Reduction methods applied to non-locally coupled oscillator systems

Nonlinear Dynamics and Chaos: Where do we go from here?

Volumn , Issue , 2002, Pages 209-227

  Kuramoto, Yoshiki ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84866286781     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (39)

1. Battogtokh D and Mikhailov A 1996 Controlling turbulence in the complex Ginzburg-Landau equation Physica D 90 84-95
2. BattogtokhD 2001 Personal communication
3. Barkley D 1992 Linear stability analysis of rotating spiral waves Phys. Rev. Lett. 682090-3
4. Chapman S and Cowling T G 1970 The Mathematical Theory of Nonuniform Gases (Lectures in Statistical Mechanics) ed G E Uhlenbeck and GW Ford (Cambridge: Cambridge University Press) chIV
5. Coullet P and Iooss G 1990 Instabilities of one-dimensional cellular patterns Phys. Rev. Lett. 64 866-9
6. Crawford J D and Davies K T R 1999 Synchronization of globally coupled phase oscillators Physica D 125 1-46
7. Cross M C and Hohenberg P C 1993 Pattern formation outside of equilibrium Rev. Mod. Phys. 65 851-1112
8. Deissler R J and Brand H R 1994 Periodic, quasi-periodic, and localized solutions of the quintic complex Ginzburg-Landau equation Phys. Rev. Lett. 72 478-81
9. Ermentrout G B and Kopell N 1984 Frequency plateaus in a chain of weakly coupled oscillators SIAM J. Math. Anal. 15 215-37
10. Golubitsky M and Schaeffer D G 1985 Singularities and Groups in Bifurcation Theory (Berlin: Springer)
11. Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Berlin: Springer)
12. Hakim V and Rappel W J 1992 Dynamics of globally coupled Ginzburg-Landau equation Phys. Rev. A 46 R7347-50
13. Hansel D, Mato G and Meunier C 1993 Phase dynamics for weakly coupled Hodgkin-Huxley neurons Europhys. Lett. 23 367-72
14. Hoppensteadt F and Izhikevich E M 1997 Weakly Connected Neural Networks (Berlin: Springer)
15. Keener J P and Tyson J J 1988 Singular perturbation theory of travelling waves in excitable media Physica D 32 327-61
16. Kuramoto Y 1975 Self-entrainment of a population of coupled nonlinear oscillators International Symposium on Mathematical Problems in Theoretical Physics (Lecture Notes in Physics 39) ed H Araki (Berlin: Springer) pp 420-2
17. Kuramoto Y 1984 Chemical Oscillations, Waves, and Turbulence (Berlin: Springer)
18. Kuramoto Y 1989 On the reduction of evolution equations in extended systems Prog. Theor. Phys. 99 244-62
19. Kuramoto Y 1991 Collective synchronization of pulse-coupled oscillators and excitable units Physica D 50 15-30
20. Kuramoto Y 1995 Scaling behaviour of turbulent oscillators with non-local interaction Prog. Theor. Phys. 94 321-30
21. Kuramoto Y in preparation
22. Kuramoto Y and Nishikawa I 1987 Statistical macrodynamics of large dynamical systems J. Stat. Phys. 49 569-605
23. Kuramoto Y and Nakao H 1996 Origin of power-law correlations in distributed oscillators and maps withnon-local coupling Phys. Rev. Lett. 76 4352-5
24. Kuramoto Y, Nakao H and BattogtokhD 2000 Multi-scaled turbulence in large populations of oscillators in a diffusive medium Physica A 288 244-64
25. Marsden J E and McCracken M 1976 The Hopf Bifurcation and Its Applications (Berlin: Springer)
26. Mikhailov A 1992 Stable autonomous pacemakers in the enlarged Ginzburg-Landau model Physica D 55 99-112
27. Nagumo J S, Arimoto S and Yoshizawa S 1962 An active pulse transmission line simulating nerve axon Proc. IRE 50 2061-71
28. Nicola E M, Or-Guil M, Wolf Wand Baer M Drifting pattern domains in a reaction- diffusion system withnon- local coupling Preprint Server nlin.PS/0005043
29. Nicolis G and Prigogine I 1977 Self-Organization in Nonequilibrium Systems (New York: Wiley)
30. Ohta T, Hayase Y and Kobayashi R 1996 Spontaneous formation of concentric waves in a two-component reaction-diffusion system Phys. Rev. E 54 6074-83
31. Pearson J E 1993 Complex patterns in a simple system Science 261 189-92
32. Sakaguchi H 1993 Phase dynamics and localized solutions to the Ginzburg-Landau type amplitude equations Prog. Theor. Phys. 89 1123-46
33. Siggia E D and Zipperius A 1981 Pattern selection in Rayleigh-Bénard convection near threshold Phys. Rev. Lett. 47 835-838
34. Skinner G S and Swinney H 1991 Periodic to quasi-periodic transition of chemical spiral rotation Physica D 48 1-16
35. Strogatz S H, Mirollo R and Matthews P C 1992 Coupled nonlinear oscillators below the synchronized threshold Phys. Rev. Lett. 68 2730-3
36. Vanak V K and Epstein I R 2001 Inwardly rotating spiral waves in a reaction- diffusion systems Science 294 835-7
37. Vanak V K and Epstein I R 2001 Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol OT microemulsion Phys. Rev. Lett.87 1228301
38. Winfree A T 1967 Biological rhythms and the behaviour of populations of coupled oscillators J. Theor. Biol. 16 15-42
39. Winfree A T 1974 Rotating chemical reactions Sci. Am. 230 82-95