Analysis, control, synchronization and spice implementation of a novel 4-D hyperchaotic Rikitake dynamo system without equilibrium

Abstract

Chaos theory has wide applications and its importance can be seen by the voluminous publications on various applications in several branches of science, commerce and engineering. Control, tracking or regulation and synchronization of different types of chaotic systems are importance areas of research in the control literature and various methods have been adopted over the past few decades for tackling these research problems. Also, the discovery of novel chaotic and hyperchaotic systems in various applications, their qualitative properties and the control of such systems are also important research areas in chaos theory. This paper announces a novel 4-D hyperchaotic Rikitake dynamo system, which is derived by adding a state feedback control to the famous 3-D Rikitake two-disk dynamo system (1958). The frequent and irregular reversals of the Earth's magnetic field inspired a number of early studies involving electrical currents within the Earth's molten core. One of the first such models to exhibit reversals was Rikitake's two-disk dynamo system (Rikitake, 1958). This paper discusses the qualitative properties of the novel hyperchaotic Rikitake dynamo system. We note that the novel hyperchaotic Rikitake dynamo system has no equilibrium points. The Lyapunov exponents of the hyperchaotic Rikitake dynamo system are found as L1 = 0.09136, L2 = 0.02198, L3 = 0 and L4 = -2.11190. The Kaplan-Yorke fractional dimension of the novel hyperchaotic Rikitake dynamo system is found as DKY = 3.05367. Next, this paper discusses control and synchronization of the novel hyperchaotic Rikitake dynamo system with unknown parameters using adaptive control method. The main results are established using Lyapunov stability theory and numerically illustrated using MATLAB. Finally, for the 4-D novel hyperchaotic system, an electronic circuit realization in SPICE has been described to confirm the feasibility of the theoretical hyperchaotic Rikitake dynamo model.