Diffusion-induced instability and chaos in random oscillator networks

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 79, Issue 3, 2009, Pages

  Nakao, Hiroya ^a   Mikhailov, Alexander S ^b  

^a KYOTO UNIVERSITY   (Japan)

^b FRITZ HABER INSTITUT DER MAX PLANCK GESELLSCHAFT   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

COMPLEX GINZBURG-LANDAU EQUATIONS; CONTINUOUS MEDIAS; DIFFUSIONAL COUPLINGS; DYNAMICAL PATTERNS; DYNAMICAL REGIMES; LIMIT-CYCLE OSCILLATORS; MEAN-FIELD THEORIES; NETWORK ANALOGS; NON-LINEAR DYNAMICS; NUMERICAL INVESTIGATIONS; OSCILLATOR NETWORKS; PARTIAL AMPLITUDES; RANDOM NETWORKS; SCALE-FREE NETWORKS;

MEAN FIELD THEORY;

OSCILLATORS (ELECTRONIC);

EID: 65449145228     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.79.036214     Document Type: Article

References (23)

1. S. H. Strogatz, Nature (London) 410, 268 (2001). 10.1038/35065725
2. Y. Moreno and A. F. Pacheco, Europhys. Lett. 68, 603 (2004). 10.1209/epl/i2004-10238-x
3. T. Ichinomiya, Phys. Rev. E 70, 026116 (2004). 10.1103/PhysRevE.70.026116
4. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Phys. Rep. 469, 93 (2008). 10.1016/j.physrep.2008.09.002
5. T.-W. Ko and G. B. Ermentrout, Phys. Rev. E 78, 016203 (2008). 10.1103/PhysRevE.78.016203
6. S. Gil and A. S. Mikhailov, Phys. Rev. E 79, 026219 (2009). 10.1103/PhysRevE.79.026219
7. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
8. M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund, and G. Ertl, Science 292, 1357 (2001). 10.1126/science.1059478
9. V. Hakim and W.-J. Rappel, Phys. Rev. A 46, R7347 (1992) 10.1103/PhysRevA.46.R7347
10. N. Nakagawa and Y. Kuramoto, Prog. Theor. Phys. 89, 313 (1993) 10.1143/PTP.89.313
11. M.-L. Chabanol, V. Hakim, and W.-J. Rappel, Physica D 103, 273 (1997) 10.1016/S0167-2789(96)00263-1
12. M. Banaji and P. Glendinning, Phys. Lett. A 251, 297 (1999). 10.1016/S0375-9601(98)00869-X
13. Y. Kuramoto, Prog. Theor. Phys. 94, 321 (1995) 10.1143/PTP.94.321
14. Y. Kuramoto and H. Nakao, Phys. Rev. Lett. 76, 4352 (1996) 10.1103/PhysRevLett.76.4352
15. H. Nakao, Chaos 9, 902 (1999). 10.1063/1.166463
16. R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002). 10.1103/RevModPhys.74.47
17. R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001). 10.1103/PhysRevLett.86.3200
18. H. Nakao and A. S. Mikhailov, e-print arXiv:0807.1230.
19. In, it is reported that the asymmetric limit cycle can coexist with the fixed points in a certain parameter region, where the transition with hysteresis occurs via a saddle-node bifurcation of the fixed points followed by a homoclinic or Hopf bifurcation of the limit cycle. In the case of Fig. 3, the transition near β=0.435 occurs by a saddle node on invariant circle (SNIC) bifurcation without hysteresis, and coexistence of fixed points with the asymmetric limit cycle does not take place. Note that coexistence of multiple attractors in the dynamics of individual oscillators, which actually occurs in the two-cluster state [Fig. 3], does not affect our argument as long as the global mean field is approximately sinusoidal.
20. Though we do not give details in the present paper, we can further develop a self-consistency analysis for the sinusoidal global mean field H (t) to determine B and Ω from the condition that the H (t) imposed on Eq. 6 coincide with the H (t) resulting from Eq. 5. This gives good agreement with direct numerical results for 0≤K<0.05, where H (t) vanishes or oscillates sinusoidally and each individual oscillator has a single attractor. For larger values of K where H (t) is not sinusoidal or some of the individual oscillators have multiple attractors (e.g., bistable fixed points), such a simple sinusoidal self-consistency analysis fails. See Chabanol for elaborate analysis of the globally coupled CGL oscillators.
21. Y. Kuramoto and D. Battogtokh, Nonlinear Phenom. Complex Syst. (Dordrecht, Neth.) 5, 380 (2002)
22. S.-i. Shima and Y. Kuramoto, Phys. Rev. E 69, 036213 (2004) 10.1103/PhysRevE.69.036213
23. D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004). 10.1103/PhysRevLett.93.174102