Chaotic behavior of discrete-time linear inclusion dynamical systems

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:

$$\begin{array}{c}\hfill x_n\in \{S_k{x}_{n-1};1\le k\le l\},x_0\in {\mathbb{R}}^d\mathrm{and}n\ge 1,\end{array}$$

we study the chaotic dynamics of the state trajectory (x_n(x₀, σ))_{n ≥ 1} with initial state $x_0\in {\mathbb{R}}^d,$ governed by a switching law $\sigma :\mathbb{N}\to \{1,\dots ,l\}$. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all $x_0,y_0\in {\mathbb{R}}^d,x_0\ne y_0,$

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; inf}\parallel x_n(x_0,\sigma )-x_n(y_0,\sigma )\parallel =0\mathrm{and}\underset{n\to +\infty }{lim\; sup}\parallel x_n(x_0,\sigma )-x_n(y_0,\sigma )\parallel =\infty .\end{array}$$

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 $x_0\in {\mathbb{R}}^d$. 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.

Introduction

Chaos is not only an interesting but also an important subject in the theory of dynamical systems. It is well known that many nonlinear systems can exhibit “chaos” that is sensitive to initial conditions. Intuitively a small perturbation in initial conditions (such as those due to rounding errors in numerical computation) yields to widely diverging outcomes and to rendering long-term prediction impossible in general. Even if a dynamical system is deterministic, i.e., their future behavior is fully determined by their initial conditions with no random elements involved, the long-term prediction of its chaotic trajectories is still impossible.

Chaotic behavior of nonlinear or piecewise-linear deterministic systems has been extensively studied using mathematical theory since Li–Yorke [37]; see, e.g., [29]. Dynamical systems that exhibit chaos with certain control actions also are never new to the control community. For example, in [53], [54] it has been demonstrated that quantization can induce chaotic behavior in digital feedback control systems; in nonlinear/piecewise-linear and adaptive control settings, chaotic behavior has also been demonstrated via continuous nonlinear/piecewise-linear feedback control (cf., e.g., [2], [40], [46]). In many setups, identifying the presence of chaos can be a great advantage for feedback control design to stabilize those chaotic trajectories; see, for example, [1], [13], [43] and references therein.

There is no doubt about that inclusion/switched systems provide a convenient method for modeling a wide variety of complex dynamical systems. Unfortunately, while the modeling paradigm itself is quite straightforward, the analysis is highly nontrivial; this is because even simple inclusion/switched dynamical systems may exhibit very complex dynamics such as chaotic behavior. In [16], Chase et al. presented an example to illustrate how chaotic behavior can arise when switching between low-dimensional linear vector fields by choosing a piecewise-linear expanding map on an interval as the transition function of switching. In [39], for a continuous-time switched system that consists of two 3-dimensional inhomogeneous linear vector fields one of which is of the form

$$\begin{array}{c}\hfill \dot{x}=\left[\begin{array}{ccc}a& b& 0\\ -b& a& 0\\ 0& 0& c\end{array}\right]x+\left[\begin{array}{c}0\\ 0\\ -d\end{array}\right]\mathrm{with}\text{an}\text{expanding}\text{equilibrium}{x}_1^{*}=\left[\begin{array}{c}0\\ 0\\ d/c\end{array}\right]\end{array}$$

and the other of which is of the form

$$\begin{array}{c}\hfill \dot{x}=\left[\begin{array}{ccc}f& 0& 0\\ 0& g& h\\ 0& -h& g\end{array}\right]x\mathrm{with}\mathrm{a}\text{contractive}\text{equilibrium}{x}_2^{*}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\end{array}$$

Liu et al. constructed a particular switching rule and gave a numerical simulation to illustrate the chaos near the contractive equilibrium $x_2^{*}$. A similar construction also leads to chaotic behavior near the equilibrium 0 for continuous-time switched linear system $\dot{x}\left(t\right)=A_{\sigma \left(t\right)}x\left(t\right)+B_{\sigma \left(t\right)}u\left(t\right)$ with certain feedback controller $u\left(t\right)=F(t,x(t\left)\right)$ as was shown in [57].

The above mentioned results are mainly based on the observation of numerical simulations. In current literature, sufficient condition that leads to chaos for linear inclusion/switched system as well as theoretical justification remain unsolved. These motivate us to study the following basic question for linear inclusion/switched systems without feedback controls, which was raised by Shorten et al. [48]:

Open problem ([48, Section 1(iv)]) Is it possible to determine if a inclusion/switched system can exhibit chaotic behavior for a given set of constituent linear subsystems?

For inclusion/switched dynamical systems, it is evident that the chaotic behaviors depend not only on the constituent subdynamics but also on the rule which orchestrates the switching.

In this paper, we shall borrow the idea of Li–Yorke chaos to give a mathematical definition of chaos for discrete-time linear inclusion dynamical system and then consider when such a system will exhibit the irregular/chaotic dynamical behavior. Our main results—Theorem 2.3, Theorem 2.4 and 2.5 in Section 2.2 solve the above Open problem of Shorten et al. [48].

The obtained results are useful for us to construct examples of discontinuity of Lyapunov exponents (cf. Remark 2.8 in Section 2.3). This is itself an interesting topic in ergodic theory.