Resumen de Global Dynamics of a Piecewise Smooth System with a Fold–Cusp and General Parameters
Zhihao Fang, Xingwu Chen
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 Poincaré map on the whole switching manifold, we obtain the bifurcation diagram in the whole parameter space and all corresponding global phase portraits in Poincaré disc. In this bifurcation diagram, the fold–fold bifurcation curve intersects the sliding homoclinic bifurcation curve and the pseudosaddle-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.
