Chaotifying Continuous-Time Nonlinear Autonomous Systems

TL;DR

This paper presents a unified method for chaotifying continuous-time autonomous nonlinear systems by decomposing them into subsystems and applying bounded nonlinear feedback, verified through numerical examples.

Contribution

It introduces a general chaotification approach applicable to both nonlinear and linear systems using state separation and bounded feedback control.

Findings

The method guarantees chaos with positive Lyapunov exponents.

Numerical examples confirm the effectiveness of the chaotification approach.

Applicable to a broad class of systems, including linear systems.

Abstract

Based on the principle of chaotification for continuous-time autonomous systems, which relies on two basic properties of chaos, i.e., globally bounded with necessary positive-zero-negative Lyapunov exponents, this paper derives a feasible and unified chaotification method of designing a general chaotic continuous-time autonomous nonlinear system. For a system consisting of a linear and a nonlinear subsystem, chaotification is achieved using separation of state variables, which decomposes the system into two open-loop subsystems interacting through mutual feedback resulting in an overall closed-loop nonlinear feedback system. Under the condition that the nonlinear feedback control output is uniformly bounded where the nonlinear function is of bounded-input/bounded-output, it is proved that the resulting system is chaotic in the sense of being globally bounded with a required placement of Lyapunov exponents. Several numerical examples are given to verify the effectiveness of the theoretical design. Since linear systems are special cases of nonlinear systems, the new method is also applicable to linear systems in general.