, for example, the situation where the parameters a, r, b vary near 10, 28, 8/3, respectively, with an uncertainty of 2%. Then we have: 49 2a = , 100 5

aM = a +

51 2a = , 100 5

rm = r −

686 2r = , 100 25

rM = r +

714 2r = , 100 25

bm = b −

196 2b = , 100 75

BM = b +

am = a −

5aM bM rM 8am bm

519

68 2b = . 100 25

Consequently, we can apply Theorem 1 to conclude that the attractor is contained in the ellipsoid: AR+µ = A96 999+1 = A97 000 = {(x, y, z) ∈ AH : rm x2 + 4am y 2 + 4am z 2 ≤ 97 000} and

(x, y, z) ∈ AH :  49 2 196 2 196 2 x + y + z ≤ 300 000 . 5 5 5

is the basin of attraction.

Now we can solve Eq. (7) in β to obtain the values β = 0, β = 0.756499 and two complex imaginary solutions. The first gives a minimum and the second gives a maximum for the function b on the ellipsoid {(x, y, z) : d(x, y, z) = 0}. Substituting the above values in (6) we obtain for the maximal point: (xM , yM , zM ) = (0, 47.6246, 9.64912). Now the maximum of the function b in the surface of the ellipsoid is given by 2 2 + 4aM zM b(xM , yM , zM ) = 4aM yM

= 96337.4. We choose R1 = 96 400 and R = 96 999, H = 300 000 and δ = 1. Since the minimum of the function d is given by

Fig. 1.

Ellipsoid that contains discrete Lorenz attractor.

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We refer to Fig. 1 for a schematic illustration of various functions and their graphs involved.

4. Applications to a Time-Periodic System The following continuous chaotic system was studied in [Afraimovich et al., 1986], where they were interested in analyzing coupled damped-excited Duffing equations

T2 (x, y, λ) = y + h[−αy − ωx − (q cos t + x2 )x], where λ := (α, ω, q, t). To find an estimate of the attractor, we use the following Liapunov function: V (x, y, λ) :=

It is easy to see that a ≤ V ≤ b, where

x ¨ + αx˙ + ωx + (q cos t + x2 )x = 0.

a(x, y) =

We are here considering a corresponding discrete system: T1 (x, y, λ) = x + hy, 1 1 1 V˙ = ωx2 + x4 + y 2 + 5xy − 2 4 2

1 2 1 4 1 2 ωx + x + y + 5xy. 2 4 2

1 1 1 ωm x2 + x4 + y 2 + 5xy, 2 4 2

1 1 1 b(x, y) := ωM x2 + x4 + y 2 + 5xy. 2 4 2 Then

1 1 ω(x + hy)2 + (x + hy)4 2 4

 1 2 2 2 + [y + h(−αy − ωx − (q cos t + x )x)] + 5(x + hy)(y + h(−αy − ωx − (q cos t + x )x)) 2

1 1 1 2 1 4 1 2 ω(x2 + 2hxy) + (x4 + 4hx3 y) = ωx + x + y − 2 4 2 2 4  1 2 2 2 + [y + 2hy(−αy − ωx − (q cos t + x )x)] + 5xy + 5xh(−αy − ωx − (q cos t + x )x) + O(h2 ) 2 = −h{ωxy + x3 y + [y(−αy − ωx − (q cos t + x2 )x)] + 5x(−αy − ωx − (q cos t + x2 )x)} + O(h2 ) = −h{ωxy + x3 y − αy 2 − ωxy − qyx cos t − yx3 − 5αxy − 5ωx2 − 5qx2 cos t − 5x4 } + O(h2 ) = h{αy 2 + qyx cos t + 5αxy + 5ωx2 + 5qx2 cos t + 5x4 } + O(h2 ) ≥ h{αm y 2 − (qM + 5αM )|x||y| + 5ωm x2 − 5qM x2 + 5x4 } + O(h2 ) 

