Subcritical Andronov-Hopf scenario for systems with a line of equilibria

TL;DR

This paper investigates the subcritical Andronov-Hopf bifurcation in systems with a line of equilibria, demonstrating self-oscillation excitation through numerical and analytical methods, with applications to memristor-based oscillators.

Contribution

It introduces the analysis of subcritical Andronov-Hopf bifurcation in systems with a line of equilibria, extending classical bifurcation theory to new system configurations.

Findings

Demonstrated self-oscillation excitation in systems with a line of equilibria

Identified hysteresis and bistability phenomena in the studied bifurcations

Analyzed memristor-based oscillator models with different characteristics

Abstract

Using numerical simulation methods and analytical approach, we demonstrate hard self-oscillation excitation in systems with infinitely many equilibrium points forming a line of equilibria in the phase space. The studied bifurcation phenomena are equivalent to the excitation scenario via the subcritical Andronov-Hopf bifurcation observed in classical self-oscillators with isolated equilibrium points. The hysteresis and bistability accompanying the discussed processes are shown and explained. The research is carried out on an example of a nonlinear memristor-based self-oscillator model. First, a simpler model including Chua's memristor with a piecewise-smooth characteristic is explored. Then the memristor characteristic is changed to a function being smooth everywhere. Finally, the action of the memristor forgetting effect is taken into consideration.