NONSMOOTH BIFURCATIONS, TRANSIENT HYPERCHAOS AND HYPERCHAOTIC BEATS IN A MEMRISTIVE MURALI–LAKSHMANAN–CHUA CIRCUIT

Abstract

In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincaré Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.