Abstract
We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.
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Svoboda, R., Tondl, A., Verhulst, F.: Autoparametric resonance by coupling of linear and nonlinear system. Int. J. Non-Linear Mech. 29, 225–232 (1994)
Tondl, A., Ruijgrok, M., Verhulst, F., Nabergoj, R.: Autoparametric Resonance in Mechanical Systems. Cambridge University Press, Cambridge (2000)
Feng, Z.C., Sethna, P.R.: Symmetry breaking bifurcations in resonant surface waves. J. Fluid Mech. 199, 495–518 (1989)
Yang, X.L., Sethna, P.R.: Non-linear phenomena in forced vibrations of a nearly square plate: antisymmetric case. J. Sound Vib. 155, 413–441 (1992)
Nayfeh, A.H., Pai, P.F.: Non-linear non-planar parametric responses of an inextensional beam. Int. J. Non-Linear Mech. 24, 139–158 (1989)
Oueini, S.S., Chin, C., Nayfeh, A.H.: Response of two quadratic coupled oscillators to a principal parametric excitation. J. Vib. Control 6, 1115–1133 (2000)
Ruijgrok, M.: Studies in parametric and autoparametric resonance. Ph.D. thesis, Universiteit Utrecht (1995)
Tien, W., Namachchivaya, N.S., Bajaj, A.K.: Non-linear dynamics of a shallow arch under periodic excitation-I. 1:2 internal resonance. Int. J. Non-Linear Mech. 29, 349–366 (1994)
Bajaj, A.K., Chang, S.I., Johnson, J.M.: Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear Dyn. 5, 433–457 (1994)
Banerjee, B., Bajaj, A.K.: Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities. Acta Mech. 124, 131–154 (1997)
Tien, W., Namachchivaya, N.S., Malhotra, N.: Non-linear dynamics of a shallow arch under periodic excitation-II.1:1 internal resonance. Int. J. Non-Linear Mech. 29, 367–386 (1994)
Feng, Z.C., Wiggins, S.: On the existence of chaos in a class of two-degree-of-freedom, damped, parametrically forced mechanical systems with broken O(2) symmetry. Z. Angew. Math. Phys. 44, 201–248 (1993)
Fatimah, S., Ruijgrok, M.: Bifurcations in an autoparametric system in 1:1 internal resonance with parametric excitation. Int. J. Non-Linear Mech. 37, 297–308 (2002)
Feng, Z.C., Sethna, P.S.: Global bifurcation and chaos in parametrically forced systems with one-to-one resonance. Dyn. Stab. Syst. 5, 201–225 (1990)
Kovacic, G., Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in and forced sine-Gordon equation. Physica D 57, 185–225 (1992)
Kovacic, G., Wettergren, T.A.: Homoclinic orbits in the dynamics of resonantly driven coupled pendula. Z. Angew. Math. Phys. 47, 221–264 (1996)
Haller, G., Wiggins, S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrödinger equation. Physica D 85, 311–347 (1995)
Haller, G., Wiggins, S.: N-pulse homoclinic orbits in perturbations of resonant Hamiltonian system. Arch. Ration. Mech. Anal. 130, 25–101 (1995)
Haller, G.: Chaos Near Resonance. Springer, New York (1999)
Malhotra, N., Sri Namachchivaya, N., McDonald, R.J.: Multipulse orbits in the motion of flexible spinning discs. J. Nonlinear Sci. 12, 1–26 (2002)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)
Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–225 (1971)
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Chen, H., Xu, Q. Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance. Nonlinear Dyn 65, 187–216 (2011). https://doi.org/10.1007/s11071-010-9883-3
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DOI: https://doi.org/10.1007/s11071-010-9883-3