Analysis of a 4-D hyperchaotic fractional-order memristive system with hidden attractors

Abstract

In 1695, G. Leibniz laid the foundations of fractional calculus, but mathematicians revived it only 300 years later. In 1971, L.O. Chua postulated the existence of a fourth circuit element, called memristor, but Williams’s group of HP Labs realized it only 37 years later. In recent years, few unusual dynamical systems, such as those with a line of equilibriums, with stable equilibria or without equilibrium, which belong to chaotic systems with hidden attractors, have been reported. By looking at these interdisciplinary and promising research areas, in this chapter, a fractional-order 4-D memristive system with a line of equilibria is introduced. In particular, a hyperchaotic behavior in a simple fractional-order memristor-based system is presented. Systematic studies of the hyperchaotic behavior in the integer and fractional-order form of the system are performed using phase portraits, Poincaré maps, bifurcation diagrams and Lyapunov exponents. Simulation results show that both integer-order and fractional-order system exhibit hyperchaotic behavior over a wide range of control parameter. Finally, the electronic circuits for the evaluation of the theoretical model of the proposed integer and fractional-order systems are presented.