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Chemistry Chimica fisica · Springer Series in Synergetics. Free Preview cover. © Chemical Oscillations, Waves, and Turbulence. Authors: Kuramoto, Y.
Table of contents
Dynamics of globally coupled oscillators: Progress and perspectives Arkady Pikovsky and Michael Rosenblum. Synchronization of chaotic systems Louis M. Pecora and Thomas L. Introduction to focus issue: Patterns of network synchronization Daniel M. Abrams , Louis M. Pecora and Adilson E. Open Published Online: 11 August Accepted: July Recommended To Librarians. Recommend to Librarian. 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.
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- Kuramoto, Y.
The adjustment of phase and frequency in large systems of oscillatory units can lead to global coherent oscillations, i. On the other hand, noise and heterogeneity in the system can weaken synchronization, or even destroy it. Synchronization in the nervous system can facilitate the transfer of information or cause epileptic seizures. Multistability and hysteresis of normal and pathological collective behavior are observed. When all oscillators are identical and the coupling tends to decrease phase differences a state of complete synchronization is asymptotically stable.
But even in random networks with uncorrelated and homogeneously distributed node degrees this absorbing state may not be reached or disappear when it is perturbed locally.
Chemical Oscillations, Waves, and Turbulence
Here we perform a detailed numerical analysis of the transition between different states of synchronization in a directed small-world network of phase oscillators. By varying the mean in-degree of the network or the nonlinearity of the phase coupling function at zero phase difference, we find discontinuous and continuous transitions with mean field critical behavior. Synchronization in spatially extended systems and complex networks is an important mechanism to create global spatiotemporal correlations from local interaction rules.
Kurths, and C. Pikovsky, M. Rosenblum, and J. Springer-Verlag , New York , Its applications range from information transfer, 5 5. Kori, K. Peters, and D. Helbing, Physica A , 39 Silber, L. Fabiny, and K. Wiesenfeld, J. B 10 , It has been studied analytically in the original texts by Kuramoto 4,8 4.
Restrepo, E. Ott, and B. Hunt, Phys. E 71 , Ott and T. Antonsen, Chaos 18 , 6 Pikovsky and M. Rosenblum, Physica D , 27 Ko and G. Ermentrout, Phys.
E 78 , Here we study the transitions from incoherence to partial synchronization and to complete synchronization in a sparse, directed small-world network of identical phase oscillators. Since the formulation of the model, 13 Watts and S. Strogatz, Nature London , Newman and D. Watts, Phys. E 60 , Ostilli and J. Mendes, Phys. Barrat and M. Weigt, Eur. B 13 , Dorogovtsev, A. Goltsev, and J.
E 66 , Hong, M. Choi, and B. Kim, Phys. E 65 , Derrida and Y. Pomeau, EPL 1 , 45 Klemm and S.
Bornholdt, Phys. E 72 , Numerical simulations are also included to investigate how different model parameters affect the synchronization.
by Y. Kuramoto
We also discuss the implementation of the model as a new approach for time synchronization in wireless sensor networks. Conflicts of Interest The authors declare no conflicts of interest.
Theory Relat. Fields , 72 , Review MR 88b A. Budhiraja, P. Dupuis, and F. Large deviation properties of weakly interacting processes via weak convergence methods. Review number not available. Collet, P. Dai Pra, and E. A simple mean field model for social interactions: dynamics, fluctuations, criticality. Dai Pra and F. McKean-Vlasov limit for interacting random processes in random media. Review MR 97f D. Critical dynamics and fluctuations for a mean-field model of cooperative behavior.
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