Global Dynamics of a Piecewise Smooth System with a Fold-Cusp and General Parameters

In this paper, for general parameters we investigate the global dynamics of a piecewise smooth system, which is a two-parametric unfolding of a normal form with a fold-cusp. The main difficulty comes from the global switching of vector fields on switching manifold and the non-locality of parameters because switching makes classic theory of qualitative analysis and bifurcations for smooth systems invalid. Analyzing the global structure of switching manifold including all singularities and determining the precise domain of Poincare map on the whole switching manifold, we obtain the bifurcation diagram in the whole parameter space and all corresponding global phase portraits in Poincare disc. In this bifurcation diagram, the fold-fold bifurcation curve intersects the sliding homoclinic bifurcation curve and the pseudo-saddle-node bifurcation curve at two certain nonlocal parameters, respectively. Such intersections correspond to a degenerate sliding homoclinic loop and a degenerate fold-fold. Moreover, a sliding limit cycle and a pseudo-equilibrium bifurcate from the former and two pseudo-equilibria bifurcate from the latter. This generalizes the bifurcation theory of sliding homoclinic loop and fold-fold to the degenerate case.

Automated summary

In this paper, we investigate the global dynamics of a piecewise smooth system with a fold-cusp using a two-parametric unfolding of a normal form. We analyze the global structure of the switching manifold and determine the precise domain of the Poincare map. The bifurcation diagram in the parameter space and the corresponding global phase portraits in the Poincare disc are obtained. Degenerate sliding homoclinic loops and fold-folds are observed at certain nonlocal parameters, leading to the emergence of sliding limit cycles and pseudo-equilibria.