Frequency and phase synchronization in stochastic systems

Chaos

Volumn 13, Issue 1, 2003, Pages 225-238

  Freund, Jan A ^a   Schimansky Geier, Lutz ^a   Hänggi, Peter ^b  

^a HUMBOLDT UNIVERSITY   (Germany)

^b UNIVERSITY OF AUGSBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037357451     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1500497     Document Type: Article

References (69)

1. V. I. Arnold, Trans. Am. Math. Soc. 42, 213 (1965); E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
2. V. I. Arnold, Trans. Am. Math. Soc. 42, 213 (1965); E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
3. R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967).
4. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); A. S. Pikovsky, Z. Phys. B: Condens. Matter 55, 149 (1984); L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
5. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); A. S. Pikovsky, Z. Phys. B: Condens. Matter 55, 149 (1984); L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
6. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); A. S. Pikovsky, Z. Phys. B: Condens. Matter 55, 149 (1984); L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
7. N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Phys. Rev. E 51, 980 (1995); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996).
8. N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Phys. Rev. E 51, 980 (1995); L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76, 1816 (1996).
9. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 78, 4193 (1997); S. Taherion and Y. C. Lai, Phys. Rev. E 59, R6247 (1999).
10. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 78, 4193 (1997); S. Taherion and Y. C. Lai, Phys. Rev. E 59, R6247 (1999).
11. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).
12. S. K. Han, T. G. Yim, D. E. Postnov, and O. V. Sosnovtseva, Phys. Rev. Lett. 83, 1771 (1999).
13. E. M. Izhikevich, Int. J. Bifur. Chaos 10. 1171 (2000); B. Hu and C. Zhou, Phys. Rev. E 63, 026201 (2001).
14. E. M. Izhikevich, Int. J. Bifur. Chaos 10. 1171 (2000); B. Hu and C. Zhou, Phys. Rev. E 63, 026201 (2001).
15. V. Anishchenko, A. Neiman, A. Astakhov, T. Vadiavasova, and L. Schimansky-Geier, Chaotic and Stochastic Processes in Dynamic Systems (Springer, Berlin, 2002).
16. L. Schimansky-Geier, V. Anishchenko, and A. Neiman, in Neuro-informatics, edited by S. Gielen and F. Moss, Vol. 4, in Handbook of Biological Physics, series editor A. J. Hoff (Elsevier Science, New York, 2001).
17. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
18. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
19. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
20. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
21. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
22. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
23. A. Neiman, X. Pei, D. Russell, W. Wojtenek, L, Wilkens, F. Moss, H. A. Braun, M. T. Huber, and K. Voigt, Phys. Rev. Lett. 82, 660 (1999); S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999); 63, 059901 (2001); W. Singer, Neuron 24, 49 (1999); R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, Phys. Rev. Lett. 81, 5692 (1998); P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, ibid. 81, 3291 (1998); R. Ritz and T. J. Sejnowski, Curr. Opin. Neurobiol. 7, 536 (1997).
24. B. Schäfer, M. G. Rosenblum, and J. Kurths, Nature (London) 392, 239 (1998).
25. B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999),
26. J. P. M. Heald and J. Stark, Phys. Rev. Lett. 84, 2366 (2000).
27. J. A. Freund, A. B. Neiman, and L. Schimansky-Geier, Europhys. Lett. 50, 8 (2000).
28. L. Callenbach, P. Hänggi, S. J. Linz, J. A. Freund, and L. Schimansky-Geier, Phys. Rev. E 65, 051110 (2002).
29. S. O. Rice, Bell Syst. Tech. J. 23/24, 1 (1944/1945); note Sec. 3.3 on pp. 57-63 in this double volume therein.
30. S. O. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), pp. 189-195 therein.
31. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
32. V. S. Anishchenko, A. B. Neiman, F. Moss, and L. Schmansky-Geier, Phys. Usp. 42, 7 (1999).
