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Stability of autoresonance models


We consider systems of two nonlinear nonautonomous differential equations on the real line which arise when averaging rapid nonlinear vibrations. We study the Lyapunov stability of solutions with infinitely increasing amplitude. Such solutions are related to the description of the initial stage of autoresonance or resonance trapping in oscillating nonlinear systems with a small perturbation. We obtain conditions on the coefficients of the equations under which the increasing solutions are stable or unstable. The problem is reduced to the analysis of an equilibrium. The stability of the equilibrium is studied by the Lyapunov second method. The construction of Lyapunov functions is based the presence of dissipative terms with coefficients moderately decaying in time.

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  1. 1.

    Bogolyubov, N.N. and Mitropol’skii, Yu.A., Asimptoticheskie metody v teorii nelineinykh kolebanii (Asymptotic Methods in the Theory of Nonlinear Vibrations), Moscow: Nauka, 1974.

    MATH  Google Scholar 

  2. 2.

    Sinclair, A.T., On the Origin of the Commensurabilities amongst the Satellites of Saturn, Month Notic Roy. Astron. Soc., 1972, vol. 160, no. 2, pp. 169–187.

    Google Scholar 

  3. 3.

    Meerson, B. and Friedland, L., Strong Autoresonance Excitation of Rydberg Atoms: the Rydberg Accelerator, Phys. Rev. A, 1990, vol. 41, pp. 5233–5236.

    Article  Google Scholar 

  4. 4.

    Khain, E. and Meerson, B., Parametric Autoresonance, Phys. Rev. E, 2001, vol. 64, pp. 1–8.

    Article  Google Scholar 

  5. 5.

    Arnol’d, V.I., Kozlov, V.V., and Neishtadt, A.I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki (Mathematical Aspects of Classical and Celestial Mechanics), Moscow: VNIITI, 1985.

    Google Scholar 

  6. 6.

    Neishtadt, A.I., Passage through a Separatrix in a Resonance Problem with a Slowly Varying Parameter, Prikl. Mat. Mekh., 1975, vol. 39, no. 4, pp. 621–632.

    MathSciNet  Google Scholar 

  7. 7.

    Kalyakin, L.A., Asymptotic Analysis of Autoresonance Models, Uspekhi Mat. Nauk, 2008, vol. 63, no. 5, pp. 3–72.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kuznetsov, A.N., On the Existence of Solutions Entering a Singular Point of an Autonomous System That Has a Formal Solution, Funktsional Anal. i Prilozhen., 1989, vol. 23, no. 4, pp. 63–74.

    MathSciNet  Google Scholar 

  9. 9.

    Kozlov, V.V. and Furta, S.D., Asimptotiki reshenii sil’no nelineinykh sistem differentsial’nykh uravnenii (Asymptotic Expansions of Solutions of Strongly Nonlinear Systems of Differential Equations), Moscow: Moskov. Gos. Univ., 1996.

    Google Scholar 

  10. 10.

    Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Certain Problems in the Theory of Stability of Motion), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1959.

    Google Scholar 

  11. 11.

    Anapol’skii, L.Yu., Irtegov, V.D., and Matrosov, V.M., Ways of Constructing Lyapunov Functions, Itogi Nauki Tekh. Obshch. Mekh., Moscow: VINITI, 1975, vol. 2, pp. 53–112.

    Google Scholar 

  12. 12.

    Chirikov, V.V., The Passage of a Nonlinear Oscillating System through Resonance, Dokl. Akad. Nauk SSSR, 1959, vol. 125, no. 5, pp. 1015–1018.

    MathSciNet  Google Scholar 

  13. 13.

    Sultanov, O.A., Lyapunov Functions for Nonautonomous Systems Close to Hamiltonian Systems, Ufimsk. Mat. Zh., 2010, vol. 2, no. 4, pp. 88–98.

    MATH  Google Scholar 

  14. 14.

    Kalyakin, L.A., Asymptotic Behavior of Solutions of Equations of Principal Resonance, Teoret. Mat. Fiz., 2003, vol. 137, no. 1, pp. 142–152.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Bryuno, A.D., Asymptotic Behavior and Expansions of Solutions of an Ordinary Differential Equation, Uspekhi Mat. Nauk, 2004, vol. 59, no. 3, pp. 31–80.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Bryuno, A.D. and Goryuchkina, I.V., Asymptotic Expansions of Solutions of the Sixth Painlevé Equation, Tr. Mosk. Mat. Obs., 2010, vol. 71, pp. 6–118.

    MathSciNet  Google Scholar 

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Correspondence to L. A. Kalyakin.

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Original Russian Text © L.A. Kalyakin, O.A. Sultanov, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 3, pp. 279–293.

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Kalyakin, L.A., Sultanov, O.A. Stability of autoresonance models. Diff Equat 49, 267–281 (2013).

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  • Quadratic Form
  • Asymptotic Expansion
  • Hamiltonian System
  • Lyapunov Function
  • Total Derivative