# Hard and soft excitation of oscillations in memristor-based oscillators with a line of equilibria

## TL;DR

This paper investigates the mechanisms of oscillation excitation in memristor-based Chuas oscillators with a line of equilibria, analyzing both hard and soft excitation scenarios through theoretical, numerical, and experimental methods.

## Contribution

It introduces a detailed analysis of bifurcation mechanisms for oscillation excitation in systems with a line of equilibria, highlighting differences between hard and soft excitation.

## Key findings

- Hard excitation involves border-collision bifurcation due to piecewise-smooth characteristics.
- Soft excitation arises from supercritical Andronov-Hopf bifurcation with smooth nonlinearities.
- Experimental results confirm theoretical and numerical predictions.

## Abstract

A model of memristor-based Chuas oscillator is studied. The considered system has infinitely many equilibrium points, which build a line of equilibria. Bifurcational mechanisms of oscillation excitation are explored for different forms of nonlinearity. Hard and soft excitation scenarios have principally different nature. The hard excitation is determined by the memristor piecewise-smooth characteristic and is a result of a border-collision bifurcation. The soft excitation is caused by addition of a smooth nonlinear function and has distinctive features of the supercritical Andronov-Hopf bifurcation. Mechanisms of instability and amplitude limitation are described for both two cases. Numerical modelling and theoretical analysis are combined with experiments on an electronic analog model of the system under study. The issues concerning physical realization of the dynamics of systems with a line of equilibria are considered. The question on whether oscillations in such systems can be classified as the self-sustained oscillations is raised.