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Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena

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Abstract

In this paper, a regular perturbation tool is suggested to bridge the gap between weakly and strongly nonlinear dynamics based on exactly solvable oscillators with trigonometric characteristics considered by Nesterov (Proc. Mosc. Inst. Power Eng. 357:68–70, 1978). It is shown that the corresponding action-angle variables linearize the original oscillators with no special functions involved. As a result, linear and strongly nonlinear areas of the dynamics are described within the same perturbation procedure. The developed tool is applied then to analyzing the nonlinear beat and energy localization phenomena in two linearly coupled Duffing oscillators. It is shown that the principal phase variable describing the beat phenomena is governed by the hardening Nesterov oscillator with some perturbation due to qubic nonlinearity and coupling between the oscillators. As a result, the above class of strongly nonlinear oscillators is given clear physical meaning, whereas a closed form analytical solution is obtained for nonlinear beat and localization dynamics. Based on this solution, necessary and sufficient conditions for onset of energy localization are obtained.

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References

  1. Coppola, V.T., Rand, R.H.: Averaging using elliptic functions: approximation of limit cycles. Acta Mech. 81, 125–142 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Xu, Z., Cheung, Y.: Averaging method using generalized harmonic functions for strongly non-linear oscillators. J. Sound Vib. 174(4), 563–576 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Nayfeh, A.H.: Perturbation methods in nonlinear dynamics. In: Nonlinear Dynamics Aspects of Particle Accelerators, Santa Margherita di Pula, 1985, pp. 238–314. Springer, Berlin (1986)

    Google Scholar 

  4. Salenger, G., Vakakis, A.F., Gendelman, O., Manevitch, L., Andrianov, I.: Transitions from strongly to weakly nonlinear motions of damped nonlinear oscillators. Nonlinear Dyn. 20(2), 99–114 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mikhlin, Y.V.: Analytical construction of homoclinic orbits of two- and three-dimensional dynamical systems. J. Sound Vib. 230(5), 971–983 (2000)

    Article  MathSciNet  Google Scholar 

  6. Andrianov, I.V.: Asymptotics of nonlinear dynamical systems with a high degree of nonlinearity. Dokl. Math. 66(2), 270–273 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    Google Scholar 

  8. Ozorio de Almeida, A.M.: Hamiltonian Systems: Chaos and Quantization. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  9. Liao, S.-J., Chwang, A.: Application of homotopy analysis method in nonlinear oscillations. ASME J. Appl. Mech. 65, 914–922 (1998)

    Article  MathSciNet  Google Scholar 

  10. Nesterov, S.V.: Examples of nonlinear Kline–Gordon equations, solvable in terms of elementary functions. In: Proceedings of the Moscow Institute of Power Engineering, vol. 357, pp. 68–70 (1978)

  11. Dimentberg, M.F., Bratus, A.S.: Bounded parametric control of random vibrations. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(2002), 2351–2363 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dimentberg, M.F., Iourtchenko, D., Bratus, A.S.: Transition from planar to whirling oscillations in a certain nonlinear system. Nonlinear Dyn. 23, 165–174 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pilipchuk, V.N.: Transient mode localization in coupled strongly nonlinear exactly solvable oscillators. Nonlinear Dyn. (2007).  http://dx.doi.org/10.1007/s11071-007-9207-4

  14. Timoshenko, S.P., Yang, D., Wiver, U.: Vibration Problems in Engineering. Wiley, New York (1974)

    Google Scholar 

  15. Kosevich, A., Kovalev, A.: Introduction to Nonlinear Physical Mechanics. Naukova Dumka, Kiev (1989) (in Russian)

    MATH  Google Scholar 

  16. Holm, D.D., Lynch, P.: Stepwise precession of the resonant swinging spring. SIAM J. Appl. Dyn. Syst. 1(1), 44–64 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Manevich, A., Manevitch, L.: The Mechanics of Nonlinear Systems with Internal Resonances. Imperial College Press, London (2005)

    MATH  Google Scholar 

  18. Uzunov, I., Muschall, R., Golles, M., Kivshar, Y., Malomed, B., Lederer, F.: Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E 51, 2527–2537 (1995)

    Article  Google Scholar 

  19. Kutz, N.: Mode-locked soliton lasers. SIAM Rev. 48(4), 629–678 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Manevitch, L.I., Musienko, A.: Limiting phase trajectory and beating phenomena in systems of coupled nonlinear oscillators. In: 2nd International Conference on Nonlinear Normal Modes and Localization in Vibrating Systems, Samos, Greece, 19–23 June 2006, pp. 25–26 (2006)

  21. Pierce, J.: Coupling of modes of propagation. J. Appl. Phys. 25(2), 179–183 (1954)

    Article  MATH  Google Scholar 

  22. Scott, L.P.S., Eilbeck, J.C.: Between the local-mode and normal-mode limits. Chem. Phys. Lett. 113(1), 29–36 (1985)

    Article  Google Scholar 

  23. Manevitch, L.: New approach to beating phenomenon in coupled nonlinear oscillatory chains. Arch. Appl. Mech. 77(5), 301–312 (2007)

    Article  MATH  Google Scholar 

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Pilipchuk, V.N. Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena. Nonlinear Dyn 52, 263–276 (2008). https://doi.org/10.1007/s11071-007-9276-4

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  • DOI: https://doi.org/10.1007/s11071-007-9276-4

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