ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Akopov A. A., Vadivasova T. E., Astakhov V. V., Matyushkin D. D. Cluster synchronization in inhomogeneous autooscillation medium. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 4, pp. 64-73. DOI: 10.18500/0869-6632-2003-11-4-64-73

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Language: 
Russian
Heading: 
Autowaves. Self-organization
Article type: 
Article
UDC: 
538.56:517.33
DOI: 
10.18500/0869-6632-2003-11-4-64-73

Cluster synchronization in inhomogeneous autooscillation medium

Autors: 
Akopov Artem Aleksandrovich, Saratov State University
Vadivasova Tatjana Evgenevna, Saratov State University
Astakhov Vladimir Vladimirovich, Yuri Gagarin State Technical University of Saratov
Matyushkin Dmitriy Dmitrievich, Saratov State University
Abstract: 

Formation of clusters of frequency synchronization is studied for a continuous extended medium with linear mismatch of the natural frequency along a spatial coordinate. We compare the behavior of the continuous medium described by the equation in partial derivatives and of its discrete analogue in the form of a chain of oscillators.

Key words: 
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Acknowledgments: 
This work was partially supported by the Civilian Research and Development Foundation (CRDF) and the Ministry of Education of the Russian Federation (grant REC-006). The authors are grateful to I.A. Khovanov and A.V. Shabunin for their useful advice and assistance in working on the article.
Reference: 
  1. Kuramoto Y. Chemical Oscillations Waves and Turbulence. Berlin: Springer; 1984. 158 p. DOI: 10.1007/978-3-642-69689-3.
  2. Romanovsky YM, Stepanova HV, Chernavsky DS. Mathematical Biophysics. Moscow: Nauka; 1984. 304 p. (in Russian).
  3. Gaponov-Grekhov AV, Rabinovich МI. Dynamic Chaos in Ensembles of Structures and Spatial Development of Turbulence in Unbounded Systems. N.Y.: Springer; 1986.
  4. Abarbanel HD, Rabinovich MI, Selverston A, Bazhenov MV, Huerta R, Sushchik MM, Rubchinskii LL. Synchronisation in neural networks. Phys. Usp. 1996;39(4):337–362. DOI: 10.1070/PU1996v039n04ABEH000141.
  5. Afraimovich VS, Nekorkin VI, Osipov GV, Shalfeev VD. Stability, Structure and Chaos in Nonlinear Synchronization Networks. Singapore: World Scientific; 1994. 260 p. DOI: 10.1142/2412.
  6. Kostin IK, Romanovsky YM. Fluctuations in systems of many coupled generators. Bulletin of Moscow State University. Ser. Physical and Astronomical. 1972;13(6):698-705 (in Russian).
  7. Aizawa Y. Synergetic approach to the phenomena of mode-locking in nonlinear systems. Progr. Theor. Phys. 1976;56(3):703-716. DOI: 10.1143/PTP.56.703.
  8. Kuramoto Y. Cooperative dynamics оf oscillator community. Progr. Theor. Phys. 1984;79:212-240. DOI: 10.1143/PTPS.79.223.
  9. Ermentrout GB, Kopell N. Frequency plateaus in a chain of weakly coupled oscillators. STAM J. Math. Ann. 1984;15(2):215-237. DOI: 10.1137/0515019.
  10. Yamaguchi Y, Shimizu H. Theory of self-synchronization in the presence of native frequency distribution and external noises. Physica D. 1984;11(1-2):212-226. DOI: 10.1016/0167-2789(84)90444-5.
  11. Sakaguchi H, Shinomoto S, Kuramoto Y. Local and global self-entrainments in oscillator lattices. Progr. Theor. Phys. 1987;77(5):1005-1010. DOI: 10.1143/PTP.77.1005.
