Zahra Monfared - Department of Mathematics & Informatics and Cluster of Excellence STRUCTURES, Heidelberg University, Heidelberg, Germany and Department of Applied Mathematics, Ferdowsi University of Mashhad (FUM), Mashhad, Iran
Zohreh Dadi - Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran
ALi Darijani - Department of Mathematics, Higher Education Complex of Bam, Bam, Iran
Yousef Qasemi Nezhad Raeini - Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

This paper studies the dynamics of a non-smooth vibrating system of the Filippov type. The main focus is on investigating the stability and bifurcation of a simple harmonic oscillator subjected to a non-smooth velocity-dependent damping force. In this way, we can analyze the effects of damping on the system's vibrations. For this purpose, we will find a parametric region for the existence of generalized Hopf bifurcation, in order to compute a branch of periodic orbits for the system. The tool for our purpose is the theoretical results about generalized Hopf bifurcation for planar Filippov systems. Some numerical simulations as examples are given to illustrate our theoretical results. Our theoretical and numerical findings indicate that the harmonic oscillator can experience different kinds of vibrations, in the presence of a non-smooth damping.

Velocity-dependent damping, Non-smooth dynamical systems, Generalized Hopf bifurcation, Vibrations, Nonlinear oscillator