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Chaotic behavior of discrete-time linear inclusion dynamical systems

Authors:	Xiongping Dai Tingwen Huang Yu Huang Yi Luo Gang Wang Mingqing Xiao

Institution:	1. Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China;2. Texas A & M University at Qatar, c/o Qatar Foundation, P.O. Box 5825 Doha, Qatar;3. Department of Mathematics, Zhongshan (Sun Yat-Sen) University, Guangzhou 510275, People’s Republic of China;4. Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, USA

Abstract:	Given any finite family of real d-by-d nonsingular matrices ${S_{1}, \dots, S_{l}},$ by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system: $\begin{matrix} x_{n} \in {S_{k} x_{n ? 1}; 1 \leq k \leq l}, x_{0} \in R^{d} and n \geq 1, \end{matrix}$ we study the chaotic dynamics of the state trajectory (x_n(x₀, σ))_{n ≥ 1} with initial state $x_{0} \in R^{d},$ governed by a switching law $σ : N \to {1, \dots, l}$ . Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all $x_{0}, y_{0} \in R^{d}, x_{0} \neq y_{0},$ $\begin{matrix} \underset{n \to + \infty}{lim inf} ∥ x_{n} (x_{0}, σ) ? x_{n} (y_{0}, σ) ∥ = 0 and \underset{n \to + \infty}{lim sup} ∥ x_{n} (x_{0}, σ) ? x_{n} (y_{0}, σ) ∥ = \infty . \end{matrix}$ This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (x_n(x₀, σ))_{n ≥ 1} are extremely sensitive to the initial states $x_{0} \in R^{d}$ . We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.

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