Synchronization transition of identical phase oscillators in a directed small-world network.

We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation.

[1]  Huzihiro Araki,et al.  International Symposium on Mathematical Problems in Theoretical Physics , 1975 .

[2]  G. Ermentrout,et al.  Bistability between synchrony and incoherence in limit-cycle oscillators with coupling strength inhomogeneity. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Monika Sharma,et al.  Chemical oscillations , 2006 .

[4]  Arkady Pikovsky,et al.  Critical properties of the synchronization transition in space-time chaos. , 2002, Physical review letters.

[5]  H. Risken.  Fokker-Planck Equation , 1984 .

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

[7]  Dirk Helbing,et al.  Decentralised control of material or traffic flows in networks using phase-synchronisation , 2006, physics/0603259.

[8]  Taylor Francis Online,et al.  Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond , 2006, cond-mat/0606771.

[9]  M. Weigt,et al.  On the properties of small-world network models , 1999, cond-mat/9903411.

[10]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[11]  D. Saad.  Europhysics Letters , 1997 .

[12]  H. Hinrichsen.  Non-equilibrium critical phenomena and phase transitions into absorbing states , 2000, cond-mat/0001070.

[13]  B. Gaveau,et al.  Finite-size scaling for mean-field percolation , 1993 .

[14]  E. Ott,et al.  Onset of synchronization in large networks of coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Alexandre Arenas,et al.  Phase Patterns of Coupled Oscillators with Application to Wireless Communication , 2008, BIOWIRE.

[16]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Beom Jun Kim,et al.  Synchronization on small-world networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Effective field theory for models defined over small-world networks: first- and second-order phase transitions. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  A. Winfree.  The geometry of biological time , 1991 .

[20]  Jon Crowcroft.  Bio-Inspired Computing and Communication , 2007, BIOWIRE.

[21]  Kurt Wiesenfeld,et al.  Stability results for in-phase and splay-phase states of solid-state laser arrays , 1993 .

[22]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[23]  B. Derrida,et al.  Random networks of automata: a simple annealed approximation , 1986 .

[24]  Arkady Pikovsky,et al.  Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators , 2009 .

[25]  István Z Kiss,et al.  Predicting mutual entrainment of oscillators with experiment-based phase models. , 2005, Physical review letters.

[26]  Sergey N. Dorogovtsev,et al.  Ising Model on Networks with an Arbitrary Distribution of Connections , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  E. Ott,et al.  Echo phenomena in large systems of coupled oscillators. , 2008, Chaos.

[28]  Dirk Helbing,et al.  Information and material flows in complex networks , 2006 .

[29]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[30]  Steven H. Strogatz,et al.  Synchronization: A Universal Concept in Nonlinear Sciences , 2003 .

[31]  J. Kurths,et al.  Synchronization in Oscillatory Networks , 2007 .

[32]  Ralf Tönjes,et al.  Perturbation analysis of complete synchronization in networks of phase oscillators. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[34]  S. Solla,et al.  Self-sustained activity in a small-world network of excitable neurons. , 2003, Physical review letters.

[35]  S. Strogatz,et al.  Chimera states for coupled oscillators. , 2004, Physical review letters.

[36]  M. Timme,et al.  Stable irregular dynamics in complex neural networks. , 2007, Physical review letters.

[37]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

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

[39]  Stefan Bornholdt,et al.  Stable and unstable attractors in Boolean networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[41]  R. Tönjes,et al.  Quasiregular concentric waves in heterogeneous lattices of coupled oscillators. , 2005, Physical review letters.