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

## TL;DR

This paper investigates how discontinuities in a memristive Murali-Lakshmanan-Chua circuit induce Hopf and Neimark-Sacker bifurcations, leading to quasiperiodic oscillations, using advanced non-smooth analysis and numerical simulations.

## Contribution

It introduces a novel analysis of bifurcations in a memristive circuit using Clarke's generalized differential and modified Floquet theory for non-smooth systems.

## Key findings

- Discontinuity induced bifurcations lead to quasiperiodic attractors.
- Numerical simulations with discontinuity mappings agree with experimental results.
- The memristive MLC circuit exhibits multiple equilibrium points and complex bifurcation behavior.

## 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 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.