Analysis, control and synchronization of a novel highly chaotic system with three quadratic nonlinearities

Abstract

In this work, we describe a novel highly chaotic system with three quadratic nonlinearities. The phase portraits of the novel highly chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel highly chaotic system has three unstable equilibrium points. We show that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are saddle foci. The novel highly chaotic system has rotation symmetry about the x3 axis. The Lyapunov exponents of the novel highly chaotic system are obtained as L1 = 6.34352, L2 = 0 and L3 = -29.26796, while the Kaplan–Yorke dimension of the novel chaotic system is obtained as DKY = 2.2167. Since the Maximal Lyapunov Exponent (MLE) of the novel chaotic system has a large value, viz. L1 = 6.34352, the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel highly chaotic system is dissipative. Next, we derive new results for the global chaos control of the novel highly chaotic system with unknown parameters via adaptive control method. We also derive new results for the global chaos synchronization of the identical novel highly chaotic systems with unknown parameters via adaptive control method. The main adaptive control results are established using Lyapunov stability theory. MATLAB simulations are shown to depict the phase portraits of the novel highly chaotic system and also the adaptive control results derived in this work.