Impact Factor

    Conditions for Phase Locking and Dephasing of Autoresonant Pumping

    2019, Vol. 15, no. 3, pp.  381-394

    Author(s): Kiselev O. M.

    We study the asymptotic behavior of nonlinear oscillators under an external driver with slowly changing frequency and amplitude. As a result, we obtain formulas for properties of the amplitude and frequency of the driver when the autoresonant behavior of the nonlinear oscillator is observed. Also, we find the measure of autoresonant asymptotic behaviors for such a driven nonlinear oscillator.
    Keywords: nonlinear oscillator, autoresonance, perturbations
    Citation: Kiselev O. M., Conditions for Phase Locking and Dephasing of Autoresonant Pumping, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp.  381-394

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    [1] Veksler, V. I., “A New Method of Acceleration of Relativistic Particles”, J. Phys. USSR, 9 (1945), 153–158  mathscinet
    [2] McMillan, E. M., “The Synchrotron: A Proposed High Energy Particle Accelerator”, Phys. Rev., 68:5–6 (1945), 143–144  crossref  adsnasa
    [3] Friedland, L., “Spatial Autoresonance: Enhancement of Mode Conversion due to Nonlinear Phase-Locking”, Phys. Fluids B, 4:10 (1992), 3199–3209  crossref  mathscinet  adsnasa
    [4] Fajans, J. and Friedland, L., “Autoresonant (Nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators”, Am. J. Phys., 69:10 (2001), 1096–1102  crossref  adsnasa
    [5] Uleysky, M. Yu., Sosedko, E. V., and Makarov, D. V., “Autoresonant Cooling of Particles in Spatially Periodic Potentials”, Tech. Phys. Lett., 36:12 (2010), 1082–1084  crossref  adsnasa  elib; Pis'ma Zh. Tekh. Fiz., 36:23 (2010), 31–38 (Russian)
    [6] Friedland, L., “Autoresonance in Nonlinear Systems”, Scholarpedia, 4:1 (2009), 5473  crossref  adsnasa
    [7] Glebov, S. G., Kiselev, O. M., and Lazarev, V. A., “The Autoresonance Threshold in a System of Weakly Coupled Oscillators”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S111–S123  crossref  zmath; Tr. Inst. Mat. i Mekh. UrO RAN, 13:2 (2007), 43–54 (Russian)  mathnet
    [8] Kalyakin, L. A., “Asymptotic Analysis of Autoresonance Models”, Russian Math. Surveys, 63:5 (2008), 791–857  mathnet  crossref  mathscinet  zmath  adsnasa  elib; Uspekhi Mat. Nauk, 63:5(383) (2008), 3–72 (Russian)  crossref  mathscinet  zmath
    [9] Glebov, S. G., Kiselev, O. M., and Tarkhanov, N. N., Nonlinear Equations with Small Parameter: Vol. 1. Oscillations and Resonances, de Gruyter Ser. Nonlinear Anal. Appl., 23/1, De Gruyter, Berlin, 2017, xviii+335 pp.  mathscinet
    [10] Kiselev, O. M., Introduction to Nonlinear Oscillations Theory, 2nd ed., KomKniga, Moscow, 2006, 208 pp.
    [11] Kalyakin, L. A. and Sultanov, O. A., “Stability of Autoresonance Models”, Differ. Equ., 49:3 (2013), 267–281  crossref  mathscinet  zmath  elib; Differ. Uravn., 49:3 (2013), 279–293 (Russian)  zmath
    [12] Sultanov, O. A., “Stability of Autoresonance Models Subject to Random Perturbations for Systems of Nonlinear Oscillation Equations”, Comput. Math. Math. Phys., 54:1 (2014), 59–73  mathnet  crossref  mathscinet  zmath  elib
    [13] Neishtadt, A. I., “Passage through a Separatrix in a Resonance Problem with a Slowly-Varying Parameter”, J. Appl. Math. Mech., 39:4 (1975), 594–605  crossref  mathscinet; Prikl. Mat. Mekh., 39:4 (1975), 621–632 (Russian)
    [14] Haberman, R., “Nonlinear Transition Layers — the Second Painlevé Transcendent”, Studies in Appl. Math., 57:3 (1977), 247–270  crossref  mathscinet  zmath
    [15] Kiselev, O. M. and Glebov, S. G., “An Asymptotic Solution Slowly Crossing the Separatrix near a Saddle-Centre Bifurcation Point”, Nonlinearity, 16:1 (2003), 327–362  crossref  mathscinet  zmath  adsnasa
