Archetypal oscillator for smooth and discontinuous dynamics

Qingjie Cao, Marian Wiercigroch, Ekaterina Evgenievna Pavlovskaia, Celso Grebogi, John Michael Tutill Thompson

Research output: Contribution to journalArticlepeer-review

238 Citations (Scopus)


We propose an archetypal system to investigate transitions from smooth to discontinuous dynamics. In the smooth regime, the system bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well. At the discontinuous limit, however, there is a substantial departure in the dynamics from the standard one. In particular, the velocity flow suffers a jump in crossing from one well to another, caused by the loss of local hyperbolicity due to the collapse of the stable and unstable manifolds of the stationary state. In the presence of damping and external excitation, the system has coexisting attractors and also a chaotic saddle which becomes a chaotic attractor when a smoothness parameter drops to zero. This attractor can bifurcate to a high-period periodic attractor or a chaotic sea with islands of quasiperiodic attractors depending on the strength of damping.

Original languageEnglish
Article number046218
Number of pages5
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Issue number4
Publication statusPublished - 30 Oct 2006


  • impact oscillator
  • systems
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  • bifurcations
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