For citation:
Koronovskii A. A., Moskalenko O. I., Popov P. V., Hramov A. E. Some approaches for chaotic synchronization analysis in coupled dynamical systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2004, vol. 12, iss. 6, pp. 159-190. DOI: 10.18500/0869-6632-2004-12-6-159-190
This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 0)
Language:
Russian
Article type:
Article
UDC:
517.9
Some approaches for chaotic synchronization analysis in coupled dynamical systems
Autors:
Koronovskii Aleksei Aleksandrovich, Saratov State University
Moskalenko Olga Igorevna, Saratov State University
Popov Pavel Viacheslavovich, Saratov State University
Hramov Aleksandr Evgenevich, Immanuel Kant Baltic Federal University
Abstract:
In this paper synchronization in coupled chaotic oscillators is investigated. New approaches proposed to the detection of synchronous behavior of oscillators are applied for the research of chaotic synchronization both in systems with a small number of freedom degrees as in spatially extended self-oscillated systems.
Key words:
Acknowledgments:
The authors are grateful to Corresponding Member of the Russian Academy of Sciences D.I. Trubetskov, Prof. B.P. Bezruchko and his research group for their interest in the work and fruitful discussions.
The work supported by the RFBR (projects 05 - 02 - 16273 and 05 - 02 - 16286), Program to support leading scientific schools of the Russian Federation (project НШ - 1250.2003.02), and Scientific and Educational Center "Nonlinear Dynamics and Biophysics" at N.G. Chernyshevsky State University of Saratov (grant ВЕС-006 оf U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF)).
The authors are also grateful to the Foundation for Nonprofit Programs "Dynasty" and the International Center for Fundamental Physics (Moscow) for financial support.
Reference:
- Afraimovich BC, Nekorkin VI, Osipov GV, Shalfeev VD. Stability, Structures and chaos in Nonlinear Synchronization Networks. Gorky: IPF AS USSR; 1989. 245 p.
- Parlitz U, Junge L, Lauterborn W. Experimental observation of phase synchronization. Phys. Rev. E. 1996;54(2):2115–2117. DOI: /10.1103/PhysRevE.54.2115.
- Tang DY, Dykstra R, Hamilton MW, Heckenberg NR. Experimental evidence of frequency entrainment between coupled chaotic oscillations. Phys. Rev. Е. 1998;57(3):3649–3651. DOI: 10.1103/PhysRevE.57.3649.
- Allaria E, Arecchi FT, Garbo AD, Meucci R. Synchronization of homoclinic chaos. Phys. Rev. Lett. 2001:86(5):791–794. DOI: 10.1103/PhysRevLett.86.791.
- Ticos CM, Rosa E, Pardo WB, Walkenstein JA, Monti M. Experimental real-time phase synchronization оf а paced chaotic plasma discharge. Phys. Rev. Lett. 2000;85(14):2929. DOI: 10.1103/PhysRevLett.85.2929.
- Rosa E, Pardo WB, Ticos CM, Wakenstein JA, Monti M. Phase synchronization of chaos in a plasma discharge tube .Int. J. Bifurcation and Chaos. 2000;10(11):2551–2563. DOI: 10.1142/S0218127400001638.
- Trubetskov DI, Hramov AE. On synchronization of chaotic auto oscillations in the distributed system helical electron flow - counter electromagnetic wave. Radiotekhnika i elektronika. 2003;48(1):116–124.
- Trubetskov DI, Koronovskii AA, Hramov AE. Synchronization of distributed autocoletic systems of electron-wave nature with a backward wave. Izv. vuzov. Radiophysics. 2004;47(5-6):343–372.
- Hramov AE, Koronovskii AA. Time scale synchronization оf chaotic oscillators. Physica D. 2005;206(3-4):252–264. DOI: 10.1016/j.physd.2005.05.008.
