Abstract
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 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 size of the low-dimensional stochastic web.
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References
Abdullaev S.S., 2006, Construction of Mappings for Hamiltonian Systems and Their Applications, Springer, Berlin, Heidelberg.
Abramovitz M. and Stegun I., 1970, Handbook of Mathematical Functions, Dover, New York.
Andronov A.A., Vitt A.A. and Khaikin S.E., 1966, Theory of Oscillators, Pergamon, Oxford.
Arnold V.I., 1964, Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl., 5, 581–585.
Bogolyubov N.N. and Mitropolsky Yu.A., 1961, Asymptotic Methods in the Theory of Nonlinear Oscillators, Gordon and Breach, New York.
Carmona H.A. et al., 1995, Two dimensional electrons in a lateral magnetic super-lattice, Phys. Rev. Lett., 74, 3009–3012.
Chernikov A.A. et al., 1987a, Minimal chaos and stochastic webs, Nature, 326, 559–563.
Chernikov A.A. et al., 1987b, Some peculiarities of stochastic layer and stochastic web formation, Phys. Lett. A, 122, 39–46.
Chernikov A.A. et al., 1988, Strong changing of adiabatic invariants, KAM-tori and web-tori, Phys. Lett. A, 129, 377–380.
Chirikov B.V., 1979, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52, 263–379.
del-Castillo-Negrete D., Greene J.M. and Morrison P.J., 1996, Area-preserving non-twist maps: periodic orbits and transition to chaos, Physica D, 61, 1–23.
Dullin H.R., Meiss J.D. and Sterling D., 2000, Generic twistless bifurcations, Non-linearity, 13, 203–224.
Dykman M.I., Soskin S.M. and Krivoglaz M.A., 1985, Spectral distribution of a nonlinear oscillator performing Brownian motion in a double-well potential, Physica A, 133, 53–73.
Elskens Y. and Escande D.F., 1991, Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory, Nonlinearity, 4, 615–667.
Fromhold T.M. et al., 2001, Effects of stochastic webs on chaotic electron transport in semiconductor superlattices, Phys. Rev. Lett, 87, 046803-1–046803-4.
Fromhold T.M. et al., 2004, Chaotic electron diffusion through stochastic webs enhances current flow in superlattices, Nature, 428, 726–730.
Gelfreich V., private communication.
Gelfreich V.G. and Lazutkin V.F., 2001, Splitting of separatrices: perturbation theory and exponential smallness, Russian Math. Surveys, 56, 499–558.
Howard J.E. and Hohs S.M., 1984, Stochasticity and reconnection in Hamiltonian systems, Phys. Rev. A, 29, 418–421.
Howard J.E. and Humpherys J., 1995, Non-monotonic twist maps, Physica, D 80, 256–276.
Landau L.D. and Lifshitz E.M., 1976, Mechanics, Pergamon, London.
Leonel E.D., 2007, Corrugated Waveguide under Scaling Investigation, Phys. Rev. Lett., 98, 114102-1–114102-4.
Lichtenberg A.J. and Lieberman M.A., 1992, Regular and Stochastic Motion, Springer, New York.
Luo A.C.J., 2004, Nonlinear dynamics theory of stochastic layers in Hamiltonian systems, Appl. Mech. Rev., 57, 161–172.
Luo A.C.J., Gu K. and Han R.P.S., 1999, Resonant-separatrix webs in stochastic layers of the Twin-Well duffing oscillator, Nonlinear Dyn., 19, 37–48.
Morozov A.D., 2002, Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom, Chaos, 12, 539–548.
Neishtadt A.I., 1986, Change in adiabatic invariant at a separatrix, Sov. J. Plasma Phys., 12, 568–573.
Neishtadt A.I., Sidorenko V.V. and Treschev D.V., 1997, Stable periodic motions in the problem on passage trough a separatrix, Chaos, 7, 2–11.
Piftankin G.N. and Treschev D.V., 2007, Separatrix maps in Hamiltonian systems, Russian Math. Surveys, 62, 219–322.
Prants S.V., Budyansky M.V., Uleysky M.Yu. and Zaslavsky G.M., 2006, Chaotic mixing and transport in a meandering jet flow, Chaos, 16, 033117-1–033117-8.
Rom-Kedar V., 1990, Transport rates of a class of two-dimensional maps and flows, Physica D, 43, 229–268.
Rom-Kedar V., 1994, Homoclinic tangles-classification and applications, Nonlinearity, 7, 441–473.
Schmelcher P. and Shepelyansky D.L., 1994, Chaotic and ballistic dynamics for two-dimensional electrons in periodic magnetic fields, Phys. Rev. B, 49, 7418–7423.
