On some mathematical topics in classical synchronization. A tutorial

Andrey Shilnikov, Leonid Shilnikov, Dmitry Turaev

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies by Cartwright and Littlewood. Today, we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive description of bifurcational phenomena in the two-parameter case. Of a particular value is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori.

Original languageEnglish
Pages (from-to)2143-2160
Number of pages18
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Issue number7
StatePublished - 1 Jan 2004


  • Homoclinic bifurcation
  • Saddle-node
  • Synchronization

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics


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