  12. Strogatz SH, Mirollo RE. Phase-locking and critical phenomena in lattices оf coupled nonlinear oscillators with random intrinsic frequencies. Physica D. 1988;31(2):143-168. DOI: 10.1016/0167-2789(88)90074-7.
  13. Strogatz SH, Mirollo RE. Collective synchronization in lattices оf nonlinear oscillators with randomness. J. Phys. А. 1988;21(13):L699-L705. DOI: 10.1088/0305-4470/21/13/005.
  14. Afraimovich VS, Nekorkin VI. Stable states in chain models of unlimited nonequilibrium media. Mathematical Modeling. 1992;4(1):83-95 (in Russian).
  15. Osipov GV, Sushchik MM. Synchronized clusters and multistability in arrays оf oscillators with different natural frequencies. Phys. Rev. E. 1998;58(6):7198-7207. DOI: 10.1103/PhysRevE.58.7198.
  16. Anishchenko VS, Aranson IS, Postnov DE, Rabinovich MI. Spatial synchronization and bifurcations of chaos development in a chain of coupled generators. Proc. Acad. Sci. USSR. 1986;286(5):1120-1124 (in Russian).
  17. Astakhov VV, Bezruchko BP, Ponomarenko VI. Multistability and isomer classification and evolution in coupled feigenbaum systems. Radiophys. Quantum Electron. 1991;34(1):28–33. DOI: 10.1007/BF01048411.
  18. Kuznetsov AP, Kuznetsov SP. Critical dynamics of coupled-map lattices at onset of chaos (review). Radiophys. Quantum Electron. 1991;34(10):845–868. DOI: 10.1007/BF01083617.
  19. Kaneko K. Clustering, coding, switching, hierarchal ordering and control in а network оf chaotic elements. Physica D. 1990;41(2):137-172. DOI: 10.1016/0167-2789(90)90119-A.
  20. Braiman Y, Ditto WL, Wiesenfeld K, Spano ML. Disorder-enhanced synchronization. Phys. Lett. А. 1995;206(1-2):54-60. DOI: 10.1016/0375-9601(95)00570-S.
  21. Kocarev L, Paralitz U. Synchronizing spatiotemporal chaos in coupled nonlinear oscillators. Phys. Rev. Lett. 1996;77(11):2206-2209. DOI: 10.1103/PhysRevLett.77.2206.
  22. Osipov GV, Pikovsky AS, Rosenblum MG, Kurths J. Phase synchronization effects in а lattice оf nonidentical Rossler oscillators. Phys. Rev. Е. 1997;55(3):2353-2361. DOI: 10.1103/PhysRevE.55.2353.
  23. Astakhov VV, Anishchenko VV, Kapitaniak T, Shabunin AV. Synchronization of chaotic oscillators by periodic parametric perturbations. Physica D. 1997;109(1-2):11-16. DOI: 10.1016/S0167-2789(97)00153-X.
  24. Pecora LM. Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems. Phys. Rev.E. 1998;58(1):347-360. DOI: 10.1103/PhysRevE.58.347.
  25. Fomin AI, Vadivasova TE, Sosnovtseva OB, Anishchenko VS. Forced phase synchronization of a chain of chaotic oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(4):103-112 (in Russian).
  26. Lindner JF, Chandramouli S, Bulsara AR, Locher M, Ditto WL. Noise enhanced propagation. Phys. Rev. Lett. 1998;81(23):5048-5051. DOI: 10.1103/PhysRevLett.81.5048.
  27. Neiman А, Schimansky-Geier L, Cornell-Bell А, Moss F. Noise-enhanced synchronization in excitable media. Phys. Rev. Lett. 1999;83(23):4896-4899. DOI: 10.1103/PhysRevLett.83.4896.
  28. Licher M, Cigna Р, Hunt ER. Noise sustained propagation оf а signal in coupled bistable electronic elements. Phys. Rev. Lett. 1998;80(23):5212-5215. DOI: 10.1103/PhysRevLett.80.5212.
  29. Hu B, Zhou C. Phase synchronization in coupled non-identical excitable systems and array-enhanced coherence resonance. Phys. Rev. Е. 2000;61(2):R1001-R1004. DOI: 10.1103/physreve.61.r1001.