Shevchenko I.I., 1998, Marginal resonances and intermittent Behavious in the motion in the vicinity of a separatrix, Phys. Scr., 57, 185–191.
Shevchenko I.I., 2008, The width of a chaotic layer, Phys. Lett. A, 372, 808–816.
Schmidt G.J.O., 1993, Deterministic diffusion and magnetotransport in periodically modulated magnetic fields, Phys. Rev. B, 47, 13007–13010.
Soskin S.M., Unpublished.
Soskin S.M. and Mannella R., 2009a, New approach to the treatment of separatrix chaos, In Macucci C. and Basso G. (eds.) Noise and Fluctuations: 20 th International Conference on Noise and Fluctuations (ICNF-2009), MP CONFERENCE PROCEEDINGS 1129, 25–28, American Institute of Physics, Melville, New York.
Soskin S.M. and Mannella R., 2009b, Maximal width of the separatrix chaotic layer, Phys. Rev. E., 80, 066212-1–006212-1F.
Soskin S.M., Mannella R., Arrayás M. and Silchenko A.N., 2001, Strong enhancement of noise-induced escape by transient chaos, Phys. Rev. E, 63, 051111-1–051111-6.
Soskin S.M., Mannella R. and McClintock P.V.E., 2003, Zero-Dispersion Phenomena in oscillatory systems, Phys. Rep., 373, 247–409.
Soskin S.M., Yevtushenko O.M. and Mannella R., 2005, Divergence of the chaotic layer width and strong acceleration of the spatial chaotic transport in periodic systems driven by an adiabatic ac force, Phys. Rev. Lett, 95, 224101-1–224101-4.
Soskin S.M., Mannella R. and Yevtushenko O.M., 2008a, Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices, Phys. Rev. E, 11, 036221-1–036221-29.
Soskin S.M., Mannella R. and Yevtushenko O.M., 2008b, Separatrix chaos: new approach to the theoretical treatment. In: Chandre C, Leoncini X., and Zaslavsky G. (eds.) Chaos, Complexity and Transport: Theory and Applications (Proceedings of the CCT-07), 119–128, World Scientific, Singapore.
Soskin S.M., Khovanov I.A., Mannella R. and McClintock P.V.E., 2009, Enlargement of a low-dimensional stochastic web, Macucci C. and Basso G. (eds.) Noise and Fluctuations: 20 th International Conference on Noise and Fluctuations (ICNF-2009), AIP CONFERENCE PROCEEDINGS 1129, 17–20, American Institute of Physics, Melville, New York.
Soskin S.M., Yevtushenko O.M. and Mannella R., 2010a, Adiabatic divergence of the chaotic layer width and acceleration of chaotic and noise-induced transport, Commun. Nonlinear Sci. Numer. Simulat, 15, 16–23.
Soskin S.M., McClintock P.V.E., Fromhold T.M., Khovamov I.A. and Mannella R., 2010b, Stochastic webs and quantium transport in superlattices: an introductory review, Contemportary Physics, 51, 233–248.
Vecheslavov V.V., 2004, Chaotic layer of a pendulum under low-and medium-frequency perturbations, Tech. Phys., 49, 521–525.
Ye P.D. et al., 1995, Electrons in a periodic magnetic field induced by a regular array of micromagnets, Phys. Rev. Lett., 74, 3013–3016.
Yevtushenko O.M. and Richter K., 1998, Effect of an ac electric field on chaotic electronic transport in a magnetic superlattice, Phys. Rev. B, 57, 14839–14842.
Yevtushenko O.M. and Richter K., 1999, AC-driven anomalous stochastic diffusion and chaotic transport in magnetic superlattices, Physica, E 4, 256–276.
Zaslavsky G.M., 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.
Zaslavsky G.M., 2007, Physics of Chaos in Hamiltonian Systems, 2nd ed., Imperial Colledge Press, London.
Zaslavsky G.M. and Filonenko N.N., 1968, Stochastic instability of trapped particles and conditions of application of the quasi-linear approximation, Sov. Phys. JETP, 27, 851–857.
Zaslavsky G.M. et al., 1986, Stochastic web and diffusion of particles in a magnetic field, Sov. Phys. JETP, 64, 294–303.
Zaslavsky G.M., Sagdeev R.D., Usikov D.A. and Chernikov A.A., 1991, Weak Chaos and Quasi-Regular Patterns, Cambridge University Press, Cambridge.
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Soskin, S.M., Mannella, R., Yevtushenko, O.M., Khovanov, I.A., McClintock, P.V.E. (2010). A New Approach to the Treatment of Separatrix Chaos and Its Applications. In: Luo, A.C.J., Afraimovich, V. (eds) Hamiltonian Chaos Beyond the KAM Theory. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12718-2_2
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