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

Abstract

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.

Dynamical Analysis and Sampled-Data Stabilization of Memristor-Based Chua’s Circuits

“…By this action, the functional relationship between the flux and charge is given in (1) is realized. A detailed description of this memristor model is given [46], [47], [52]. A memristive Chua's oscillator circuit with a monotoneincreasing and piecewise linear memristor is presented in Fig.…”

“…To the best of the authors' knowledge, the existing published literature does not provide the results of sampled-data stabilization of LPV memristorbased Chua's circuits (MCCs), and this is the motivation of this article. Moreover in this manuscript, the considered system has peculiar and very interesting nonlinear characteristics nature i.e, non smooth boundary switching type [46], [47]. These are another important motivation which leads to coin few future works like to apply the MCCs in the FPAA [48], fractional order [49] and other memristor emulators based switching type nonlinear systems studies.…”

Associate Submersions and Qualitative Properties of Nonlinear Circuits with Implicit Characteristics

“…We illustrate the form of this polynomial in terms of the example depicted in Figure 1, taken from [16]. Specifically, our purpose is to detail in an example how the different coefficients in the multihomogeneous characteristic polynomial (29) can be examined in terms of the spanning-tree structure of the circuit, and show the way in which eventual restrictions in the controlling variables are captured via dehomogenization.…”

Input–Output Characterization of the Dynamical Properties of Circuits with a Memelement

“…15(a) (see, e.g. [Ahamed & Lakshmanan, 2017]) where w(t) is a voltage generator. It can be readily verified that L is described by (126) with L i (D) as in (10) and…”

“…This circuit is the unforced version of the well-known Murali-Lakshmanan-Chua oscillatory memristive circuit (see, e.g [Ahamed & Lakshmanan, 2017]…”