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A new approach to the treatment of Separatrix Chaos and its applications

By S. M. Soskin, R. Mannella, O. M. Yevtushenko, I. A. Khovanov and P. V. E. McClintock


We consider time-periodically perturbed 1D Hamiltonian systems possessing one or more separatrices. If the perturbation is weak, then the separatrix chaos is most developed when the perturbation frequency lies in the logarithmically small or moderate ranges: this corresponds to the involvement of resonance dynamics into the separatrix chaos. We develop a method matching the discrete chaotic dynamics of the separatrix map and the continuous regular dynamics of the resonance Hamiltonian. The method has allowed us to solve the long-standing problem of an accurate description of the maximum of the separatrix chaotic layer width as a function of the perturbation frequency. It has also allowed us to predict and describe\ud new phenomena including, in particular: (i) a drastic facilitation of the onset of global chaos between neighbouring separatrices, and (ii) a huge increase in the\ud size of the low-dimensional stochastic web

Topics: QA, QC
Publisher: Springer
Year: 2010
OAI identifier: oai:wrap.warwick.ac.uk:3312

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  1. (1979). A universal instability of many-dimensional oscillator systems. doi
  2. (1999). AC-driven anomalous stochastic diffusion and chaotic transport in magnetic superlattices. doi
  3. (2010). Adiabatic divergence of the chaotic layer width and acceleration of chaotic and noise-induced transport. doi
  4. (1984). and Hohs S.M.: Stochasticity and reconnection in Hamiltonian systems. doi
  5. (2001). and Silchenko A.N.: Strong enhancement of noiseinduced escape by transient chaos. doi
  6. (1986). Change in adiabatic invariant at a separatrix. doi
  7. (1994). Chaotic and ballistic dynamics for two–dimensional electrons in periodic magnetic fields. doi
  8. (2004). Chaotic layer of a pendulum under low-and medium-frequency perturbations. doi
  9. (2006). Chaotic mixing and transport in a meandering jet flow. doi
  10. (2006). Construction of Mappings for Hamiltonian Systems and Their Applications. doi
  11. (2007). Corrugated Waveguide under Scaling Investigation. doi
  12. (2002). Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom. doi
  13. (1996). del-Castillo-NegreteD.,GreeneJ.M.,MorrisonP.J.:Area-preservingnon-twistmaps:periodic orbits and transition to chaos.
  14. (1993). Deterministic diffusion and magnetotransport in periodically modulated magnetic fields. doi
  15. (2005). Divergence of the Chaotic Layer Width and Strong Acceleration of the Spatial Chaotic Transport in Periodic Systems Driven by an Adiabatic ac Force. doi
  16. (1998). Effect of an ac electric field on chaotic electronic transport in a magnetic superlattice. doi
  17. (2004). et al.: Chaotic electron diffusion through stochastic webs enhances current flow in superlattices. doi
  18. (1995). et al.: Electrons in a periodic magnetic field induced by a regular array of micromagnets. doi
  19. (1987). et al.: Minimal chaos and stochastic webs. doi
  20. (1987). et al.: Some peculiarities of stochastic layer and stochastic web formation. doi
  21. (1988). et al.: Strong changing of adiabatic invariants, KAM-tori and web-tori. doi
  22. (2000). Generic twistless bifurcations. doi
  23. (2008). Hamiltonian Chaos and Fractional Dynamics. doi
  24. (1994). Homoclinic tangles – classification and applications. doi
  25. (1964). Instability of dynamical systems with several degrees of freedom. doi
  26. (2001). Lazutkin V.F.: Splitting of separatrices: perturbation theory and exponential smallness. doi
  27. (1998). Marginal resonances and intermittent Behavious in the motion in the vicinity of a separatrix. doi
  28. (2009). Maximal width of the separatrix chaotic layer. doi
  29. (1961). Mitropolsky Yu.A.: Asymptotic Methods in the Theory of Nonlinear Oscillators.
  30. New Approach To The Treatment Of Separatrix Chaos. doi
  31. (1995). Non-monotonic twist maps. doi
  32. (2004). Nonlinear dynamics theory of stochastic layers in Hamiltonian systems. doi
  33. (2010). P.V.E.: Enlargement of a lowdimensional stochastic web. In: Macucci
  34. (2007). Physics of Chaos in Hamiltonian systems, 2nd edn. doi
  35. private communication.
  36. (1992). Regular and Stochastic Motion. doi
  37. (1991). Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory. doi
  38. (1985). Spectral distribution of a nonlinear oscillator performing Brownian motion in a double-well potential. Physica A 133, doi
  39. (1997). Stable periodic motions in the problem on passage trough a separatrix. Chaos 7, doi
  40. (1970). Stegun I.: Handbook of Mathematical Functions. doi
  41. (1968). Stochastic instability of trapped particles and conditions of application of the quasi-linear approximation.
  42. (1986). Stochastic web and diffusion of particles in a magnetic field.
  43. (2008). The width of a chaotic layer. doi
  44. (1966). Theory of Oscillators. Pergamon, doi
  45. (1990). Transport rates of a class of two-dimensional maps and flows. doi
  46. (2007). Treschev D.V.: Separatrix maps in Hamiltonian systems. doi
  47. (1995). Two dimensional electrons in a lateral magnetic superlattice. doi
  48. (1991). Weak Chaos and QuasiRegular Patterns. doi
  49. (2008). Yevtushenko O.M.: Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices. doi
  50. (2008). Yevtushenko O.M.: Separatrix chaos: new approach to the theoretical treatment. In: Chandre doi
  51. (2003). Zero-Dispersion Phenomena in oscillatory systems. doi

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