We report using Clarke's concept of generalised differential and a modification of Floquet theory to non-smooth oscillations, the occurrence of discontinuity induced Hopf bifurcations and Neimark-Sacker bifurcations leading to quasiperiodic attractors in a memristive MuraliLakshmanan-Chua (memristive MLC) circuit. The above bifurcations arise because of the fact that a memristive MLC circuit is basically a nonsmooth system by virtue of having a memristive element as its nonlinearity. The switching and modulating properties of the memristor which we have considered endow the circuit with two discontinuity boundaries and multiple equilibrium points as well. As the Jacobian matrices about these equilibrium points are non-invertible, they are non-hyperbolic, some of these admit local bifurcations as well. Consequently when these equilibrium points are perturbed, they lose their stability giving rise to quasiperiodic orbits. The numerical simulations carried out by incorporating proper discontinuity mappings (DMs), such as the Poincaré discontinuity map (PDM) and zero time discontinuity map (ZDM), are found to agree well with experimental observations.

A negative conductance forced LCR circuit exhibiting strong chaos via the torus breakdown route as well as period-doubling route is described. The strong chaoticity is evidenced by the high value of the largest Lyapunov exponent and statistical studies. The dual nature of this circuit exhibiting the rich dynamics of both the Murali-Lakshmanan-Chua (MLC) circuit [Murali et al., 1994] and the circuit due to Inaba and Mori [1991] is also explored. The performance of the circuit is investigated by means of laboratory experiments, PSpice circuit simulation, numerical integration of appropriate mathematical model and explicit analytical studies, which all agree well with each other.

In this paper, a time varying resistive circuit realizing the action of an active three segment piecewise linear flux controlled memristor is proposed. Using this as the nonlinearity, a driven Chua's circuit is implemented. The phenomenon of chaotic beats in this circuit is observed for a suitable choice of parameters. The memristor acts as a chaotically time varying resistor (CTVR), switching between a less conductive OFF state and a more conductive ON state. This chaotic switching is governed by the dynamics of the driven Chua's circuit of which the memristor is an integral part. The occurrence of beats is essentially due to the interaction of the memristor aided self-oscillations of the circuit and the external driving sinusoidal forcing. Upon slight tuning/detuning of the frequencies of the memristor switching and that of the external force, constructive and destructive interferences occur leading to revivals and collapses in amplitudes of the circuit variables, which we refer as chaotic beats. Numerical simulations and Multisim modeling as well as statistical analyses have been carried out to observe as well as to understand and verify the mechanism leading to chaotic beats.

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.

In this paper, we report the occurrence of sliding bifurcations admitted by the memristive Murali–Lakshmanan–Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their nonlinear element. The three-segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three subregions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the subregions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.