√ (qM + 5αM ) 2 2 2 4 |x|| αm y| + 5ωm x − 5qM x + 5x + O(h2 ) = h αm y − √ αm

 (qM + 5αM )2 2 αm 2 2 2 2 4 ≥ h αm y − y + 5ωm x − 5qM x + 5x + O(h2 ) x − 2αm 2   

(qM + 5αM )2 αm 2 2 4 y − + 5qM − 5ωm x + 5x + O(h2 ). ≥h 2 2αm Let d(x, y) := (αm /2)y 2 − [((qM + 5αM )2 / 2αm ) + 5qM − 5ωm ]x2 + 5x4 and let D := {(x, y) : d(x, y) ≤ 0}. If we let J := ((qM + 5αM )2 /2αm ) + 5qM − 5ωm , we have d(x, y) = (αm /2)y 2 − Jx2 + 5x4 . Now we consider the parameters varying with 2% of uncertainty from the basic values α = 1, ω = 1, q = 50, that is:

αm = 1 −

49 2 = , 100 50

αM = 1 +

51 2 = , 100 50

ωm = 1 −

49 2 = , 100 50

ωM = 1 +

51 2 = , 100 50

qm = 50 − 1 = 49,

qM = 50 + 1 = 51.

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521

 2   25αm (20p − 1) − 2(pαm − 1) (20p − 1)J   

+ 10(pαm − 1)[25 + (pαm − 1)(wM + 2pJ)] = 0. (11)

Using Mathematica we solve numerically the above equation and we find the following Lagrange multipliers: p = 1.1437812609853224, p = 0.9037535609612867, p = 0.09355631580936598. The maximum will be obtained using the first Lagrange multiplier. Using (10) and (9) we obtain the maximum point: (xM , yM ) = (14.267595144549922, 590.0301939691485). This gives b(xM , yM ) = 226 623. As in the previous example, we take R1 = 226 624 and R = 226 625. We can also compute the minimum of the function d, and use Theorem 1 to conclude that for sufficiently small h the attractor is contained in the set Fig. 2.

Ellipsoid and discrete Lorenz attractor.

As in the previous example, using Lagrange multipliers we compute the maximum of the function b on the boundary of D. We denote by p the Lagrange multiplier. Then bx = pdx if and only if wM x + x3 + 5y = −2pJx + 20px3 , or 1 y = [−wM x − x3 − 2pJx + 20px3 ]. 5

(8)

Similarly, by = pdy if and only if, y=

5x . pαm − 1

(9)

For x = 0, from (8) and (9) we obtain x2 =

25 + (pαm − 1)(wM + 2pJ) . (pαm − 1)(20p − 1)

(10)

Using x and y given respectively in (9) and (10) in the equation d(x, y) = 0, we obtain

AR+1 = {(x, y) : a(x, y) ≤ 226 625}

1 1 1 = (x, y) : ωm x2 + x4 + y 2 + 5xy 2 4 2  ≤ 226 625 . Figure 3 illustrates the estimate and the above argument.

4.1. Nonlinear perturbations of parametrized linear stable systems In this section, we consider nonlinear perturbations of a linear stable system with parameters, and we wish to obtain a uniform bound for the attractor. We start with some notations. Let X be a Banach space and let L(X) be the space of bounded operators from X to X, with its usual norm, and let A ∈ L(X) and λ ∈ C. If A − λ := A − λI is invertible with Rλ (A) := (A − λ)−1 ∈ L(X), the resolvent operator of A, then we say that λ ∈ ρ(A), the resolvent set of A. We indicate its complement in C by σ(A), the spectrum of A. It is well known that σ(A) is compact and limn→∞ |An |1/n = inf n=1,2,... |An |1/n = max{|λ|, λ ∈ σ(A)} := r(A), the spectral radius of A. See [Kato, 1976, p. 27].

