The European Physical Journal Special Topics

, Volume 222, Issue 10, pp 2497–2507 | Cite as

Multistable randomly switching oscillators: The odds of meeting a ghost

  • I. BelykhEmail author
  • V. Belykh
  • R. Jeter
  • M. Hasler
Regular Article Nonlinear Dynamics of Stochastic Systems


We consider oscillators whose parameters randomly switch between two values at equal time intervals. If random switching is fast compared to the oscillator’s intrinsic time scale, one expects the switching system to follow the averaged system, obtained by replacing the random variables with their mean. The averaged system is multistable and one of its attractors is not shared by the switching system and acts as a ghost attractor for the switching system. Starting from the attraction basin of the averaged system’s ghost attractor, the trajectory of the switching system can converge near the ghost attractor with high probability or may escape to another attractor with low probability. Applying our recent general results on convergent properties of randomly switching dynamical systems [1, 2], we derive explicit bounds that connect these probabilities, the switching frequency, and the chosen initial conditions.


Ghost Lyapunov Function European Physical Journal Special Topic Stochastic Resonance Stable Limit Cycle 
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© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Department of Mathematics & Statistics and Neuroscience InstituteGeorgia State UniversityAtlantaUSA
  2. 2.Department of MathematicsVolga State AcademyNizhny NovgorodRussia
  3. 3.Advanced School of General and Applied Physics, University of Nizhny NovgorodNizhny NovgorodRussia
  4. 4.School of Computer and Communication Sciences, Ecole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland

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