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

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Postnov D. E., Balanov A. G. «Chaotic hierarchy» in the model map. Izvestiya VUZ. Applied Nonlinear Dynamics, 1999, vol. 7, iss. 6, pp. 26-34. DOI: 10.18500/0869-6632-1999-7-6-26-34

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Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-1999-7-6-26-34

«Chaotic hierarchy» in the model map

Autors: 
Postnov Dmitrij Engelevich, Saratov State University
Balanov Aleksandr Gennadevich, Loughborough University
Abstract: 

We investigate the model that describes dynamics of global coupled oscillators. We demonstrate how the Lyapunov dimension grows with increasing numbers of interacting units. Development of additional unstable directions of chaotic attractor is traced both ш the spectrum of Lyapunov exponents and in transformation of phase portraits.

Key words: 
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Acknowledgments: 
The work was supported by the grant of RFBR № 99-02-17732.
Reference: 
  1. Kaplan JL,Yorke JA. Chaotic behavior оf multidimensional difference equations. In: Peitgen HO, Walther HO, editors. Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics. Vol 730. Berlin: Springer; 1979. P. 204-227. DOI: 10.1007/BFb0064319.
  2. Neimark YuI, Landa PS. Stochastic and chaotic fluctuations. M.: Nauka; 1987. 424 p.
  3. Anishchenko VS. Complex oscillations in simple systems. М.: Nauka; 1990. 312 p. (in Russian).
  4. Rossler OE. An equation for hyperchaos. Phys. Lett. А. 1979;17:155-157. DOI: 10.1016/0375-9601(79)90150-6. Baier G, Klein М. Maximum hyperchaos in generalized Hénon maps. Phys. Lett. А. 1990;151(6-7):281-284. DOI: 10.1016/0375-9601(90)90283-T.
  5. Anishchenko VS, Kapitaniak T, Safonova MA, Sosnovtseva OV. Birth of double—double scroll attractor in coupled Chua’s circuits. Phys. Lett. А. 1994;192(2-4):207-214. DOI: 10.1016/0375-9601(94)90245-3.
  6. Pikovsky AS, Rosenblum MG, Kurths J. Synchronization in a population of globally coupled chaotic oscillators. Europhys. Lett. 1996;34(3):165-170.
  7. Osipov GV, Pikovsky AS, Rosenblum MG, Kurths J. Phase synchronization effects in а lattice оf nonidentical Rossler oscillators. Phys. Rev. E. 1997;55(3):2353-2361. DOI: 10.1103/PhysRevE.55.2353.
  8. Anishchenko VS, Aranson IS, Postnov DE, Rabinovich MI. Spatial synchronization and bifurcation of chaos development in the chain of connected generators. Soviet Physics. Doklady. 1986;286(5):1120-1124. (in Russian).
  9. Anishchenko VS, Postnov DE, Safonova MA. Dimension and physical properties of chaotic attractors in the chain of coupled generators. Tech. Phys. Lett. 1985;11(24):1505-1509. (in Russian).
  10. Levin BR, Stewart FM, Chao L. Resource—limited growth, competition and predation: а model and experimental studies with bacteria and bacteriophage. American Naturalist. 1977;111(977):3-24.
  11. Baier G, Thomsen JS, Mosekilde E. Chaotic hierarchy in а model оf competing populations. J. Theor. Biol. 1993;165:593-607.
  12. Mosekilde Е. Topics in Nonlinear Dynamics: Applications to Physics, Biology and Economic Systems. Singapore: World Scientific; 1997. 392 p.
  13. Baier G, Sahle S. Design of hyperchaotic flows. Phys. Rev. Е. 1995;51(4):2712-2714. DOI: 10.1103/PhysRevE.51.R2712.
  14. Meyer Th, Bunner MJ, Kittel A, Parisi J. Hyperchaos in the generalized Rossler system. Phys. Rev. Е. 1997;56:5069-5082. DOI: 10.1103/PhysRevE.56.5069.
  15. Kaneko K. Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements. Physica D. 1990;41(2):137-172. DOI: 10.1016/0167-2789(90)90119-A.
  16. Belykh VN, Mosekilde E. One—dimensional map lattices: Synchronization, bifurcation and chaotic structures. Phys. Rev. E. 1996;54(4):3196-3203. DOI: 10.1103/PhysRevE.54.3196.
  17. Kaneko K. Globally coupled circle maps. Physica D. 1990;54(1-2):5-19. DOI: 10.1016/0167-2789(91)90103-G.
  18. Willeboordse FH. Encoding of travelling waves in a coupled map lattice. Int. J. Bifurc. Chaos. 1994;4(6):1667-1673. DOI: 10.1142/S0218127494001271.
  19. Afraimovich VS, Nekorkin VI. Chaos of travelling waves in a discrete chain of diffusively coupled maps. Int. J. Bifurc. Chaos. 1994;4(3):631-637. DOI: 10.1142/S0218127494000459.
  20. Afraimovich VS, Nekorkin VI, Osipov GV, Shalfeev VD. Stability, Structures and Chaos in Nonlinear Synchronization Networks. Singapore: World Scientific; 1995. 260 p. DOI: 10.1142/2412.
  21. Kuznetsov SP. Theory and Applications оf Coupled Мар Lattices. N.Y.: Wiley, 1993. 192 p.
  22. Mattews PC, Strogatz SН. Phase diagram for the collective behavior оf limit— cycle oscillators. Phys. Rev. Lett. 1990;65(14):1701-1704. DOI: 10.1103/PhysRevLett.65.1701.
  23. Baesens C, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: mode—locking, global bifurcations and toroidal chaos. Physica D. 1991;49(3):387-475. DOI: 10.1016/0167-2789(91)90155-3.
  24. Grebogi C, Ott E, Yorke J. Attractors on an N—torus: quasiperiodicity versus chaos. Physica D. 1985;15(3):354-373. DOI: 10.1016/S0167-2789(85)80004-X.
  25. Rend D, Ostlund S, Sethna J, Siggia ED. Universal transition from quasiperiodisity to chaos in dissipative systems. Physica D. 1983;8(3):303-342. DOI: 10.1016/0167-2789(83)90229-4.
  26. Ruelle D, Takens F. On the nature of turbulence. Commun. Math. Phys. 1971;20:167-192. DOI: 10.1007/BF01646553.
  27. Newhouse S, Ruelle D, Takens F. Occurrance of strange axiom A attractors near quasi—periodic flows on Tm, m>=3. Commun. Math. Phys. 1978;64:35-40. DOI: 10.1007/BF01940759.
  28. Grebogi C, Ott E, Yorke JA. Chaotic attractors in crisis. Phys. Rev. Lett. 1982;48(22):1507-1510. DOI: 10.1103/PhysRevLett.48.1507.
  29. Anishchenko VS. Interaction of strange attractors, intermittance of “chaos-chaos”. Tech. Phys. Lett. 1984;10(10):629-633. (in Russian).
Received: 
22.04.1999
Accepted: 
09.11.1999
Published: 
01.02.2000
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