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

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.