Abstract
Slow-fast dynamics and resonant phenomena can be found in a wide range of physical systems, including problems of celestial mechanics, fluid mechanics, and charged particle dynamics. Important resonant effects that control transport in the phase space in such systems are resonant scatterings and trappings. For systems with weak diffusive scatterings the transport properties can be described with the Chirikov standard map, and the map parameters control the transition between stochastic and regular dynamics. In this paper we put forward the map for resonant systems with strong scatterings that result in nondiffusive drift in the phase space, and trappings that produce fast jumps in the phase space. We demonstrate that this map describes the transition between stochastic and regular dynamics and find the critical parameter values for this transition.
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Arnol’d, V. I., Kozlov, V. V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.
Artemyev, A. V., Neishtadt, A. I., and Vasiliev, A. A., Kinetic Equation for Nonlinear Wave-Particle Interaction: Solution Properties and Asymptotic Dynamics, Phys. D, 2019, vol. 393, pp. 1–8.
Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A., and Mourenas, D., Kinetic Equation for Nonlinear Resonant Wave-Particle Interaction, Phys. Plasmas, 2016, vol. 23, no. 9, 090701, 4 pp.
Bénisti, D., Morice, O., Gremillet, L., Friou, A., and Lefebvre, E., Nonlinear Kinetic Modeling of Stimulated Raman Scattering in a Multidimensional Geometry, Phys. Plasmas, 2012, vol. 19, no. 5, 056301, 8 pp.
Chirikov, B. V., Passage of Nonlinear Oscillatory System through Resonance, Sov. Phys. Dokl., 1959, vol. 4, pp. 390–394; see also: Dokl. Akad. Nauk SSSR, 1959, vol. 125, no. 5, pp. 1015–1018.
Chirikov, B. V., A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep., 1979, vol. 52, no. 5, pp. 264–379.
Dewald, E. L., Hartemann, F., Michel, P., Milovich, J., Hohenberger, M., Pak, A., Landen, O. L., Divol, L., Robey, H. F., Hurricane, O. A., Döppner, T., Albert, F., Bachmann, B., Meezan, N. B., MacKinnon, A. J., Callahan, D., and Edwards, M. J., Generation and Beaming of Early Hot Electrons onto the Capsule in Laser-Driven Ignition Hohlraums, Phys. Rev. Lett., 2016, vol. 116, no. 7, 075003, 5 pp.
Dolgopyat, D., Repulsion from Resonances, Mém. Soc. Math. France (N. S.), no. 128, Paris: Soc. Math. France, 2012.
Kasahara, S., Miyoshi, Y., Yokota, S., Mitani, T., Kasahara, Y., Matsuda, S., Kumamoto, A., Matsuoka, A., Kazama, Y., Frey, H. U., Angelopoulos, V., Kurita, S., Keika, K., Seki, K., and Shinohara, I., Pulsating Aurora from Electron Scattering by Chorus Waves, Nature, 2018, vol. 554, pp. 337–340.
Krasnoselskikh, V., Balikhin, M., Walker, S. N., Schwartz, S., Sundkvist, D., Lobzin, V., Gedalin, M., Bale, S. D., Mozer, F., Soucek, J., Hobara, Y., and Comisel, H., The Dynamic Quasiperpendicular Shock: Cluster Discoveries, Space Sci. Rev., 2013, vol. 178, nos. 2–4, pp. 535–598.
Lichtenberg, A. J. and Lieberman, M. A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992.
Morbidelli, A., Modern Celestial Mechanics: Aspects of Solar System Dynamics, London: Taylor & Francis, 2002.
Neishtadt, A. I., On Adiabatic Invariance in Two-Frequency Systems, in Hamiltonian Systems with Three or More Degrees of Freedom (S’Agaro, 1995), C. Simó (Ed.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Dordrecht: Kluwer, 1999, pp. 193–212.
Omura, Y., Miyashita, Y., Yoshikawa, M., Summers, D., Hikishima, M., Ebihara, Y., and Kubota, Y., Formation Process of Relativistic Electron Flux through Interaction with Chorus Emissions in the Earth’s Inner Magnetosphere, J. Geophys. Res. Space Phys., 2015, vol. 120, pp. 9545–9562.
Rom-Kedar, V., Leonard, A., and Wiggins, S., An Analytical Study of Transport, Mixing and Chaos in an Unsteady Vortical Flow, J. Fluid Mech., 1990, vol. 214, pp. 347–394.
Sagdeev, R. Z., Usikov, D. A., and Zaslavsky, G. M., Nonlinear Physics: From the Pendulum to Turbulence and Chaos, Chur: Harwood Acad. Publ., 1990.
Thorne, R. M., Li, W., Ni, B., Ma, Q., Bortnik, J., Chen, L., Baker, D. N., Spence, H. E., Reeves, G. D., Henderson, M. G., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Blake, J. B., Fennell, J. F., Claudepierre, S. G., and Kanekal, S. G., Rapid Local Acceleration of Relativistic Radiation-Belt Electrons by Magnetospheric Chorus, Nature, 2013, vol. 504, pp. 411–414.
van Kampen, N. G., Stochastic Processes in Physics and Chemistry, 3rd ed., Amsterdam: North-Holland, 2007.
Wilson, L. B. III, Cattell, C., Kellogg, P. J., Goetz, K., Kersten, K., Hanson, L., MacGregor, R., and Kasper, J. C., Waves in Interplanetary Shocks: A Wind/WAVES Study, Phys. Rev. Lett., 2007, vol. 99, no. 4, 041101, 4 pp.
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This work was supported by the Russian Science Foundation, project 19-12-00313.
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Artemyev, A.V., Neishtadt, A.I. & Vasiliev, A.A. A Map for Systems with Resonant Trappings and Scatterings. Regul. Chaot. Dyn. 25, 2–10 (2020). https://doi.org/10.1134/S1560354720010025
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DOI: https://doi.org/10.1134/S1560354720010025