  30. Kopeikin AS, Matyushkin DD, Vadivasova TE, Sosnovtseva OB, Anishchenko VS. Synchronization effects in a chain of stochastic bistable oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(3):61-72 (in Russian).
  31. Loskutov AY, Mikhailov AS. Introduction to Synergetics. Moscow: Nauka; 1990. 272 p. (in Russian).
  32. Elphick C, Hagberg А, Meron Е. Phase front instability in periodically forced oscillatory systems. Phys. Rev. Lett. 1998;80(22):5007-5010. DOI: 10.1103/PhysRevLett.80.5007.
  33. Boccaletti S, Bragard J, Arecchi FT. Controlling and synchronizing space time chaos. Phys. Rev. E. 1999;59(6):6574-6578. DOI: 10.1103/PhysRevE.59.6574.
  34. Chaté H, Pikovsky A, Rudzick O. Forcing oscillatory media: phase kinks vs synchronization. Physica D. 1999;131(1-4):17-30. DOI: 10.1016/S0167-2789(98)00215-2.
  35. Junge L, Parlitz U. Synchronization and control of coupled Ginzburg-Landau equations using local coupling. Phys. Rev. Е. 2000;61(4):3736-3642. DOI: 10.1103/PhysRevE.61.3736.
  36. Park H-K. Frequency locking in spatially extended systems. Phys. Rev. Lett. 2001;86(6):1130-1133. DOI: 10.1103/PhysRevLett.86.1130.
  37. Ermentrout GB, Troy WC. Phase locking in а reaction-diffusion system with а linear frequency gradient. SIAM J. Appl. Math. 1986;39:623-660.
  38. Osipov GB, Sushchik MM. Synchronization and control in chains of connected self-oscillators. In: Bulletin of Nizhny Novgorod State University. Nonlinear Dynamics - Synchronization and Chaos - II. Nizhny Novgorod: Nizhny Novgorod State University Publishing; 1997. P. 5-23 (in Russian).
  39. Vadivasova ТЕ, Strelkova GL, Anishchenko VS. Phase-frequency synchronization in a chain of periodic oscillators in the presence of noise and harmonic forcings. Phys. Rev. Е. 2001;63(3):036225. DOI: 10.1103/PhysRevE.63.036225.
  40. Nekorkin VI, Makarov VA, Velarde MG. Clustering in а chain of bistable nonisochronous oscillators. Phys. Rev. Е. 1998;58(5):5742-5747. DOI: 10.1103/PhysRevE.58.5742.
  41. Sosnovtseva ОМ, Fomin AA, Postnov DE, Anishchenko VS. Clustering оf noise-induced oscillations. Phys. Rev. Е. 2001;64(2):026204. DOI: 10.1103/PhysRevE.64.026204.
  42. Coullet Р, Gil L, Lega J. Defect-mediated turbulence. Phys.Rev.Lett. 1989;62(14):1619-1622. DOI: 10.1103/PhysRevLett.62.1619.
  43. Gil L. Space and time intermittency behaviour of a one-dimensional complex Ginzburg-Landau equation. Nonlinearity. 1991;4(4):1213-1222. DOI: 10.1088/0951-7715/4/4/009.
  44. Shraiman ВL, Pumir А, Van Saarlos W, Hohenberg PC, Chaté H, Holen M. Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation. Physica D. 1992;57(3-4):241-248. DOI: 10.1016/0167-2789(92)90001-4.
  45. Bazhenov MV, Rabinovich MI, Fabrikant AL. The amplitude-phase turbulence transition in а Ginzburg-Landau model аз а critical phenomenon. Phys. Lett. А. 1992;163:87-94.
  46. Cross MG, Hohenberg PC. Pattern formation outside оf equilibrium. Rev. Mod. Phys. 1993;65(3):851-1112. DOI: 10.1103/RevModPhys.65.851.
  47. Chaté H. Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation. Nonlinearity. 1994;7(1):185-204. DOI: 10.1088/0951-7715/7/1/007.
  48. Samarsky AA, Gulin AB. Numerical Methods. Moscow: Nauka; 1989. 432 p. (in Russian).
Received: 
17.12.2002
Accepted: 
28.03.2003
Available online: 
30.11.2023
Published: 
31.12.2003
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