Behavior of harmonics generated by a duffing type equation with a nonlinear damping. Part I

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 15, Issue 10, 2005, Pages 3181-3221

  Khammari, Hédi ^a   Mira, Christian ^b,c   Carcassés, Jean Pierre ^d  

^a UNIVERSITY OF TUNIS EL MANAR   (Tunisia)

^b CESNLA   (France)

^c UNIVERSITY OF URBINO   (Italy)

^d NONE   (France)

Author keywords

Bifurcation; Harmonics; Nonlinear dynamics; Nonlinear oscillations; Qualitative methods; Resonance

Indexed keywords

EID: 29244432518     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127405014076     Document Type: Article

References (39)

1. Carcassès, J. P., Mira, C., Simo, C. & Tatjer, J. C. [1991] "Crossroad area-spring area transition. (I) parameter plane representation," Int. J. Bifurcation and Chaos 1, 183-196.
2. Carcassès, J. P. [1992] "An algorithm to determine the nature and the transitions of communication areas generated by one dimensional map," Proc. ECIT 91, Lisbon, Portugal, 15-21 September 1991, (World Scientific, Singapore), pp. 27-38.
3. Carcassès, J. P. [1993] "Determination of different configurations of fold and flip bifurcation curves of one, or two-dimensional map," Int. J. Bifurcation and Chaos 1, 869-902.
4. Carcassès, J. P. [1997] "A new kind of parametric singularities and their use for the study of the bifurcation structure of a n-dimensional map," Nonlin. Anal. Th. Meth. Appl. 28, 917-946.
5. Carcassès, J. P. & Kawakami, H. [1999a] "Appearance and disappearance of a dovetail structure in the parameter plane of a n-dimensional map," Int. J. Bifurcation and Chaos 9, 769-783.
6. Carcassès, J. P. & Kawakami, H. [1999b] "Existence of a cusp point on a fold bifurcation curve and stability of the associated fixed point. Case of a n-dimensional map," Int. J. Bifurcation and Chaos 9, 875-894.
7. Guckenheimer, J. [1980] "The bifurcations of quadratic functions," N.Y. Acad. Sci. 75, 343-347.
8. Gumowski, I. & Mira, C. [1980] Dynamique Chaotique. Transformations Ponctuelles. Transition Ordre-désordre (Cépadues Editions Toulouse), 480 pp.
9. Hayashi, C. [1964] Nonlinear Oscillations in Physical Systems (McGraw Hill Book Co, NY); [1985] New edition (Princeton University Press).
10. Hayashi, C., Ueda, Y. & Kawakami, H. [1969] "Periodic solutions of the Duffing equation with reference to doubly asymptotic solutions," Proc. 5th Int. Conf. Nonlinear Oscillations, Kiev, Vol. 2, pp. 507-521.
11. Hayashi, C. & Kawakami, H. [1984] "Behavior of solutions for Duffing-Van der Pol equation," Proc. 10th Int. Conf. Nonlinear Oscillations, Varna, pp. 137-140.
12. Kawakami, H. [1973] "Qualitative study on the solution of Duffing's equation," Thesis of Kyoto University, December 1973.
13. Kawakami, H. & Matsuo, J. [1982] "Bifurcation of doubly asymptotic motions in nonlinear systems," IEICE Trans. J65-A, 647-654.
14. Kawakami, H. [1983] "Bifurcations and chaotic states of forced dynamic nonlinear circuits," Proc. Int. Symp. Circuits and Systems, Vol. 1, pp. 378-381.
15. Kawakami, H. & Sekaï, M. [1983] "Synchronization phenomena observed in a Duffing-Rayleigh equation," Publication NLP83-37, Tokushima University (in Japanese).
16. Kawakami, H. [1984] "Bifurcation of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters," IEEE Trans. Circuits Syst. CAS 31, 248-259.
17. Kawakami, H. & Sekaï, M. [1984] "Synchronization phenomena and Hopf bifurcation," Publication NLP84-7, Tokushima University (in Japanese), pp. 25-33.
18. Kawakami, H. [1986] "Strange attractors in Duffing's equation," Publication of the Faculty of Sciences of Tokushima University.
19. Kawakami, H. & Sekaï, M. [1988] "Synchronization and chaos in a Duffing-Rayleigh oscillator," in Asymptotic Methods of Mathematical Physics (Collection of Scientific. Papers), (Kiev, Naukova Dumka), pp. 83-92.