33. A. Neiman, A. Silchenko, V. Anishchenko, and L. Schimansky-Geier, Phys. Rev. E 58, 7118 (1998).
34. P. Jung, Phys. Rep. 234, 175 (1995).
35. N. N. Bogoliubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1961).
36. The explicit expression for the phase involving the arctan has to be understood in the sense of adding multiples of π to make it a continuously growing function of time.
37. P. Hänggi and P. Riseborough, Am. J. Phys. 51, 347 (1983).
38. N = π/2 + n π, n ∈ N; this, however, is no severe restriction.
39. A. T. Winfree, J. Theor. Biol. 16, 15 (1967); J. Guckenheimer, J. Math. Biol. 1, 259 (1975).
40. A. T. Winfree, J. Theor. Biol. 16, 15 (1967); J. Guckenheimer, J. Math. Biol. 1, 259 (1975).
41. D. Gabor, J. IEE (London) 93(III), 429 (1946).
42. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
43. L. A. Vainstein and D. E. Vakman, Frequency Analysis in the Theory of Oscillations and Waves (in Russian) (Nauka, Moscow, 1983).
44. R. Carmona, W.-L. Hwang, and B. Torresani, Practical Time-Frequency Analysis (Academic, San Diego, 1998).
45. J.-P. Lachaux, E. Rodriguez, M. Le Van Quen, A. Lutz, J. Martinerie, and F. Varela, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2429 (2000); M. Le Van Quen. J. Foucher, J.-P. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie, and F. Varela, J. Neurosci. Methods 111, 83 (2001).
46. J.-P. Lachaux, E. Rodriguez, M. Le Van Quen, A. Lutz, J. Martinerie, and F. Varela, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2429 (2000); M. Le Van Quen. J. Foucher, J.-P. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie, and F. Varela, J. Neurosci. Methods 111, 83 (2001).
47. D. J. DeShazer, R. Breban, E. Ott, and R. Roy, Phys. Rev. Lett. 87, 044101 (2001).
48. B. McNamara and K. Wiesenfeld, Phys. Rev. A 39, 4854 (1989).
49. D. R. Cox, Renewal Theory (Chapman and Hall, London, 1967).
50. A. Freund, A. B. Neiman, and L. Schimansky-Geier, in Stochastic Climate Models: Progress in Probability, Vol. 49, edited by P. Imkeller and J. von Storch (Birkhäser, Boston, 2001).
51. N. G. van Kämpen, Stochastic Processes in Physics and Chemistry, revised and enlarged edition (North-Holland, Amsterdam, 1992).
52. R. Adler, Proc. IRE 34, 351 (1946); P. Hänggi and P. Riseborough, Am. J. Phys. 51, 347 (1983).
53. R. Adler, Proc. IRE 34, 351 (1946); P. Hänggi and P. Riseborough, Am. J. Phys. 51, 347 (1983).
54. C. Van den Broeck, Phys. Rev. E 47, 4579 (1993); U. Zürcher and C. R. Doering, ibid. 47, 3862 (1993).
55. C. Van den Broeck, Phys. Rev. E 47, 4579 (1993); U. Zürcher and C. R. Doering, ibid. 47, 3862 (1993).
56. F. Marchesoni, F. Apostolico, and S. Santucci, Phys. Rev. E 59, 3958 (1999).
57. A. Neiman, L. Schimansky-Geier, F. Moss, B. Shulgin, and J. J. Collins, Phys. Rev. E 60, 284 (1999).
58. B. Shulgin, A. Neiman, and V. Anishchenko, Phys. Rev. Lett. 75, 4157 (1995).
59. R. Rozenfeld, J. A. Freund. A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001).
60. P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
61. P. Hänggi and H. Thomas, Phys. Rep. 88, 207 (1982); see Sec. 2.4 therein.
62. P. Hänggi and P. Jung, Adv. Chem. Phys. 89, 239 (1995); P. Hänggi, F. Marchesoni, and P. Grigolini, Z. Phys. B: Condens. Matter 56, 333 (1984).
63. P. Hänggi and P. Jung, Adv. Chem. Phys. 89, 239 (1995); P. Hänggi, F. Marchesoni, and P. Grigolini, Z. Phys. B: Condens. Matter 56, 333 (1984).
64. R. Benzi, G. Parisi, and A. Vulpiani, J. Phys. A 14, L453 (1981); C. Nicolis, Sol. Phys. 74, 473 (1981).
65. R. Benzi, G. Parisi, and A. Vulpiani, J. Phys. A 14, L453 (1981); C. Nicolis, Sol. Phys. 74, 473 (1981).
66. A. Neiman, L. Schimansky-Geier, F. Moss, B. Shulgin, and J. J. Collins, Phys. Rev. E 60, 284 (1999).
67. B. Lindner, M. Kostur, and L. Schimansky-Geier, Fluct. Noise Lett. 1, R25 (2001).
68. J. A. Freund, J. Kienert, L. Schimansky-Geier, B. Beisner, A. Neiman, D. F. Russell, T. Yakusheva, and F. Moss, Phys. Rev. E 63, 031910 (2001).
69. L. Callenbach, Synchronization Phenomena in Chaotic and Noisy Oscillatory Systems (Logos Verlag, Berlin, 2001).