Three-period quasi-periodic solutions in the self-excited quasi-periodic Mathieu oscillator. (English) Zbl 1098.70018
Summary: Using perturbation method, we investigate three-period quasi-periodic (QP) oscillations in a vicinity of 2:2:1 resonance for a self-excited QP Mathieu equation. Two successive averaging are performed to reduce the original QP equation to an autonomous amplitude and phase system describing the modulation of the slow flow dynamic. Approximation of three-period QP solution is obtained via the study of limit cycle of the reduced autonomous system. The efficiency of the method is illustrated by comparison between analytical approximations and numerical integration. The double reduction procedure, applied in previous works to construct two-period QP solution, can be implemented to approximate excplicit analytical three-period QP solutions.
MSC:
| 70K43 | Quasi-periodic motions and invariant tori for nonlinear problems in mechanics |
| 70K70 | Systems with slow and fast motions for nonlinear problems in mechanics |
| 70K65 | Averaging of perturbations for nonlinear problems in mechanics |
References:
| [1] | Schmidt, G., ?Application of the theory of nonlinear oscillations: Interaction of self-excited forced and parametrically excited vibrations?, in The 9th International Conference on Non-linear Oscillations, 3 Kiev, Naukowa Dumka, 1984. |
| [2] | Szabelski, K. and Warminski, J., ?Self-excited system vibrating with parametric and external excitations?, Journal of Sound and Vibration 187(4), 1995, 595-607. · doi:10.1006/jsvi.1995.0547 |
| [3] | Belhaq, M. and Houssni, M., ?Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations?, Nonlinear Dynamics 18, 1999, 1-24. · Zbl 0969.70017 · doi:10.1023/A:1008315706651 |
| [4] | Belhaq, M., Guennoun, K., and Houssni, M., ?Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation?, International Journal of Non-Linear Mechanics 37, 2002, 445-460. · Zbl 1346.34030 · doi:10.1016/S0020-7462(01)00020-8 |
| [5] | Rand, R., Guennoun, K., and Belhaq, M., ?2:2:1 Resonance in the quasi-periodic Mathieu equation?, Nonlinear Dynamics 31, 2003, 367-374. · Zbl 1062.70596 · doi:10.1023/A:1023216817293 |
| [6] | Weidenhammer, F., ?Nicht-linear Schwingungen mit fast-periodischer Parametererregten?, Zeitschrift f?r Angewandte Mathematik und Mechanik 61, 1961, 633-638. · Zbl 0496.70035 · doi:10.1002/zamm.19810611205 |
| [7] | Weidenhammer, F., ?Intabilit?ten eines ged?mpften Schwingers mit fast-periodischer Parametererregung?, Ingeniour-Archiv 49, 1980, 187-193. · Zbl 0437.70020 · doi:10.1007/BF01351332 |
| [8] | Zounes, R. and Rand, R., Transition curves for the quasi-periodic Mathieu equation?, SIAM Journal on Applied Mathematics 58, 1998, 1094-1115. · Zbl 0916.34031 · doi:10.1137/S0036139996303877 |
| [9] | Zounes, R. and Rand, R., ?Global behavior of a nonlinear quasi-periodic Mathieu equation?, Nonlinear Dynamics 27, 2002, 87-105. · Zbl 1003.34041 · doi:10.1023/A:1017931712099 |
| [10] | Lakrad, F. and Belhaq, M., ?Solutions of a shallow arch under fast and slow excitations?, in IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, 8-13 June 2003, Rome, Italy. |
| [11] | Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973. · Zbl 0265.35002 |
| [12] | Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979. · Zbl 0418.70001 |
| [13] | Nayfeh, A.H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley, New York, 1995. · Zbl 0848.34001 |
| [14] | Guennoun, K., Houssni, M., and Belhaq, M., ?Quasi-periodic solutions and stability for a weakly damped nonlinear quasi-periodic Mathieu equation?, Nonlinear Dynamics 27, 2002, 211-236. · Zbl 1013.70018 · doi:10.1023/A:1014496917703 |
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