Abstract
As mentioned above, resonance energy transfer represents one of the most effective ways of the response enhancement for a broad range of physical and engineering systems. In this chapter, the notion of resonance energy transfer is extended to the oscillators subjected to harmonic forcing with a slowly varying frequency. We investigate capture into resonance of a Klein–Gordon chain of identical linearly coupled Duffing oscillators excited by a harmonic force with a slowly varying frequency applied at an edge of the chain.
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References
Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Dover, New York (1966)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Springer, Berlin (2006)
Barth, I., Friedland, L.: Two-photon ladder climbing and transition to autoresonance in a chirped oscillator. Phys. Rev. A 87(1–4), 053420 (2013)
Ben-David, O., Assaf, M., Fineberg, J., Meerson, B.: Experimental study of parametric autoresonance in Faraday waves. Phys. Rev. Lett. 96(1–4), 154503 (2006)
Bohm, D., Foldy, L.L.: Theory of the synchro-cyclotron. Phys. Rev. 72, 649–661 (1947)
Chacón, R.: Energy-based theory of autoresonance phenomena: application to duffing-like systems. Europhys. Lett. 70, 56–62 (2005)
Dodin, I.Y., Fisch, N.J.: Adiabatic nonlinear waves with trapped particles. III. Wave dynamics. Phys. Plasmas 19, 012104 (2012)
Friedland L.: http://www.phys.huji.ac.il/~lazar/
Friedland, L.: Efficient capture of nonlinear oscillations into resonance. J. Phys. A: Math. Theor. 41(1–8), 415101 (2008)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)
Kalyakin, L.A.: Asymptotic analysis of autoresonance models. Russ. Math. Surv. 63, 791–857 (2008)
Kivshar, Y.S.: Intrinsic localized modes as solitons with a compact support. Phys. Rev. E 48, R43–R45 (1993)
Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers, 2nd edn. Dover Publications, New York (2000)
Kovaleva A.: Resonance energy transport in an oscillator chain. http://arXiv:1501.00552. (2015)
Kovaleva, A., Manevitch, L.I.: Classical analog of quasilinear Landau-Zener tunneling. Phys. Rev. E. 85(1–8), 016202 (2012)
Kovaleva, A., Manevitch, L.I.: Resonance energy transport and exchange in oscillator arrays. Phys. Rev. E 88(1–10), 022904 (2013a)
Kovaleva, A., Manevitch, L.I.: Emergence and stability of autoresonance in nonlinear oscillators. Cybern. Phys. 2, 25–30 (2013b)
Kovaleva, A., Manevitch, L.I.: Limiting phase trajectories and emergence of autoresonance in nonlinear oscillators. Phys. Rev. E 88(1–6), 024901 (2013c)
Kovaleva, A., Manevitch, L.I.: Autoresonance energy transfer versus localization in weakly coupled oscillators. Phys D Nonlinear Phenom 320, 1–8 (2016)
Kovaleva, A., Manevitch, L.I., Manevitch, E.L.: Intense energy transfer and superharmonic resonance in a system of two coupled oscillators. Phys. Rev. E 81(1–12), 056215 (2010)
Marcus, G., Friedland, L., Zigler, A.: From quantum ladder climbing to classical autoresonance. Phys. Rev. A 69(1–5), 013407 (2004)
McMillan, E.M.: The synchrotron—a proposed high energy particle accelerator. Phys. Rev. 68, 143–144 (1945)
Murch, K.W., Vijay, R., Barth, I., Naaman, O., Aumentado, J., Friedland, L., Siddiqi, I.: Quantum fluctuations in the chirped pendulum. Nat. Phys. 7, 105–108 (2011)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-VCH, Weinheim (2004)
Neishtadt, A.I.: Passage through a separatrix in a resonance problem with slowly varying parameter. J. Appl. Math. Mech. 39, 594–605 (1975)
Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, 2nd edn. Springer, Berlin (2007)
Shalibo, Y., Rofe, Y., Barth, I., Friedland, L., Bialczack, R., Martinis, J.M., Katz, N.: Quantum and classical chirps in an anharmonic oscillator. Phys. Rev. Lett. 108(1–5), 037701 (2012)
Veksler, V.I.: Some new methods of acceleration of relativistic particles. ComptesRendus (Dokaldy)de l’Academie Sciences de l’URSS 43, 329–331 (1944)
Zelenyi, L.M., Neishtadt, A.I., Artemyev, A.V., Vainchtein, D.L., Malova, H.V.: Quasiadiabatic dynamics of charged particles in a space plasma. Physics-Uspekhi 56, 347–394 (2013)
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Manevitch, L.I., Kovaleva, A., Smirnov, V., Starosvetsky, Y. (2018). Limiting Phase Trajectories and the Emergence of Autoresonance in Anharmonic Oscillators. In: Nonstationary Resonant Dynamics of Oscillatory Chains and Nanostructures. Foundations of Engineering Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4666-7_8
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DOI: https://doi.org/10.1007/978-981-10-4666-7_8
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