Periodic orbits for planar piecewise smooth systems with a line of discontinuity
The paper deals with bifurcation phenomena in the context of Filipov's theory for discontinuous planar systems of ODEs. More precisely, the study is mainly focused on the investigation of the following family of piecewise linear systems: \[ \dot{u}= A_1u,\text{ if }y-qx-m<0\text{ and }\dot{u}= A_2u\text{ when }y-qx-m>0, \] where \(u=(x,y)^T\in \mathbb{R}^2\), \(A_1,A_2\) are \(2\times2\) matrices with constant coefficients and complex conjugate eigenvalues. The existence of periodic orbits is examined. The distance from the equilibrium \(u=0\) to the line of dicontinuity \(y-qx-m=0\) plays the role of a bifurcation parameter. Finally, the results are extended for a weak non linear perturbation of the system.
scientific article; zbMATH DE number 6389985