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600 400 200 0 -200 -400 -600

-30

-20

-10

0

10

20

30

Fig. 3. Set D (butterfly like), set AR+1 (largest set) and discrete Duffing attractor.

f (γn ) > 0, ∀ n ∈ N. Since Γ ⊂ X is compact, we can also suppose that γn → γ0 ∈ Γ. Let ε > 0. As f is upper semicontinous, there exists δ > 0 such that f (γ) < f (γ0 ) + ε if γ ∈ Bδ (γR0 ). Since γn → γ0 , there exists n0 ∈ N such that for n ≥ n0 , we have γn ∈ Bδ (γ0 ), which implies that f (γn ) < f (γ0 ) + ε and this is a contradiction. Then f is bounded from above. Let S := supγ∈Γ f (γ) ∈ R. Then, S ≥ f (γ), ∀ γ ∈ Γ. Then there exists γn ∈ Γ such that f (γn ) → S. We can suppose that γn → γ ∈ Γ. Let us show that f (γ) = S. In fact let ε > 0 be a real number. Since f is upper semicontinous, there exists δ > 0 such that f (γ) < f (γ) + ε, if γ ∈ Bδ (γ). Since γn → γ, let n0 ∈ N be such that for n ≥ n0 , we have γn ∈ Bδ (γ), which implies that f (γn ) < f (γ) + ε. Then by leting n → ∞, we have S ≤ f (γ) + ε for every ε > 0. Therefore, S ≤ f (γ) and so, S = f (γ).  In a similar way, we obtain

We will also need the following technical lemma (see [Kato, 1976, p. 280]): Lemma 1. A map L(X)  A → σ(A) is upper semicontinous, that is, given A ∈ L(X) and ε > 0, there exists δ > 0 such that σ(B) ⊂ Vε (σ(A)) if |B − A| < δ and B ∈ L(X). Proposition 2. The map L(X)  A → r(A) is upper semicontinous, that is, given A ∈ L(X) and ε > 0, there exists δ > 0 such that r(B) < r(A) + ε if |B − A| < δ and B ∈ L(X). Proof. Let A ∈ L(X) and ε > 0. Since the map L(X)  A → σ(A) is upper semicontinuous, there exists δ > 0 such that σ(B) ⊂ Vε (σ(A)) if |B − A| < δ and B ∈ L(X). There exists λ1 ∈ σ(B) such that r(B) = |λ1 |. Since λ1 ∈ Vε (σ(A)), there exists λ0 ∈ σ(A) such that |λ1 − λ0 | < ε. Therefore, r(B) < r(A) + ε, because

|λ1 | − |λ0 | ≤ |λ1 − λ0 | < ε ⇒ |λ1 | < |λ0 | + ε ≤ r(A) + ε.



Proposition 3. Let X be a Banach space, Γ a com-

pact subset of a Banach space and f : Γ → R an upper semicontinuous function. Then there exists γ ∈ Γ such that f (γ) ≥ f (γ), ∀ γ ∈ Γ. We claim that f is bounded from above in Γ. In fact, let us suppose that there exists a sequence γn ∈ Γ such that f (γn ) → ∞. We can assume that

Proof.

Proposition 4. Let X be a Banach space, Γ a compact subset of another Banach space and f : Γ → R a lower semicontinuous function, that is, given A ∈ L(X) and ε > 0, there exists δ > 0 such that f (B) > f (A)−ε if |B−A| < δ and B ∈ L(X). Then there exists γ ∈ Γ such that f (γ) ≤ f (γ), ∀ γ ∈ Γ. Theorem 3. Let X be a Banach space, Γ a compact subset of another Banach space. We assume that the map Γ  γ → Aγ ∈ L. We also suppose that |λ| < 1, ∀ λ ∈ σ(Aγ ). Then there exists θ ∈ (0, 1) such that r(Aγ ) ≤ θ < 1, ∀ γ ∈ Γ. Proof. From Proposition 1, the map L(X)  Aγ → r(Aγ ) is upper semicontinuous. Then the map γ ∈ Γ → r(Aγ ) is also upper semicontinuous. By Proposition 3, there exists γ ∈ Γ such that r(Aγ ) ≥ r(Aγ ), ∀ γ ∈ Γ. Taking θ := r(Aγ ) = supλ∈σ(Aγ ) |λ|, we have r(Aγ ) ≤ θ < 1, ∀ γ ∈ Γ. 

