Global dynamics and unfolding of planar piecewise smooth quadratic quasi-homogeneous differential systems

TL;DR

This paper analyzes the global behavior and bifurcations of planar piecewise smooth quadratic quasi-homogeneous differential systems, revealing conditions for centers and limit cycle bifurcations, and highlighting differences from smooth systems.

Contribution

It provides necessary and sufficient conditions for centers in these systems and studies their global structures and limit cycle bifurcations using Melnikov method.

Findings

Centers can be global and non-isochronous in piecewise systems.

Conditions for the existence of centers are established.

Maximum number of bifurcating limit cycles is determined.

Abstract

In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi--homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi--homogeneous systems. Moreover, the center is global and non-isochronous if it exists, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi--homogeneous but non-homogeneous systems are studied. Finally we investigate limit cycle bifurcations of the piecewise smooth quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from the periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.