The boundary focus–saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators. (October 2018)

• Record Type: Journal Article
• Authors: Ponce, Enrique; Ros, Javier; Vela, Elísabet
• Is Part Of: Nonlinear analysis. Volume 43(2018)
• Page Start: 495
• Page End: 514
• Publication Date: 2018-10
• Subjects: Qualitative theory of odes -- Piecewise linear systems -- Limit cycles -- Bifurcations; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2018.03.011
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: Among the boundary equilibrium bifurcations in planar continuous piecewise linear systems with two zones separated by a straight line, the focus–saddle bifurcation corresponds with a one-parameter transition from a situation without equilibria to a configuration with two equilibria, namely a focus and a saddle point. Depending on the dynamics of the two linear systems involved, the focus can appear surrounded by a limit cycle, by a saddle-loop (homoclinic connection) or by nothing else. After introducing a criticality coefficient whose sign discriminates the different possible situations, the focus–saddle bifurcation is quantitatively characterized for the first time. The analysis requires to work in a more general framework, as is the family of planar refracting linear systems with two zones, for which a new result about existence and uniqueness of limit cycles and saddle-loops is also shown. The achieved results are applied to the study of oscillations in an electronic circuit involving a single memristor cell, showing rigorously the appearance of limit cycles via the focus–saddle bifurcation analyzed in the paper.