Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T11:44:21.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Subharmonic orbits in an anharmonic oscillator

Published online by Cambridge University Press:  17 February 2009

J. R. Christie ,
K. Gopalsamy  and
M. P. Panizza
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a recent paper, Christie and Gopalsamy [2] used Melnikov's method to establish a sufficient condition for the existence of chaotic behaviour, in the sense of Smale, in a particular time-periodically perturbed planar autonomous system of ordinary differential equations. They then concluded with an application to the dynamics of a one-dimensional anharmonic oscillator. In this paper, the same system is considered and a condition for the existence of subharmonic orbits in the perturbed system is deduced, using the subharmonic Melnikov theory. Finally, an application is given to the dynamical behaviour of the one-dimensional anharmonic oscillator system.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 3 , January 1997 , pp. 307 - 315
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Byrd, P. F. and Friedman, M. D., Handbook of elliptic integrals for engineers and scientists (Springer, New York, 1971).CrossRefGoogle Scholar
[2]Christie, J. R. and Gopalsamy, K., “Chaos in an anharmonic oscillator”, J. Austral. Math. Soc. Ser. B 37 (1995) 186–207.CrossRefGoogle Scholar
[3]Christie, J. R., Gopalsamy, K. and Li, J., “Chaos in sociobiology”, Bull. Austral. Math. Soc. 51 (1995) 439–451.CrossRefGoogle Scholar
[4]DiFilippo, F., Natarajan, V., Boyce, K. R. and Pritchard, D. E., “Classical amplitude squeezing for precision measurements”, Phys. Rev. Lett. 68 (1992) 2859–2862.CrossRefGoogle ScholarPubMed
[5]Greenspan, B. D. and Holmes, P. J., “Homoclinic orbits, subharmonics and global bifurcations in forced oscillations”, in Nonlinear dynamics and turbulence (eds. Barenblatt, G., Iooss, G. and Joseph, D. D.), (Pitman, London, 1983) 172–214.Google Scholar
[6]Greenspan, B. D. and Holmes, P. J., “Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators”, SIAM J. Math. Anal. 15 (1984) 69–97.CrossRefGoogle Scholar
[7]Grimshaw, R. and Tian, X., “Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation”, Proc. R. Soc. Lond. A 445 (1994) 1–21.Google Scholar
[8]Guckenheimer, J. and Holmes, P. J., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer, New York, 1983).CrossRefGoogle Scholar
[9]Huilgol, R. R., Christie, J. R. and Panizza, M. P., “The motion of a mass hanging from an overhead crane”, Chaos Solitons & Fractals 5 (1995) 1619–1631.CrossRefGoogle Scholar
[10]Langebartel, R. G., “Fourier expansions of rational fractions of elliptic integrals and Jacobian elliptic functions”, SIAM J. Math. Anal. 11 (1980) 506–513.CrossRefGoogle Scholar
[11]Li, J., Chaos and Melnikov's method (Chongqing University, Chongqing, 1989).Google Scholar
[12]Melnikov, V. K., “On the stability of the center for time-periodic perturbations”, Trans. Moscow Math. Soc. 12 (1963) 1–57.Google Scholar
[13]Wielinga, B. and Milburn, G. J., “Quantum tunneling in a Kerr medium with parametric pumping”, Phys. Rev. A 48 (1993) 2494–2496.CrossRefGoogle Scholar
[14]Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (Springer, New York, 1990).CrossRefGoogle Scholar
[15]Zhao, X., Kwek, K., Li, J. and Huang, K., “Chaotic and resonant streamlines in the ABC flow”, SIAM J. Appl. Math. 53 (1993) 71–77.CrossRefGoogle Scholar