A Novel 3-D Circulant Highly Chaotic System with Labyrinth Chaos

Abstract

In this work, we describe a novel 3-D circulant highly chaotic system with labyrinth chaos. The novel chaotic system is a nine-term polynomial system with six sinusoidal nonlinearities. The phase portraits of the novel circulant chaotic system are illustrated and the dynamic properties of the novel circulant chaotic system are discussed. The novel circulant chaotic system has infinitely many equilibrium points and it exhibits labyrinth chaos. We show that all the equilibrium points of the novel circulant chaotic system are saddle-foci and hence they are unstable. The Lyapunov exponents of the novel circulant chaotic system are obtained as \(L_1 = 10.3755\), \(L_2 = 0\) and \(L_3 = -10.4113\). Thus, the Maximal Lyapunov Exponent (MLE) of the novel chaotic system is obtained as \(L_1 = 10.3755\), which is a large value. This shows that the novel 3-D circulant chaotic system is highly chaotic. Also, the Kaplan-Yorke dimension of the novel circulant highly chaotic system is obtained as \(D_{KY} = 2.9966\). Since the Kaplan-Yorke dimension of the the novel circulant chaotic system has a large value and close to three, the novel circulant chaotic system with labyrinth chaos exhibits highly complex behaviour. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we derive new results for the global chaos control of the novel circulant highly chaotic system with unknown parameters using adaptive control method. We also derive new results for the global chaos synchronization of the identical novel circulant highly chaotic systems with unknown parameters using adaptive control method. The main control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate the phase portraits of the novel circulant highly chaotic system and also the adaptive control results derived in this work.