Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations

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

Volumn 72, Issue 1, 2005, Pages

  Frank, T D ^a  

^a UNIVERSITY OF MÜNSTER   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BOLTZMANN DISTRIBUTIONS; DELAY FOKKER-PLANCK EQUATIONS; NOVIKOV'S THEOREM; TIME-DELAYED STOCHASTIC SYSTEMS;

APPROXIMATION THEORY; FUNCTIONS; SPURIOUS SIGNAL NOISE; TIME SERIES ANALYSIS;

STOCHASTIC CONTROL SYSTEMS;

EID: 27244434621     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.72.011112     Document Type: Article

References (71)

1. H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.46.474 46, 474 (1981)
2. D. Lenstra, Opt. Commun. OPCOB8 0030-4018 10.1016/0030-4018(91)90640-Y 81, 209 (1991)
3. I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.73.2188 73, 2188 (1994)
4. M. Bestehorn, E. V. Grigorieva, H. Haken, and S. A. Kaschenko, Physica D PDNPDT 0167-2789 10.1016/S0167-2789(00)00106-8 145, 110 (2000)
5. S. Wieczorek, B. Krauskopf, and D. Lenstra, Opt. Commun. OPCOB8 0030-4018 10.1016/S0030-4018(00)00867-1 183, 215 (2000)
6. A. N. Pisarchik, R. Meucci, and F. T. Arecchi, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.62.8823 62, 8823 (2000)
7. V. S. Udaltsov, J. P. Goedgebuer, L. Larger, and W. T. Rhodes, Opt. Commun. OPCOB8 0030-4018 10.1016/S0030-4018(01)01343-8 195, 187 (2001)
8. C. Masoller, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.88. 034102 88, 034102 (2002).
9. F. T. Arecchi, G. Giacomelli, A. Lapucci, and A. Politi, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.45.R4225 45, R4225 (1992).
10. G. Giacomelli, R. Meucci, A. Politi, and F. T. Arecchi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.73.1099 73, 1099 (1994).
11. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. IEJQA7 0018-9197 10.1109/JQE.1980.1070479 16, 347 (1980)
12. K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.45.709 45, 709 (1980)
13. T. Erneux, L. Lager, M. W. Lee, and J. Goedgebuer, Physica D PDNPDT 0167-2789 10.1016/j.physd.2004.01.038 194, 49 (2004).
14. C. Masoller, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.90. 020601 90, 020601 (2003)
15. J. Houlihan, D. Goulding, T. Busch, C. Masoller, and G. Huyet, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.050601 92, 050601 (2004)
16. T. Ackemann, M. Sondermann, A. Naumenko, and N. A. Loiko, Appl. Phys. B: Lasers Opt. APBOEM 0946-2171 10.1007/s00340-003-1348-7 77, 739 (2003).
17. W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, and J. Groslambert, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.49.203 49, 203 (1994).
18. L. Larger and J. Goedgebuer, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.69.036210 69, 036210 (2004).
19. E. Villermaux, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.75. 4618 75, 4618 (1995).
20. S. Boccaletti, D. Maza, H. Mancini, R. Genesio, and F. T. Arecchi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.79.5246 79, 5246 (1997).
21. H. Sompolinsky, D. Golomb, and D. Kleinfeld, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.43.6990 43, 6990 (1991)
22. E. Niebur, H. G. Schuster, and D. M. Kammen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.67.2753 67, 2753 (1991)
23. S. Kim, S. H. Park, and H. B. Pyo, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.82.1620 82, 1620 (1999)
24. M. K. Stephen Yeung and S. H. Strogatz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.82.648 82, 648 (1999)
25. H. Haken, Brain Dynamics (Springer, Berlin, 2002)
26. D. Goldobin, M. Rosenblum, and A. Pikovsky, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.67.061119 67, 061119 (2003)
27. A. Hutt, M. Bestehorn, and T. Wennekers, Network Comput. Neural Syst. NTWREZ 0954-898X 10.1088/0954-898X/14/2/310 14, 351 (2003)
28. V. K. Jirsa and M. Ding, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.070602 93, 070602 (2004)
29. M. G. Rosenblum and A. S. Pikovsky, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.114102 92, 114102 (2004)
30. H. Hasegawa, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.021911 70, 021911 (2004)
31. H. Hasegawa, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.021912 70, 021912 (2004).
32. M. C. Mackey and L. Glass, Science SCIEAS 0036-8075 197, 287 (1977)