- Anishchenko VS, Balanov AG, Janson NB, Igosheva NB, Bordyugov GV. Entrainment between heart rate and weak nonlinear forcing. Int. J. Bifurcation аnd Chaos. 2000;10(10)2339–2348. DOI: 10.1142/S0218127400001468.
- Elson RC, Selverston AI, Huerta R, Rulkov FN, Rabinovich MI, Abarbanel HDI. Synchronous behavior оf two coupled biological neurons. Phys. Rev. Lett. 1998;81(25):5692–5695. DOI: 10.1103/PhysRevLett.81.5692.
- Rulkov NF. Modeling оf spiking-bursting neural behavior using two- dimensional map. Phys. Rev. E. 2002;65(1):041922. DOI: 10.1103/PhysRevE.65.041922.
- Tass PA, Fieseler T, Dammers J, Dolan K, Morosan P, Majtanik M, BoersF, Muren A, Zilles K, Fink GR. Synchronization tomography: a method for three-dimensional localization of phase synchronized neuronal populations in the human brain using magnetoencephalography. Phys. Rev. Lett. 2003;90(8)088101. DOI: 10.1103/PhysRevLett.90.088101.
- Prokhorov MD, Ponomarenko VI, Gridnev VI, Bodrov MB, Bespyatov AB. Synchronization between main rhytmic processes in the human cardiovascular system. Phys. Rev. Е. 2003;68(4):041913. DOI: 10.1103/PhysRevE.68.041913.
- Pikovsky А, Rosenblum M, Kurths J. Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge University Press. 2001. 433 p. DOI: 10.1017/CBO9780511755743.
- Anishchenko VS, Astakhov V, Neiman А, Vadivasova T, Schimansky-Geier L. Nonlinear Dynamics of Сhaotic and Stochastic Systems. Tutorial and Modern Developments. Verlag, Heidelberg: Springer; 2001. 373 p.
- Pikovsky A., Rosenblum M., Kurths J. Phase synchronisation in regular and chaotic systems. Int. J. Bifurcation and Chaos. 2000;10(10):2291–2305. DOI: 10.1142/S0218127400001481.
- Anishchenko VS, Vadivasova TE. Synchronization оf self-oscillations and noiseinduced oscillations. Journal of Communications Technology and Electronics. 2002;47(2):117–148.
- Rosenblum MG, Pikovsky AS, Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 1997;78(22):4193–4196. DOI: 10.1103/PhysRevLett.78.4193.
- Zheng Z, Hu С. Generalized synchronization versus phase synchronization. Phys. Rev. Е. 2000;62(6);7882–7885. DOI: 10.1103/PhysRevE.62.7882.
- Taherion S, Lai YC. Observability of lag synchronization оf coupled chaotic oscillators. Phys. Rev. Е. 1999;59(6):R6247–R6250. DOI: 10.1103/physreve.59.r6247.
- Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys. Rev. Lett. 1990;64(8):821–824. DOI: 10.1103/PhysRevLett.64.821.
- Pecora LM, Carroll TL. Driving systems with chaotic signals. Phys. Rev. А. 1991;44(4):2374–2383. DOI: 10.1103/PhysRevA.44.2374.
- Murali K, Lakshmanan M. Drive-response scenario of chaos synchronization in identical nonlinear systems. Phys. Rev. Е. 1994;49(6):4882–4885. DOI: 10.1103/PhysRevE.49.4882.
- Murali K, Lakshmanan M. Transmission of signals by synchronization in a chaotic Van der Pol-Duffing oscillator. Phys. Rev. Е. 1994;48(3):R1624–R1626. DOI: 10.1103/PhysRevE.48.R1624.
- Rulkov МЕ, Sushchik MM, Tsimring LS, Abarbanel НРD. Generalized synchronization оf chaos in directionally coupled chaotic systems. Phys. Rev. Е. 1995;51(2):980–994. DOI: 10.1103/PhysRevE.51.980.
