Discontinuity Induced Hopf and Neimark-Sacker Bifurcations in a Memristive Murali-Lakshmanan-Chua Circuit

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 Murali-Lakshmanan-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 Poincar\'{e} discontinuity map (PDM) and zero time discontinuity map (ZDM), are found to agree well with experimental observations.Comment: 28 pages,18 figure

Sliding Bifurcations in the Memristive Murali-Lakshmanan-Chua Circuit and the Memristive Driven Chua Oscillator

In this paper we report the occurrence of sliding bifurcations admitted by the memristive Murali-Lakshmanan-Chua circuit \cite{icha13} and the memristive driven Chua oscillator \citep{icha11}. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their non-linear element. The three segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three sub-regions. 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 \textit{Filippov} systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the sub-regions, 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.Comment: 23 pages, 14 figures, Accepted for publication in the Int. J. Bifurcation and Chao

Homogeneous Models of Nonlinear Circuits

This paper develops a general approach to nonlinear circuit modelling aimed at preserving the intrinsic symmetry of electrical circuits when formulating reduced models. The goal is to provide a framework accommodating such reductions in a global manner and without any loss of generality in the working assumptions; that is, we avoid global hypotheses imposing the existence of a classical circuit variable controlling each device. Classical (voltage/current but also flux/charge) models are easily obtained as particular cases of a general homogeneous model. Our approach extends the results introduced for linear circuits in a previous paper, by means of a systematic use of global parametrizations of smooth planar curves. This makes it possible to formulate reduced models in terms of homogeneous variables also in the nonlinear context: contrary to voltages and currents (and also to fluxes and charges), homogeneous variables qualify as state variables in reduced models of uncoupled circuits without any restriction in the characteristics of devices. The inherent symmetry of this formalism makes it possible to address in broad generality certain analytical problems in nonlinear circuit theory, such as the state-space problem and related issues involving impasse phenomena, as well as index analyses of differential-algebraic models. Our framework applies also to circuits with memristors, and can be extended to include controlled sources and coupling effects. Several examples illustrate the results.Comment: Updated versio