Basins of attraction in piecewise smooth Hamiltonian systems

Ying-Cheng Lai, D R He, Y M Jiang

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


Piecewise smooth Hamiltonian systems arise in physical and engineering applications. For such a system typically an infinite number of quasiperiodic "attractors" coexist. (Here we use the term "attractors" to indicate invariant sets to which typically initial conditions approach, as a result of the piecewise smoothness of the underlying system. These "attractors" are therefore characteristically different from the attractors in dissipative dynamical systems.) We find that the basins of attraction of different "attractors" exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales. This practically prevents prediction of "attractors" from specific initial conditions and parameters. The mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems. We demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation.

Original languageEnglish
Article number025201
Number of pages4
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Issue number2
Publication statusPublished - Aug 2005


  • chaotic dynamical systems
  • riddled basins
  • transverse stability
  • boundaries
  • intermittency
  • bifurcation
  • oscillator


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