Generating Chaos via Feedback Control From a Stable Ts Fuzzy System Through a Sinusoidal Nonlinearity

Abstract

An approach is proposed for making chaotic a given stable Takagi–Sugeno (TS) fuzzy system using state feedback control of arbitrarily small magnitude. The feedback controller chosen among several candidates is a simple sinusoidal function of the system states, which can lead to uniformly bounded state vectors of the controlled system with positive Lyapunov exponents, and satisfy the chaotic mechanisms of stretching and folding, thereby yielding chaotic dynamics. This approach is mathematically proven for rigorous generation of chaos from a stable TS fuzzy system, where the chaos is in the sense of Li and Yorke. A numerical example is included to visualize the theoretical analysis and the controller design.

Robustification of Chaos in 2d Maps

“…This paper proposes a chaotification (robustification) algorithm for 2D discrete mappings, which is a topic that has been extensively studied over the last decade as indicated by [1,2,[6][7][8][9][16][17][18][19][20] and references therein. The significance of our work relative to the existing literature is that we present a new method for robustifying chaos in two-dimensional discrete maps based on the construction of a simple piecewise smooth feedback controller.…”

“…Examples include high-performance circuits and devices, liquid mixing, chemical reactions, biological systems, crisis management, secure information processing, and critical decision-making in politics, economics, and military applications. In other words, anticontrolling chaos produces chaotic behavior in a system that would not otherwise be chaotic [1,2,[6][7][8][9][16][17][18][19][20]. For example, these chaotification schemes were presented for discrete mappings using Lyapunov exponents, or by the use of several modified versions of the Marotto theorem [4,5,[9][10][11][12][13][14][15], or by the use of the Li-Yorke definition of chaos [14].…”

Generating independent chaotic attractors by chaos anticontrol in nonlinear circuits

“…The application of the classical method ( [9][10][11]) of anticontrol of chaos to obtain a chaotic dynamics in the controlled system (1) uses a simple nonlinear feedback with a small amplitude. We are interested in a simple sine function of the system state, as in [10], but with large variations of the sine amplitude.…”

“…Another technique to create chaos is the anticontrol of chaos (sometimes called chaotification), using time-delay feedback perturbation on a system parameter or employing an exogenous time-delay state-feedback input. The anticontrol reference method designs a simple nonlinear feedback controller with an arbitrarily small amplitude obtaining a chaotic dynamic in the controlled system [9][10][11].…”

A new technique to generate independent periodic attractors in the state space of nonlinear dynamic systems

“…Another technique to create chaos is to use a time-delay feedback perturbation on a system parameter or to employ an exogenous time-delay statefeedback input. This chaotic reference method designs a simple nonlinear feedback controller with an arbitrarily small amplitude leading to a chaotic dynamics in the controlled system [13,14].…”