Role of saddle tori in the mutual synchronization of periodic oscillations.

We show that in the mutual synchronization of periodic oscillators, besides an attracting torus, there is also a saddle torus that plays an equally important role. We demonstrate that the saddle and stable tori form an elegant structure, allowing for a variety of phenomena, both known and new, related to the origin and evolution of coexisting synchronous regimes (phase multistability).

[1]  Hans G. Othmer,et al.  An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators , 1987 .

[2]  Alexander B. Neiman,et al.  Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments , 2003 .

[3]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[4]  E. Mosekilde,et al.  Bifurcations in two coupled Ro¨ssler systems , 1996 .

[5]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[6]  K. Bar-Eli,et al.  On the stability of coupled chemical oscillators , 1985 .

[7]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[8]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[9]  H. Pyo,et al.  Multistability analysis of phase locking patterns in an excitatory coupled neural system. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Phase Multistability in Coupled Oscillator Systems , 2003 .

[11]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[12]  E. Mosekilde,et al.  Chaotic Synchronization: Applications to Living Systems , 2002 .

[13]  Modes of oscillation and their evolution in dissipatively coupled Feigenbaum systems , 1990 .

[14]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[15]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[16]  T. Vadivasova,et al.  Phase Multistability of Synchronous Chaotic Oscillations , 2000 .

[17]  Yury I. Bychkov,et al.  Sov Phys Tech Phys , 1975 .

[18]  P. McClintock.  Synchronization:a universal concept in nonlinear science , 2003 .