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Slipushenko S. V., Tur A. V., Yanovsky V. V. Origin of intermittency in singular hamiltonian systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 4, pp. 91-110. DOI: 10.18500/0869-6632-2010-18-4-91-110

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Origin of intermittency in singular hamiltonian systems

Slipushenko Sergej Vasilevich, Institute of Monocrystals of NAS of Ukraine
Tur Anatolij Valentinovich, Institut de Recherche en Astrophysique et Planetologie
Yanovsky Vladimir Vladimirovich, Institute of Monocrystals of NAS of Ukraine

In the paper we studied properties of conservative singular maps. It was found that under some conditions the intermittency without chaotic phases can be observed in these maps. The alternative mechanism of the intermittency origin in Hamiltonian singular systems was considered. Its general properties were discussed. We studied special properties of phase space structure in these systems. It is shown that Hamiltonian intermittency can be characterized by zero Lyapunov exponents. It gives us the possibility to classify it as pseoudochaos dynamics.

  1. Kolmogorov AN. La theorie generale des systemes dynamiques et la mecanique classique. Amsterdam Congress. 1954;1:315–333.
  2. Arnol'd VI. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math. Surveys. 1963;18(6):85–191.
  3. Moser JK. Stable and random motions in dynamical systems: with special emphasis on celestial mechanics. Princeton: Princeton Univ. Press Ann. of Math. Studies; 1973.
  4. Chirikov VV. Nonlinear resonance. NSU; 1977. (in Russian).
  5. Arnold VI, Avez A. Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin; 1968.
  6. Lichtenberg AJ, Lieberman MA. Regular and Stochastic Motion. New York: Springer; 1983.
  7. Moser Yu. Lectures on Hamiltonian systems. Moscow: Mir; 1973. (in Russian).
  8. Fan R, Zaslavsky GM. Pseudochaotic dynamics near global periodicity. Communications in Nonlinear Science and Numerical Simulation. 2007;12(6):1038–1052. DOI: 10.1016/j.cnsns.2005.09.002.
  9. Scott AJ, Holmesa CA, Milburnb GJ. Hamiltonian mappings and circle packing phase spaces. Physica D: Nonlinear Phenomena. 2001;155:34–50. DOI: 10.1016/S0167-2789(01)00263-9.
  10. Zaslavsky GM, Sagdeev RZ, Usikov DA, Chernikov AA. Weak Chaos and Quasi-Regular Patterns. Cambridge: Cambridge University Press; 1991. 
  11. Zaslavsky GM, Edelman M. Pseudochaos; 2001. arXiv:nlin/0112033v2.
  12. Grebogi C, Ott E, Yorke JA. Fractal basin boundaries, long-lived chaotic transients, and unstable-unstaible pair bifurcation. Phys. Rev. Lett. 1983;50(13):935–938. DOI: 10.1103/PhysRevLett.50.935.
  13. Sinai YaG. Introduction to ergodic theory. Princeton: Princeton Univ. Press; 1977. 144 p.
  14. Rosenfeld BA, Sergeeva ND. Stereographic projection. Moscow: Nauka; 1973. (in Russian).
  15. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  16. Schuster G. Deterministic Chaos. An Introduction. Moscow: Mir; 1988. (in Russian).
  17. Slipushenko SV., Tur AV, Yanovsky VV. Intermittency without chaotic phases. Functional Materials. 2006;13(4):551–557.
  18. Slipushenko SV, Tur AV, Janovskij VV. Intermittency concurrence. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(4):3–19. DOI: 10.18500/0869-6632-2008-16-4-3-19.
  19. Schwarz AS. Quantum field theory and topology. Grundlehren der Math. Wissen. New York: Springer. 1993;307. 
  20. Shenker SJ. Scaling behavior in a map of a circle into itself: Empirical results. Physica D. 1982;5:405–411. DOI: 10.1016/0167-2789(82)90033-1.
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