Bifurcation sets of continuous piecewise linear systems with two zones. (English) Zbl 0996.37065

The authors study planar continuous piecewise linear vector fields with two zones in case when the straight line dividing the plane coincides with the vertical axis. The Lum-Chua conjecture that a continuous piecewise linear vector field with one boundary condition has at most one limit cycle and if it exists then is either attracting or repelling is proved. A canonical form which captures the most interesting oscillatory behavior, the analysis of the canonical form with only one equilibrium point and the corresponding analysis for the two equilibria case is done and their bifurcation sets are drawn. Certain Hopf-like bifurcations are considered and the quantitative differences between these bifurcations and the well-known Hopf bifurcation in smooth systems is stated.


37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G05 Normal forms for dynamical systems
34C23 Bifurcation theory for ordinary differential equations
37C10 Dynamics induced by flows and semiflows
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