Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-26T20:33:20.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbed generalised Hamiltonian systems and some advection models

Published online by Cambridge University Press:  17 April 2009

Jibin Li ,
J.R. Christie  and
K. Gopalsamy
Jibin Li
Affiliation:
Institute of Applied Maths of Yunnan ProvinceDepartment of MathematicsKunming University of Science & TechnologyYunnan 650093People's Republic of China e-mail: jibinli@ynu.edu.cn
J.R. Christie
Affiliation:
Department of Mathematics and StatisticsThe Flinders University of South AustraliaGPO Box 2100Adelaide SA 5001Australia e-mail: johnc@ist.flinders.edu.au
K. Gopalsamy
Affiliation:
Department of Mathematics and StatisticsThe Flinders University of South AustraliaGPO Box 2100Adelaide SA 5001Australia e-mail: gopal@ist.flinders.edu.au
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 this paper, it is shown that the theory of perturbed generalised Hamiltonian systems provides an effective method for understanding the description of flow patterns of some three-dimensional flows. Firstly, theorems for the persistence of periodic solutions of three-dimensional generalised Hamiltonian systems under perturbation are given by developing Melnikov's method. Then, three different systems of three-dimensional steady fluid flows are discussed and the existence or non-existence of periodic solutions of these systems is proved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 1 - 24
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Armbruster, D., Guckenheimer, J. and Holmes, P., ‘Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry’, Phys. D 29 (1988), 257–282.CrossRefGoogle Scholar
[2]Arter, W., ‘Ergodic stream-lines in steady convection’, Phys. Lett. A 97 (1983), 171–174.CrossRefGoogle Scholar
[3]Bajer, K. and Moffatt, H.K., ‘On a class of steady confined Stokes flows with chaotic streamlines’, J. Fluid Mech. 212 (1990), 337–363.CrossRefGoogle Scholar
[4]Byrd, P.F. and Friedman, M.D., Handbook of elliptic integrals for engineers and scientists (Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
[5]Chandrasekhar, S., Hydrodynamic and hydromagnetic stability (Clarendon, Oxford, 1961).Google Scholar
[6]Holm, D.D. and Kimura, Y., ‘Zero-helicity Lagrangian kinematics of three-dimensional advection’, Phys. Fluids A 3 (1991), 1033–1038.CrossRefGoogle Scholar
[7]Jones, C.A. and Proctor, M.R.E., ‘Strong spatial resonance and travelling waves in Bénard convection’, Phys. Lett. A 121 (1987), 224–228.CrossRefGoogle Scholar
[8]Li, J., Zhao, X. and Liu, Z., Theory and application of generalized Hamiltonian systems, (in Chinese) (Science Press, Beijing, 1994).Google Scholar
[9]Melnikov, V.K., ‘On the stability of the center for time periodic perturbations’, Trans. Moscow Math. Soc. 12 (1963), 1–57.Google Scholar
[10]Moffatt, H.K. and Proctor, M.R.E., ‘Topological constraints associated with fast dynamo action’, J. Fluid Mech. 154 (1985), 493–507.CrossRefGoogle Scholar
[11]Olver, P.J., Applications of Lie groups to differential equations, (Second edition) (Springer-Verlag, New York, 1993).CrossRefGoogle Scholar
[12]Proctor, M.R.E. and Jones, C.A., ‘The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance’, J. Fluid Mech. 188 (1988), 301–335.CrossRefGoogle Scholar
[13]Smale, S., ‘Dynamics retrospective: Great problems, attempts that failed’, Phys. D 51 (1991), 267–273.CrossRefGoogle Scholar
[14]Weinstein, A., ‘The local structure of Poisson manifolds’, J. Differential Geom. 18 (1983), 523–557.CrossRefGoogle Scholar
[15]Wiggins, S. and Holmes, P., ‘Periodic orbits in slowly varying oscillators’, SIAM J. Math. Anal. 18 (1987), 592–611.CrossRefGoogle Scholar