The discontinuous limit case of an archetypal oscillator with constant excitation and van der Pol damping: A single equilibrium

TL;DR

This paper completes the analysis of a special oscillator with constant excitation and a single equilibrium, revealing how its limit cycles and dynamics are affected in the discontinuous limit case.

Contribution

It provides a comprehensive analysis of the global dynamics and bifurcation structure of the oscillator in the case of a single equilibrium, extending previous work.

Findings

Presence of additional limit cycles around the single equilibrium

Enrichment of dynamics despite fewer equilibria

Completion of the global bifurcation diagram for this case

Abstract

This paper investigates the global dynamics of the discontinuous limit case of an archetypal oscillator with constant excitation that exhibits a single equilibrium. For parameter regions in which this oscillator possesses two or three equilibria, the global bifurcation diagram and the corresponding phase portraits on the Poincare disc have been presented in [Phys. D, 438 (2022) 133362]. The present work completes the global structure of the discontinuous limit case of an archetypal oscillator with constant excitation. Although the dynamical phenomena are less rich compared to systems with more than one equilibrium, the presence of a single equilibrium gives rise to additional limit cycles surrounding it, thereby enriching the overall dynamics and making the analysis substantially more intricate than in the previously studied cases.