Dynamics of a harmonic oscillator perturbed by a non-smooth velocity-dependent damping force

Document Type : Research Paper


1 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

2 Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran

3 Department of Mathematics, Higher Education Complex of Bam, Bam, Iran

4 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.


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Volume 10, Issue 2
Special Issue Dedicated to Professor M. Radjabalipour on the occasion of his 75th birthday.
October 2021
Pages 145-162
  • Receive Date: 25 April 2021
  • Accept Date: 06 October 2021