- Kocarev L, Parlitz U. Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 1996;76(11):1816–1819. DOI: 10.1103/PhysRevLett.76.1816.
- Pyragas K. Weak and strong synchronization of chaos. Phys. Rev. Е. 1996;54(5):R4508–R4511. DOI: 10.1103/PhysRevE.54.R4508.
- Rosenblum MG, Pikovsky AS, Kurths J. Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 1996;76(11):1804–1807. DOI: 10.1103/PhysRevLett.76.1804.
- Osipov GV, Pikovsky AS, Rosenblum MG, Kurth J. Phase synchronization effect in а lattice оf nonidentical Rossler oscillators. Phys. Rev. Е. 1997;55(3):2353–2361. DOI: 10.1103/PhysRevE.55.2353.
- Boccaletti S, Valladares DL. Characterization оf intermittent lag synchronization. Phys. Rev. Е. 2000;62(5):7497–7500. DOI: 10.1103/PhysRevE.62.7497.
- Hramov AE, Koronovskii AE. Intermitted generalized synchronization in unidirectionally coupled chaotic oscillators. Europhysics Letters. 2005;70(2):169–175. DOI:10.1209/epl/i2004-10488-6.
- Landa PS. To the question of partial synchronization. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004;12(4):48–59.
- Abarbanel HDI, Rulkov МЕ, Sushchik М. Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. Е. 1996;53(5):4528–4535. DOI: 10.1103/PhysRevE.53.4528.
- Pecora LM, Carroll TL, Heagy JF. Statistics for mathematical properties of maps between time series embeddings. Phys. Rev. Е. 1995;52(4):3420–3439. DOI: 10.1103/PhysRevE.52.3420.
- Pyragas K. Conditional Lyapunov exponents from time series. Phys. Rev. В. 1997;56(5):5183–5188. DOI: 10.1103/PhysRevE.56.5183.
- Vadivasova TE, Anishchenko BC. Interrelation of frequency and phase characteristics of chaos. Two criteria of synchronization. Radiotekhnika i elektronika. 2004;49(1):76–82.
- Pikovsky A, Rosenblum M, Osipov G, Kurths J. Phase synchronization of chaotic oscillators by external driving. Physica D. 1997;104(4):219–238. DOI: 10.1007/978-94-010-0217-2_9.
- Lachaux JP, Eugenio R, Quyen MLV, Lutz A. Studying single-trials оf the phase synchronization activity in the brain. Int. J. Bifurcation and Chaos. 2000;10(10):2429–2439. DOI: 10.1142/S0218127400001560.
- Quiroga RQ, Kraskov А, Kreuz Т, Grassberger P. Performance of different synchronization measures in real data: a case study on electroencephalographic signals. Phys. Rev. E. 2002;65:041903. DOI: 10.1103/PhysRevE.65.041903.
- Pikovsky AS, Rosenblum MG, Kurths J. Synchronization in а population of globally coupled chaotic oscillators. Europhysics Letters. 1996;34(3):165–170. DOI: 10.1209/epl/i1996-00433-3.
- Rosenblum MG, Pikovsky AS, Kurths J. Locking-based frequency measurement and synchronization оf chaotic oscillators with complex dynamics. Phys. Rev. Lett. 2002;89(26):264102. DOI: 10.1103/PhysRevLett.89.264102.
- Koronovskii AA, Hramov AE. Analysis of chaotic synchronization of dynamical systems by means of wavelet transform. Tech. Phys. Lett. 2004;79(7):391–395.
- Hramov AE, Koronovskii AA. An approach to chaotic synchronization. Chaos. 2004;14(3):603–610. DOI: 10.1063/1.1775991.
- Koronovskii AA, Moskalenko OI, Hramov AE. A new type of universality in chaotic synchronization of dynamical systems. Tech. Phys. Lett. 2004;80(1):25–28.
- Koronovskii AA, Moskalenko OI, Hramov AE. Synchronization of spectral components of coupled chaotic oscillators. Tech. Phys. Lett. 2004;30(18):56–64.
