The boundary focus–saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators. (October 2018) - Ponce, Enrique; Ros, Javier; Vela, Elísabet
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.