A no-equilibrium novel 4-D highly hyperchaotic system with four quadratic nonlinearities and its adaptive control

Abstract

In this work, we describe an eleven-term novel 4-D highly hyperchaotic system with four quadratic nonlinearities. The phase portraits of the eleven-term novel highly hyperchaotic system are depicted and the qualitative properties of the novel highly hyperchaotic system are discussed. We shall show that the novel hyperchaotic system does not have any equilibrium point. Hence, the novel 4-D hyperchaotic system exhibits hidden attractors. The Lyapunov exponents of the novel hyperchaotic system are obtained as L1 = 15.06593, L2=0.03551, L3= 0 and L4= -42.42821. The Maximal Lyapunov Exponent (MLE) of the novel hyperchaotic system is found as L1 = 15.06593, which is large. Thus, the novel 4-D hyperchaotic system proposed in this work is highly hyperchaotic. Also, the Kaplan–Yorke dimension of the novel hyperchaotic system is derived as DKY = 3.3559D. Since the sum of the Lyapunov exponents is negative, the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel highly hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve global chaos synchronization of the identical novel highly hyperchaotic systems with unknown parameters. MATLAB simulations are depicted to illustrate all the main results derived in this work.