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Complex dynamics in Duffing-Van der Pol equation. (English) Zbl 1083.37061

Summary: The Duffing-Van der Pol equation with fifth-degree nonlinear-restoring force and two external forcing terms is investigated. The threshold values of the existence of chaotic motions are obtained under periodic perturbation. By the second-order averaging method and the Melnikov method, we prove a criterion for the existence of chaos in averaged systems under quasiperiodic perturbation for \(\omega_{2} = n\omega_{1} + \varepsilon \sigma\), \(n = 1, 3, 5,\) and cannot prove a criterion for the existence of chaos in second-order averaged systems under quasiperiodic perturbation for \(\omega_{2} = n\omega_{1} + \varepsilon \sigma\), \(n = 2, 4, 6, 7, 8, 9, 10,\) where \(\sigma\) is not rational to \(\omega_{1}\), but we can show the occurrence of chaos in the original system by numerical simulation.
Numerical simulations, including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponents, phase portraits and the Poincaré map, not only show the consistence with theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleave occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange nonchaotic attractors, nonattracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly different from the behavior of the Duffing-Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.

MSC:

37N05 Dynamical systems in classical and celestial mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C29 Averaging method for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
70L05 Random vibrations in mechanics of particles and systems
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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[1] Guckenheimer, J.; Hoffman, K.; Weckesser, W., The forced Van der Pol equation. I. The slow flow and its bifurcations, SIAM J Appl Dyn Syst, 2, 1, 1-35 (2003) · Zbl 1088.37504
[2] Guckenheimer, J.; Holmes, P., Dynamical systems and bifurcations of vector fields (1997), Springer-Verlag: Springer-Verlag New York
[3] Jing, Z. J.; Wang, R. Q., Chaos in Duffing system with two external forcings, Chaos, Solitons & Fractals, 23, 399-411 (2005) · Zbl 1077.34048
[4] Kao, Y. H.; Wang, C. S., Analog study of bifurcation structures in a Van der Pol oscillator with a nonlinear restoring force, Phys Rev E, 48, 2514-2520 (1993)
[5] Kakmeni, F. M.M.; Bowong, S.; Tchawoua, C.; Kaptouom, E., Strange attractors and chaos control in a Duffing-Van der Pol oscillator with two external periodic forces, J Sound Vibr, 277, 783-799 (2004) · Zbl 1236.93085
[6] Kapitaniak, T., Analytical condition for chaotic behavior of the Duffing oscillation, Phys Lett A, 144, 6,7, 322-324 (1990)
[7] Kapitaniak, T., Chaotic oscillations in mechanical systems (1991), Manchester University Press: Manchester University Press UK · Zbl 0786.58027
[8] Kapitaniak, T.; Steeb, W. H., Transition to chaos in a generalized Van der Pol’s equation, J Sound Vibr, 143, 1, 167-170 (1990) · Zbl 0735.34034
[9] Kenfark, A., Bifurcation structure of two coupled periodically driven double-well Duffing oscillators, Chaos, Solitons & Fractals, 15, 205-218 (2003)
[10] Lakshianan, M.; Murall, K., Chaos in nonlinear oscillations (1996), World Scientific
[11] Leung, A. Y.T.; Zhang, Q. C., Complex normal form for strongly non-linear vibration systems exemplified by Duffing-Van der Pol equation, J Sound Vibr, 213, 5, 907-914 (1998) · Zbl 1235.34116
[12] Li, G. X.; Moon, F. C., Criteria for chaos of a three-well potential oscillator with homoclinic and heteroclinic orbits, J Sound Vibr, 136, 1, 17-34 (1996) · Zbl 1235.74096
[13] Liang, Y.; Namachchuvaya, N. S., P-bifurcations in the noise Duffing-Van der Pol equation (1999), Stochastic Dynamics: Stochastic Dynamics Springer, New York, pp. 49-70 · Zbl 0940.60085
[14] NamachchiVaga, N. S.; Sowers, R. B.; Vedula, L., Non-standard reduction of noisy Duffing-Van der Pol equation, Dyn Syst, 16, 3, 223-245 (2001) · Zbl 1077.34065
[15] Parlitz, U.; Lauterborn, W., Period-doubling cascades and Peril’s staircases of the Driven Van der Pol oscillator, Phys Rev A, 36, 1428-1434 (1987)
[16] Rajasikar, S.; Parthasarathy, S.; Lakshmanan, M., Prediction of horseshoe chaos in BVP and DVP oscillators, Chaos, Solitons & Fractals, 2, 271-280 (1992) · Zbl 0768.58032
[17] Wakako, M.; Chieko, M.; Koi-ichi, H.; Yoshi H, I., Integrable Duffing’s maps and solutions of the Duffing equation, Chaos, solitons & Fractals, 15, 3, 425-443 (2003) · Zbl 1031.37047
[18] Wiggins, S., Introduction to applied nonlinear dynamical system and chaos (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0701.58001
[19] Yagasaki, K., Second-order averaging and chaos in quasi-periodically forced weakly nonlinear oscillators, Physica D, 44, 445-458 (1990) · Zbl 0706.58037
[20] Yagasaki, K., Homoclinic tangles, phase locking, chaos in a two frequency perturbation of Duffing’s equation, J Nonlinear Sci, 9, 131-148 (1999) · Zbl 0939.34039
[21] Yagasaki, K., Detection of bifurcation structures by higher-order averaging for Duffing’s equation, Nonlinear Dyn, 18, 121-158 (1999) · Zbl 0944.70017
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