    [16] Poincaré, H., Les méthodes nouvelles de la mécanique céleste, v. 3, Gauthier-Villars, Paris; Dover, New York, 1899, 430 pp.  mathscinet  adsnasa
    [17] Krylov, N. M. and Bogolyubov, N. N., Introduction to Non-Linear Mechanics, Princeton Univ. Press, Princeton, 1949, 106 pp.  mathscinet
    [18] Wasow, W. R., Asymptotic Expansion for Ordinary Differential Equations, Pure Appl. Math., 14, Interscience, New York, 1965, 362 pp.  mathscinet
    [19] Olver, F. W. J., Asymptotics and Special Functions. Computer Science and Applied Mathematics, Acad. Press, New York, 1974, xvi+572 pp.  mathscinet
    [20] Arnol'd, V. I., Kozlov, V. V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., 3, 3rd ed., Springer, Berlin, 2006, xiv+518 pp.  crossref  mathscinet  zmath
    [21] Neishtadt, A. I., “Capture into Resonance and Scattering on Resonances in Two-Frequency Systems”, Proc. Steklov Inst. Math., 250 (2005), 183–203  mathnet  mathscinet  zmath; Tr. Mat. Inst. Steklova, 250 (2005), 198–218 (Russian)  zmath
    [22] Treschev, D. and Zubelevich, O., Introduction to the Perturbation Theory of Hamiltonian Systems, Springer, Berlin, 2010, x+211 pp.  mathscinet  zmath
    [23] Neishtadt, A., “Averaging Method for Systems with Separatrix Crossing”, Nonlinearity, 30:7 (2017), 2871–2917  crossref  mathscinet  zmath  adsnasa  elib
    [24] Friedland, L., “Subharmonic Autoresonance”, Phys. Rev. E, 61:4 (2000), 3732–3735  crossref  adsnasa
    [25] Kalyakin, L. A., “Asymptotic Solution of the Autoresonance Problem”, Theoret. and Math. Phys., 133:3 (2002), 1684–1691  mathnet  crossref  mathscinet  zmath; Teoret. Mat. Fiz., 133:3 (2002), 429–438 (Russian)  crossref  mathscinet  zmath
    [26] Garifullin, R. N., “Asymptotic Analysis of a Subharmonic Autoresonance Model”, Proc. Steklov Inst. Math. (Suppl.), suppl. 1 (2003), S75–S83  mathscinet  zmath; Tr. Inst. Mat. i Mekh. UrO RAN, 9:1 (2003), 56–63 (Russian)  mathnet  mathscinet
    [27] Chirikov, B. V., “Resonance Processes in Magnetic Traps”, J. Nucl. Energy: Part C, 1:4 (1960), 253–260  crossref; Soviet J. Atom. Energy, 6:6 (1960), 464–470  crossref
    [28] Chirikov, B. V., “A Universal Instability of Many-Dimensional Oscillator Systems”, Phys. Rep., 52:5 (1979), 264–379  crossref  mathscinet  adsnasa
    [29] Chirikov, B. V., “Passage of Nonlinear Oscillatory System through Resonance”, Sov. Phys. Dokl., 4 (1959), 390–394  mathscinet  zmath  adsnasa; Dokl. Akad. Nauk SSSR, 125:5 (1959), 1015–1018 (Russian)  zmath
    [30] Molchanov, A. M., “The Resonant Structure of the Solar System”, Icarus, 8:1–3 (1968), 203–215  crossref  adsnasa
    [31] Kevorkian, J., “On a Model for Reentry Roll Resonance”, SIAM J. Appl. Math., 26:3 (1974), 638–669  crossref  zmath
    [32] Murdock, J., “Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods”, Dynamics Reported, v. 1, eds. U. Kirchgraber, H. O Walther, Vieweg+Teubner, Wiesbaden, 1988, 91–172  crossref  mathscinet
    [33] Kiselev, O. M. and Tarkhanov, N., “The Capture of a Particle into Resonance at Potential Hole with Dissipative Perturbation”, Chaos Solitons Fractals, 58 (2014), 27–39  crossref  mathscinet  zmath  adsnasa  elib
    [34] Mel'nikov, V. K., “On the Stability of the Center for Time Periodic Perturbations”, Trans. Moscow Math. Soc., 12 (1963), 1–57  mathnet; Tr. Mosk. Mat. Obs., 12 (1963), 3–52 (Russian)  zmath
    [35] Kiselev, O. and Tarkhanov, N., “Scattering of Trajectories at a Separatrix under Autoresonance”, J. Math. Phys., 55:6 (2014), 063502, 24 pp.  crossref  mathscinet  zmath  adsnasa  elib

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