- Hramov AE, Koronovskii AA, Kurovskaya MK, Moskalenko ОI. Synchronization оf spectral components and its regularities in chaotic dynamical systems. Phys. Rev. E. 2005;71(5):056204. DOI: 10.1103/PhysRevE.71.056204.
- Hramov AE, Koronovskii AA, Levin Y. Synchronization оf chaotic oscillator time scales. Tech. Phys. 2005;127(4):886–897.
- Koronovskii A.A, Hramov AE. Continuous wavelet analysis and its applications. Moscow: Fizmatlit; 2003. 176 p. (in Russian).
- Daubechies I. Ten Lectures оn Wavelets. USA: SIAM; 1992. 376 p.
- Kaiser С. A Friendly Guide to Wavelets. New-York:Springer-Verlag; 1994. 300 p.
- Torresani В. Continuous Wavelet Transform. Paris: Savoire; 1995. 676 p.
- Astafieva NM. Wavelet analysis: basics of theory and examples of application. Phys. Usp. 1996;166(11):1145–1170.
- Anfinogentov VG, Koronovskii AA, Hramov AE. Wavelet analysis and its use for analyzing the dynamics of nonlinear dynamical systems of different nature. Izvestiya RAN. Ser. physic. 2000;64(12):2383–2390.
- Koronovskii AA, Hramov AE. Introduction to continuous wavelet analysis for specialists in nonlinear dynamics. Part 1. Basic provisions, numerical realization and model signals. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(4-5):3–43.
- Grossman А. Morlet J. Decomposition оf hardy function into square integrable wavelets of constant shape. SIAM J. Math. Anal. 1984;15(4):273.
- Koronovskii AA, Hramov AE. Analysis of phase chaotic synchronization using continuous wavelet analysis. Tech. Phys. Lett. 2004;30(14):29–36.
- Anishchenko VS, Vadivasova TE, Postnov DE, Safonova МА. Synchronization of chaos. Int. J. Bifurcation and Chaos. 1992;2(3):633–644
- Sosnovtseva ОV, Balanov АG, Vadivasova ТE, Astakhov VV, Mosekilde Е. Loss of lag synchronization in coupled chaotic systems. Phys. Rev. Е. 1999;60(6):6560–6565. DOI: 10.1103/PhysRevE.60.6560/
- Koronovskii AA, Popov PW, Hramov AE. Chaotic synchronization of single directionally coupled electron media with a counter wave. Tech. Phys. 2005;75(4):1–9.
- Hramov AE, Koronovskii AA, Popov РК, Rempen IS. Chaotic synchronization оf coupled electron-wave systems with backward waves. Chaos. 2005;15(1):013705. DOI: 10.1063/1.1857615.
- Kuznetsov SP, Chetverikov AP. K theory of the inverse wave lamp with a transverse field. Radiotekhnika i elektronika. 1978;23(2):385.
- Trubetskov DI, Chetverikov AP. Auto oscillations in distributed systems "electron flow - counter (reverse) electromagnetic wave". Izv. VUZ. Applied nonlinear dynamics. 1994;2(5):3.
- Chetverikov AP. Nonlinear dynamics of the system of interacting counter electromagnetic wave and electron wave with cubic phase nonlinearity. Izv. VUZ. Applied nonlinear dynamics. 1994;2(5):46.
- Torrence C, Compo GP. А practical guide to wavelet analysis. Bulletin of the American Meteorological Society. 1998;79(1):61-78. DOI: 10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2.
- Gusev BA, Koronovskii AA, Hramov AE. Application of adaptive wavelet bases to the analysis of nonlinear systems with chaotic dynamics. Tech. Phys. Lett. 2003;29(18):61–69.
Received:
15.12.2004
Accepted:
16.04.2005
Published:
15.06.2005
Journal issue:
- 399 reads