Invariant torus and its destruction for an oscillator with dry friction

Xiaoming Zhang, Chao Zeng, Denghui Li* (Corresponding Author), Jianhua Xie, Celso Grebogi

*Corresponding author for this work

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Mechanical systems with dry friction are typical Filippov systems. Such class of systems have complicated dynamical behaviors due to the existence of sliding motion. In this work, we consider a one-degree-of-freedom oscillator with dry friction force. The phase map is derived to reduce the system to a circle map, and then the existence of forward invariant torus is proved under suitable assumptions. Moreover, the typical resonance phenomenon and the grazing bifurcation of invariant torus are discussed. We find that the destruction of invariant tori is due to a loss of transversality for sufficiently large perturbation, which is different from the usual smooth tori dynamics.
Original languageEnglish
Pages (from-to)3467–3477
Number of pages11
JournalNonlinear Dynamics
Early online date20 May 2021
Publication statusPublished - 1 Jun 2021

Bibliographical note

This work is supported by the National Natural Science Foundation of China (11732014).


  • dry friction oscillator
  • phase map
  • invariant torus
  • grazing-sliding bifurcation


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