WIAS Preprint No. 892, (2003)

On some mathematical topics in classic synchronization


  • Shilnikov, Andrey
  • Shilnikov, Leonid
  • Turaev, Dmitry

2010 Mathematics Subject Classification

  • 37G15 37E15 37C27 34C26


  • synchronization, saddle-node, global bifurcations, stability boundaries, blue sky bifurcation


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 done 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 descriptiob of bifurcational phenomena in the two-parameter case. Of a particular value among ones 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.

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