Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems

Younghae Do, Ying-Cheng Lai

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

Original languageEnglish
Article number043107
Number of pages9
Issue number4
Publication statusPublished - Dec 2008


  • bifurcation
  • Jacobian matrices
  • nonlinear dynamical systems
  • stability
  • border-collision bifurcations
  • grazing bifurcations
  • impact oscillators
  • linear-oscillator
  • attractors
  • mode
  • maps


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