20. Khammari, H., Carcasses, J. P. & Benrejeb, M. [1998] "Bifurcations of periodic solutions and higher harmonic oscillations in the RLC-circuit," Int. Conf. CESA '98 Computational Engineering in System Applications, Nabeul-Hammamet (Tunisia), Vol. 2, 2nd IMACS, pp. 19-23.
21. Khammari, H. [1999] "Sur le comportement complexe des oscillations forcées de systèmes non linéaires soumis à une excitation sinusoïdale," Thèse de l'Ecole Nationale d'Ingénieurs de Tunis (12/02/99).
22. Mira, C. [1975] "Accumulations de bifurcations et structures boites-emboitées dans les récurrences et transformations ponctuelles," Proc. 7th Int. Conf. Nonlinear Oscillations, Berlin, Sep. 1975 (Akademic Verlag, Berlin) Band I 2, pp. 81-93.
23. Mira, C. [1978] "Sur la structure des bifurcations des difféomorphismes du cercle," Comptes Rendus Acad. Se. Paris, Série A, 287, 883-886.
24. Mira, C. [1987] Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism (World Scientific, Singapore).
25. Mira, C. & Carcassès, J. P. [1991] "On the crossroad area-saddle area and crossroad area-spring area transitions," Int. J. Bifurcation and Chaos 1, 641-655.
26. Mira, C., Carcassès, J. P., Simo, C. & Tatjer, J. C. [1991] "Crossroad area-spring area transition. (II) Foliated parametric representation," Int. J. Bifurcation and Chaos 1, 339-348.
27. Mira, C. & Taha, A. K. [1991] "Isoordinal cascade of cusps with monotonic convergence generated by two-dimensional diffeomorphisms," Proc. European Conf. Iteration Theory, ECIT 1989, Batschums, Austria (World Scientific, Singapore).
28. Mira, C. & Kawakami, H. [1992] "Qualitative modifications of the lip bifurcation structure," Proc. European Conf. Iteration Theory, ECIT'92, Batschuns, 13-19 Sept. 1992, Austria, eds. Förg-Rob et al. (World Scientific, Singapore), pp. 199-203.
29. Mira, C. & Djellit, I. [1993a] "Bifurcations structure in a model of frequency modulated CO2 laser," Int. J. Bifurcation and Chaos 3, 97-129.
30. Mira, C. & Qriouet, M. [1993b] "On a 'Crossroad area-spring area' transition occurring in a Duffing-Rayleigh equation with periodical excitation," Int. J. Bifurcation and Chaos 3, 1029-1037.
31. Mira, C., Gardini, L., Barugola, A. & Cathala, J. C. [1996] Chaotic Dynamics in Two-Dimensional Non-invertible Maps, World Scientific Series on Nonlinear Sciences, Series A, Vol. 20, 630 pp.
32. Mira, C., Kawakami, H. & Touzani-Qriouet, M. [1999] "Bifurcations structures generated by the non-autonomous Duffing equation," Int. J. Bifurcation and Chaos 9, 1363-1379.
33. Qriouet, M. & Mira, C. [1994] "Fractional harmonic synchronization in the Duffing-Rayleigh differential equation," Int. J. Bifurcation and Chaos 4, 411-426.
34. Shilnikov, L. P. [1997] "Mathematical problem of non-linear dynamics: A tutorial," Int. J. Bifurcation and Chaos 7, 1953-2001.
35. Touzani, M. & Mira, C. [2000] "Reducible fractional harmonics generated by the nonautonomous Duffing-Rayleigh equation. Pockets of reducible harmonics and Arnold's tongues," Int. J. Bifurcation and Chaos 10, 1345-1366.
36. Ueda, Y. [1980] "Explosion of strange attractors exhibited by the Duffing's equation," Ann. New York Acad. Sci. 357, 422-434.
37. Ueda, Y. [1985] "Random phenomena resulting from the nonlinearity in the system described by the Duffing's equation," Int. J. Nonlin. Mech. 20, 481-491.
38. Ueda, Y. [1991] "Survey of regular and chaotic phenomena in the forced Duffing oscillator," Chaos Solit. Fract. 1, 199-231.
39. Ueta, T. &: Kawakami, H. [1996] "Unstable saddle-node connecting orbits in the averaged Duffing-Rayleigh equation," Proc. ISCAS 96, Atlanta USA, Vol. III, pp. 288-291.