We note that for any compact K ⊂ C, K compact, there exists a Hilbert space H and an operator A ∈ L(H) such that σ(A) = K. This result can be found in [Taylor, 1958, p. 263]. We also note that for a given Banach space (X, | · |), and given A ∈ L(X) and ε > 0, there exists  ·  =  · A,ε , a norm equivalent to | · |, such that r(A) ≤ A ≤ r(A) + ε. This result can be found in [Irwin, 1980] or in [Rodrigues & Sol` a-Morales, 2004] for a more general case. As a consequence,

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So, we have

we have Corollary 1. Let (X, | · |) be a Banach space and

A ∈ L(X). Suppose |λ| < 1, ∀ λ ∈ σ(A). Then A is a contraction with respect to some norm equivalent to | · | on X.

  k0 ∞   |F (γ1 ) − F (γ2 )| =  fk (γ1 ) + fk (γ1 ) k=0 k=k0 +1    fk (γ2 ) − fk (γ2 ) −  k=0 k=k0 +1 k0 

Proposition 5. Let X be a Banach space, Γ a compact subset of another Banach space. Suppose that the map Γ  γ → Aγ ∈ L(X) is continuous. Then there exist M > 0 and θ ∈ (0, 1) such that |Akγ | ≤ M θ k for every γ ∈ Γ and k ∈ N.

Let γ ∈ Γ, k ∈ N, A = Aγ ∈ L(X), M := supλ∈Cθ |(Aγ − λ)−1 | = maxλ∈Cθ |(Aγ − λ)−1 |, Cθ , the circle in the complex plane with center at the origin and radius θ, with the positive orientation. Then by [Taylor, 1958, p. 287], we have  1 λk Rλ dλ Ak = 2πi Cθ  1 k λk (A − λ)−1 dλ ⇒A = 2πi Cθ Proof.

⇒ |Ak | ≤

1 sup |λk (Aλ )−1 |2πθ ≤ M θ k+1 2π λ∈Cθ

≤ M θk.



∞ 

 k 0     |fk (γ1 ) − fk (γ2 )| + 2ε. =   k=0

We consider now fk , k = 0, . . . , k0 . Then, ∃ δk = δk (ε) > 0 for k = 0, . . . , k0 |fk (γ1 ) − fk (γ2 )|
0 such that |Akγ | ≤ M θ k for every γ ∈ Γ, θ ∈ (0, 1) and k ∈ N. Then, Proof.

|Bγ | ≤

∞ 

|(A∗γ )k ||C||Akγ | = |C|

k=0

∞ 

|Aγ |2k

k=0

≤ |C|M 2

∞ 

θ 2k < ∞.

k=0

∞

∀ γ ∈ Γ.



From now on we will assume that H = Rp or H = Cp and consider the p × p matrices  ∞    (A∗γ )k CAkγ , for γ ∈ Γ, Aγ , C and Bγ := k=0

k=0

523

Then the series k=0 (A∗γ )k CAkγ is absolutely and uniformly convergent and the map γ → Bγ is continuous by Proposition 5. 

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Recall that for a given V : Rp → R, we say that V is positive definite if V (0) = 0 and V (x) > 0 for x = 0 in a neighborhood of the origin. Let D be a p × p matrix. We say that D is positive definite if the quadratic form (Dx|x) := x∗ Dx is positive definite. The following results can be verified easily. ∞ ∗ k k Proposition 7. If Bγ = k=0 (Aγ ) CAγ then the following statements hold: (i) A∗γ Bγ Aγ − Bγ = −C; (ii) If C is positive definite so is Bγ ; (iii) If C is self-adjoint so is Bγ .