33. J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics (Springer, Berlin, 1977)
34. G. A. Bocharov and F. A. Rihan, J. Comput. Appl. Math. JCAMDI 0377-0427 10.1016/S0377-0427(00)00468-4 125, 183 (2000).
35. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, U.K., 1997).
36. M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.54.R3082 54, R3082 (1996);
37. V. I. Ponomarenko and M. D. Prokhorov, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.66.026215 66, 026215 (2002);
38. T. D. Frank, P. J. Beek, and R. Friedrich, Phys. Lett. A PYLAAG 0375-9601 10.1016/j.physleta.2004.06.012 328, 219 (2004);
39. S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, Physica A PHYADX 0378-4371 351, 133 (2005).
40. J. Hale, Theory of Functional Differential Equations (Springer, Berlin, 1977)
41. C. Simmendinger, A. Wunderlin, and A. Pelster, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.59.5344 59, 5344 (1999)
42. M. Schanz and A. Pelster, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.67.056205 67, 056205 (2003).
43. J. D. Farmer, Physica D PDNPDT 0167-2789 10.1016/0167-2789(82)90042-2 4, 366 (1982).
44. J. L. Cabrera and J. G. Milton, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.89.158702 89, 158702 (2002).
45. K. Vasilakov and A. Beuter, J. Theor. Biol. JTBIAP 0022-5193 10.1006/jtbi.1993.1196 165, 389 (1993);
46. P. Tass, J. Kurths, M. G. Rosenblum, G. Guasti, and H. Hefter, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.54.R2224 54, R2224 (1996)
47. R. Engbert, C. Scheffczyk, R. T. Krampe, M. Rosenblum, J. Kurths, and R. Kliegl, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.56.5823 56, 5823 (1997)
48. Y. Chen, M. Ding, and J. A. Scott Kelso, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.79.4501 79, 4501 (1997)
49. F. Ishida and Y. E. Sawada, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.168105 93, 168105 (2004).
50. A. Longtin, J. G. Milton, J. E. Bos, and M. C. Mackey, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.41.6992 41, 6992 (1990).
51. C. W. Eurich and J. G. Milton, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.54.6681 54, 6681 (1996)
52. T. Ohira and Y. Sato, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.82.2811 82, 2811 (1999)
53. T. Ohira and T. Yamane, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.61.1247 61, 1247 (2000)
54. R. J. Peterka, Biol. Cybern. BICYAF 0340-1200 10.1007/s004220050587 82, 335 (2000).
55. U. Küchler and B. Mensch, Stoch. Stoch. Rep. SSTREY 1045-1129 40, 23 (1992).
56. M. C. Mackey and I. G. Nechaeva, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.52.3366 52, 3366 (1995).
57. S. Guillouzic, I. L'Heureux, and A. Longtin, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.59.3970 59, 3970 (1999).
58. T. D. Frank and P. J. Beek, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.64.021917 64, 021917 (2001).
59. T. D. Frank, P. J. Beek, and R. Friedrich, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.68.021912 68, 021912 (2003).
60. E. I. Verreist, in Advances in Time-Delay Systems, edited by, S. I. Niculescu, and, K. Gu, (Springer, Berlin, 2004), pp. 390-420.
61. A. A. Budini and M. O. Caceres, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.046104 70, 046104 (2004).
62. L. S. Tsimring and A. Pikovsky, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.87.250602 87, 250602 (2001)
63. D. Huber and L. S. Tsimring, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.91.260601 91, 260601 (2003).
64. S. Guillouzic, I. L'Heureux, and A. Longtin, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.61.4906 61, 4906 (2000).
65. T. D. Frank, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.66.011914 66, 011914 (2002)
66. T. D. Frank, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.69.061104 69, 061104 (2004).
67. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1989).
68. V. V. Konotop and L. Vazquez, Nonlinear Random Waves (World Scientific, Singapore, 1994).
69. N. Goldenfeld, Lecture on Phase Transitions and the Renormalization Group (Addison-Wesley, New York, 1992).
70. T. D. Frank, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.71.031106 71, 031106 (2005).
71. R. L. Stratonovich, Topics in the Theory of Random Noise, (Gordon and Beach, New York, 1963), Vol. 1.