Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T19:45:47.812Z Has data issue: false hasContentIssue false
  • English
  • Français

On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter

Published online by Cambridge University Press:  16 July 2009

John G. Byatt-Smith
John G. Byatt-Smith
Affiliation:
Department of Mathematics, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 2 , June 1991 , pp. 133 - 158
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amick, C. J. & McLeod, B. 1990 Small internal waves in two fluid systems. Arch. Rat. Mech. 109, 139–171.Google Scholar
Amick, C. J. & Kirchgässner, K. 1989 A theory of solitary water waves in the presence of surface tension. Arch. Rat. Mech. 105, 1–49.CrossRefGoogle Scholar
Beale, J. T. 1991 Exact solitary water waves with capillary ripples at infinity. Comm. Pure and App. Math. (to appear).CrossRefGoogle Scholar
Dashen, R. F., Kessler, D. A., Levine, H. & Savit, R. 1986 The geometric model of dendritic growth. Physica 21D, 371–380.Google Scholar
Glicksman, M. E. 1985 In: Kaldis, F., ed., Crystal Growth in Electronic Materials. Elsevier.Google Scholar
Hammersley, J. M. & Mazzerino, G. 1989 A differential equation connected with the dendritic growth of crystals. IMA J. of Appl. Math. 42, 43–75.CrossRefGoogle Scholar
Hooper, A. P. & Grimshaw, R. 1985 Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28, 27.Google Scholar
Hooper, A. P. & Grimshaw, R. 1988 Travelling wave solution of the kuramoto–shivashinsky equation. Wave motion 10, 405–420.CrossRefGoogle Scholar
Hunter, J. K. & Vanden-Broeck, J. M. 1983 Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 136, 63.CrossRefGoogle Scholar
Kruskal, M. D. & Segur, H. 1991 Asymptotics beyond all orders in a model of crystal growth. (to appear).CrossRefGoogle Scholar
Kuramoto, Y. & Tsuzuki, T. 1976 Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progress in Theor. Phys. 55, 356–369.CrossRefGoogle Scholar
Sun, S. M. & Shen, M. C. 1991 Exact solitary waves with capillary ripples at infinity. J. Math. Anal. Appl. (to appear).Google Scholar
Toland, J. F. 1988 Proc. Roy. Soc. Edin. Existence and uniqueness of heterodinic orbits for the equation λu‴ + u′ = f(u). 109A, 23–36.Google Scholar
Troy, W. C. 1990 Non existence of monotonic solutions in a model of dendritic growth. Q. App. Math. 48, 209–216.CrossRefGoogle Scholar
Xu, J. J. 1990 Global wave mode theory for formation of dendritic structure of a growing needle crystal. Phys. Stat. Sol. (b) 157, 577.Google Scholar
Xu, J. J. 1989 Interfacial wave theory for dendritic structure of a growing needle crystsl I and II. Phys. Rev. A 40, 1599.CrossRefGoogle Scholar