Bifurcation of a non-smooth planar limit cycle from a vertex

Abstract

We investigate non-smooth planar systems of differential equations with discontinuous right-hand sides. Discontinuity sets intersect at a vertex, and are of quasilinear nature. By means of the B -equivalence method, which was introduced in [M. Akhmetov, Asymptotic representation of solutions of regularly perturbed systems of differential equations with a nonclassical right-hand side, Ukrainian Math. J. 43 (1991) 1209–1214; M. Akhmetov, On the expansion of solutions to differential equations with discontinuous right-hand side in a series in initial data and parameters, Ukrainian Math. J. 45 (1993) 786–789; M. Akhmetov, N.A. Perestyuk, Differential properties of solutions and integral surfaces of nonlinear impulse systems, Differential Equations 28 (1992) 445–453] (see also [E. Akalın, M.U. Akhmet, The principles of B -smooth discontinuous flows, Math. Comput. Simul. 49 (2005) 981–995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Anal. 60 (2005) 163–178]), these systems are reduced to impulsive differential equations. Sufficient conditions are established for the existence of foci and centers both in the noncritical and critical cases. Hopf bifurcation is considered from a vertex, which unites several curves, in the critical case. An appropriate example is provided to illustrate the results.