Then, c0 ≤ y ∗ Cy ⇒ c0 ≤ ⇒ c0 ≤

x x∗ C ∗ |x | |x|

1 ∗ x Cx ⇒ x∗ Cx ≥ c0 |x|2 . |x|2

We now prove (ii). By the continuity of Vγ and the compactness of S, it follows that there exist constants c1 , c2 > 0 such that c1 := min(x,γ)∈S×Γ x∗ Bγ x and c2 := max(x,γ)∈S×Γ x∗ Bγ x. Let x = 0 and y = (x/|x|) ∈ S. Then, c1 ≤ Vγ (y) ≤ c2 ⇒ c1 ≤ y ∗ Bγ y ≤ c2

We can now consider the linear parametrized system:

⇒ c1 ≤

x∗ x Bγ ≤ c2 ∗ |x | |x|

xn+1 = Aγ xn

⇒ c1 ≤

1 ∗ x Bγ x ≤ c2 |x|2

Theorem 5. Let C, Aγ , Bγ be defined as above. Let C be positive definite and suppose |λ| < 1, ∀ λ ∈ σ(Aγ ). Let Vγ (x) = (Bγ x|x) = x∗ Bγ x, ∀ x. Then for every x and every γ ∈ Γ, we have:

(i) Vγ is positive definite in the whole space and there exists a constant c0 > 0 such that −V˙ γ (x) = x∗ Cx ≥ c0 |x|2 , ∀ x ∈ Rp ; (ii) There exist constants c1 , and c2 > 0 such that c1 |x|2 ≤ Vγ (x) ≤ c2 |x|2 , ∀ x ∈ Rp . Proof. We first prove (i). By Theorem 4, Bγ is a continuous function of γ. Since C is positive definite then so is Bγ thanks to Proposition 7(ii). This implies that V is positive definite. To simplify the notations we set A := Aγ , B := Bγ and V := Vγ . Since from Proposition 7(i), we have A∗ BA − B = −C, we have

V˙ (x) = V (Ax) − V (x) = (Ax)∗B(Ax) − x∗Bx = x∗ (−C + B)x − x∗ Bx = −x∗ Cx. Since C is positive definite it follows that −V is definite positive in the whole space. Therefore, V is a Liapunov function for the dynamical system defined by xn+1 = Aγ (xn ). Let us prove now that there exists constant c0 > 0 such that −V˙ γ (x) = x∗ Cx ≥ c0 |x|2 , ∀ x ∈ Rp , ∀ γ ∈ Γ. In fact by the continuity of the function x → x∗ Cx and the compactness of S := {x ∈ Rp : |x| = 1}, it follows that there exists a constant c0 > 0 such that c0 := min(x,γ)∈S×Γ x∗ Cx. Let x = 0 and y = x/|x| ∈ S.

and so, c1 |x|2 ≤ Vγ (x) ≤ c2 |x|2 , ∀ x ∈ Rp .



We can now state the main result of this section for nonlinear perturbations of a linear parametrized system: Theorem 6. Let Aγ and Bγ be real matrices as in (12), Vγ (x) = (Bγ x|x) = x∗ Bγ x, x ∈ Rp , C positive definite and self-adjoint. We suppose that |λ| < 1, ∀ λ ∈ σ(Aγ ). Let fγ :∈ Rp →∈ Rp be continuous with |fγ (x)| ≤ L, ∀ x ∈ Rp , e Tγ := Aγ +fγ .

(i) Then there exists a function c :∈ Rp → R such that −V˙ γ (x) ≥ c(x), ∀ x ∈ Rp and there exists a constant M > 0 such that c(x) > 0, if |x| ≥ M, and c(x) → ∞ as |x| → ∞. (ii) There exist constants c1 , c2 > 0 such that c1 |x|2 ≤ Vγ (x) ≤ c2 |x|2 , ∀ x ∈ Rp , γ ∈ Γ. We first prove (i). By Theorem 4, Bγ is a continuous function of γ. To simiplify the notations we let A := Aγ , B := Bγ , f := fγ , T := Tγ and V := Vγ . By Proposition 7, we have A∗BA−B = −C, B is positive definite and self-adjoint. Next we are going to show that function c : ∈ Rp →∈ R such that −V˙ γ (x) ≥ c(x). In fact, Proof.

