Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators

Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. We investigate such a heteroclinic network between partially synchronized states where the phases cluster into three groups. For the coupling considered there exist 30 different three-cluster states in the case of five oscillators. We study the structure of the heteroclinic network and demonstrate that it is possible to navigate around the network by applying small impulsive inputs to the oscillator phases. This paper shows that such navigation may be done reliably even in the presence of noise and frequency detuning, as long as the input amplitude dominates the noise strength and the detuning magnitude, and the time between the applied pulses is in a suitable range. Furthermore, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states and replace the heteroclinic network...

[1]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[2]  P. Holmes,et al.  Random perturbations of heteroclinic attractors , 1990 .

[3]  S. Chow,et al.  Homoclinic bifurcation at resonant eigenvalues , 1990 .

[4]  P. Ashwin,et al.  The dynamics ofn weakly coupled identical oscillators , 1992 .

[5]  Hansel,et al.  Clustering and slow switching in globally coupled phase oscillators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  K. Okuda.  Variety and generality of clustering in globally coupled oscillators , 1993 .

[7]  D. Hansel,et al.  Phase Dynamics for Weakly Coupled Hodgkin-Huxley Neurons , 1993 .

[8]  M. Krupa.  Robust heteroclinic cycles , 1997 .

[9]  R. Menzel,et al.  The glomerular code for odor representation is species specific in the honeybee Apis mellifera , 1999, Nature Neuroscience.

[10]  Martin Hasler,et al.  On Strong and Weak Chaotic Partial Synchronization , 2000, Int. J. Bifurc. Chaos.

[11]  Y. Kuramoto,et al.  Slow switching in globally coupled oscillators: robustness and occurrence through delayed coupling. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  M. Golubitsky,et al.  The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space , 2002 .

[13]  A. Katok,et al.  The development of dynamics in the 20th century and the contribution of Jürgen Moser , 2002, Ergodic Theory and Dynamical Systems.

[14]  P. Holmes,et al.  Globally Coupled Oscillator Networks , 2003 .

[15]  Pablo Varona,et al.  Heteroclinic Contours in Neural Ensembles and the Winnerless Competition Principle , 2004, Int. J. Bifurc. Chaos.

[16]  P. Ashwin,et al.  Encoding via conjugate symmetries of slow oscillations for globally coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Eric Shea-Brown,et al.  On the Phase Reduction and Response Dynamics of Neural Oscillator Populations , 2004, Neural Computation.

[18]  P. Ashwin,et al.  Discrete computation using a perturbed heteroclinic network , 2005 .

[19]  Marc Timme,et al.  Nonlinear dynamics: When instability makes sense , 2005, Nature.

[20]  G. Stépán,et al.  Subcritical Hopf bifurcations in a car-following model with reaction-time delay , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[21]  A. Selverston,et al.  Dynamical principles in neuroscience , 2006 .

[22]  P. Ashwin,et al.  Extreme sensitivity to detuning for globally coupled phase oscillators. , 2006, Physical review letters.

[23]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .