The simplest 4-D autonomous hyperchaotic system coined: Theoretical analysis and analog circuit design

Abstract

In this paper, a new fourth-order autonomous hyperjerk system capable of producing hyperchaotic signals (FHHO hereafter) is proposed. The proposed model features a single nonlinear term represented by the hyperbolic sine of the weighted sum of two state variables (i.e., the fundamental variable and the jerk). The FHHO system is dissipative and symmetric and has a single unstable equilibrium point located at the origin of the state space. To describe the mechanisms leading to chaos and subsequent hyperchaos, a systematic study is carried out using appropriate analysis tools, such as Lyapunov exponent graphs, phase portraits, Poincaré maps, and bifurcation diagrams. We highlight rich and varied dynamics marked by periodic, tori, chaotic or hyperchaotic attractors and, even more interestingly, offset control and symmetry control properties. The electronic simulator of the proposed FHHO model is built using only five operational amplifiers (i.e., four integrators and a summing amplifier) and a pair of diodes mounted head to tail. The experimental results confirm the presence of hyperchaotic signals as well as the bifurcation modes predicted by the theoretical study. To the best of our knowledge, the hyperchaotic model studied combines the two forms of simplicity rarely encountered, namely, the simplicity of the evolution equations and the simplicity of electronic realization.