V˙ (x) = V (T x) − V (x) = V (Ax + f (x)) − V (x) = (Ax + f (x))∗B(Ax + f (x)) − x∗Bx = (x∗A∗ + f ∗ (x))B(Ax + f (x)) − x∗Bx

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= x∗ A∗BAx + x∗ A∗Bf (x) + f ∗ (x)BAx + f ∗ (x)Bf (x) − x∗Bx = x∗ A∗BAx + x∗ A∗Bf (x) + (x∗ A∗Bf (x))∗ + f ∗ (x)Bf (x) − x∗Bx = x∗ A∗BAx + 2x∗ A∗Bf (x) + f ∗ (x)Bf (x) − x∗Bx = x∗ (−C + B)x + (2x∗A∗Bf (x) + f ∗ (x)Bf (x)) − x∗Bx = −x∗ Cx + (2x∗A∗Bf (x) + f ∗ (x)Bf (x)). Then, V˙ (x) = −x∗ Cx + (2x∗ A∗ Bf (x) + Since f is bounded there exist constants P > 0 and Q > 0 such that 2x∗ A∗ Bf (x) + f ∗ (x)Bf (x) ≤ P |x| + Q, because f ∗ (x)Bf (x)).

2x∗ A∗Bf (x) + f ∗ (x)Bf (x) ≤ |2x∗ A∗Bf (x) + f ∗ (x)Bf (x)|

525

of application problems including the synchronization of coupled chaotic systems arising from secured communications, the main issue that motivates this series of papers. Obtaining such an estimate is highly nontrivial since the desired chaotic behavior (of the master system) are usually associated with the coexistence of both expansion and contraction in different subspaces. Not to mention that any such reasonable estimation should involve small step size of the discretization if the considered discrete system is the discretization of a continuous system (of differential equations). In this paper, we establish some general results based on the Liapunov direct method, for the uniform estimation of the attractor of nonautonomous discrete systems with parameters, and we also show how the classical Lagrange multiplier method can be effectively used to construct the required Liapunov function for two important examples from secured communications. This is the basis for some general synchronization results of a discrete Master-slave system, to be presented in a future study.

≤ |2x∗ A∗Bf (x)| + |f ∗ (x)Bf (x)| ≤ |2x∗A∗Bf (x)| + |B| ≤ P |x| + Q. As in Theorem 5, there exists a constant c0 > 0 such that x∗ Cx ≥ c0 |x|2 . Now, −V˙ (x) = x∗Cx − (2x∗A∗Bf (x) + f ∗ (x)Bf (x)) ≥ x∗Cx − |2x∗A∗Bf (x) + f ∗ (x)Bf (x)| ≥ c0 |x|2 − P |x| − Q =: c(x). Then, −V˙ γ (x) ≥ c(x) for c(x) = c0 |x|2 − P |x| − Q, with positive constants c0 , P and Q. Moreover, c(x) → ∞ as |x| → ∞. Then there exists η > 0 such that c(x) > 0 for |x| ≥ η. The proof for (ii) is the same as that of Theorem 5.  We conclude with a simple remark that since −V˙ (x) ≥ c(x), we can use Theorems 1 and 2 to obtain uniform dissipativeness and uniform invariance for the nonlinear system.

5. Conclusions Estimations of the global attractors for dynamical systems with parameters is important for a number

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Ott, E., Sauer, T. & Yorke, J. A. [1994] Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems, Wiley Series in Nonlinear Science. Rodrigues, H. M. [1996] “Abstract methods for synchronization and applications,” Appl. Anal. 62, 263–296. Rodrigues, H. M., Alberto, L. F. C. & Bretas, N. C. [2000] “On the invariance principle, generalizations and aplications to synchronism,” IEEE Trans. Circuit Syst.-I: Fund. Th. Appl. 47, 730–739. Rodrigues, H. M., Alberto, L. F. C. & Bretas, N. C. [2001] “Uniform invariance principle and synchronization, robustness with respect to parameter variation,” J. Diff. Eqs. 169, 228–254. Rodrigues, H. M. & Sol` a-Morales, J. [2004] “Linearization of class C 1 for contractions on Banach spaces,” J. Diff. Eqs. 201, 351–382. Taylor, A. E. [1958] Introduction to Functional Analysis (John Wiley & Sons, NY).