Analysis and Adaptive Synchronization of Two Novel Chaotic Systems with Hyperbolic Sinusoidal and Cosinusoidal Nonlinearity and Unknown Parameters

Abstract

This research work describes the modelling of two novel 3-D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A)) and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B)). In this work, a detailed qualitative analysis of the novel chaotic systems (A) and (B) has been presented, and the Lyapunov exponents and Kaplan-Yorke dimension of these chaotic systems have been obtained. It is found that the maximal Lyapunov exponent (MLE) for the novel chaotic systems (A) and (B) has a large value, viz. for the system (A) and for the system (B). Thus, both the novel chaotic systems (A) and (B) display strong chaotic behaviour. This research work also discusses the problem of finding adaptive controllers for the global chaos synchronization of identical chaotic systems (A), identical chaotic systems (B) and nonidentical chaotic systems (A) and (B) with unknown system parameters. The adaptive controllers for achieving global chaos synchronization of the novel chaotic systems (A) and (B) have been derived using adaptive control theory and Lyapunov stability theory. MATLAB simulations have been shown to illustrate the novel chaotic systems (A) and (B), and also the adaptive synchronization results derived for the novel chaotic systems (A) and (B).

Keywords

• Chaos
• chaotic attractors
• Lyapunov exponents
• synchronization
• adaptive control.