Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-16T20:19:09.956Z Has data issue: false hasContentIssue false
  • English
  • Français

Homoclinic orbits in the dynamic phase-space analogy of an elastic strut

Published online by Cambridge University Press:  16 July 2009

C. J. Amick  and
J. F. Toland
C. J. Amick
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
J. F. Toland
Affiliation:
School of Mathematical Sciences, University of Bath Claverton Down, Bath BA2 7AY, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The equation

is a possible dimensionless version of a model for the configuration of a very long strut resting on a nonlinear elastic foundation with axial loading P. By seeking to establish the existence of homoclinic orbits connecting the zero equilibrium of (*), now regarded as defining a four dimensional dynamical system, to itself one is pursuing the so-called ‘dynamical phase-space analogy’ for the spatial configuration suggested by the form of the equation. The existence of homoclinic solutions is then interpreted as indicating the presence of spatially localized buckling of the deformed strut.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 2 , June 1992 , pp. 97 - 114
Copyright
Copyright © Cambridge University Press 1992

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. & Toland, J. F. 1992 Global uniqueness of homoclinic orbits for a class of fourth order equations. (To appear in ZAMP.).CrossRefGoogle Scholar
Hille, E. 1972 Methods in Classical and Functional Analysis. Barton, MA: Addison-Wesley.Google Scholar
Hofer, H. & Toland, J. F. 1984 Homoclinic heteroclinic and periodic orbits for a class of indefinite Hamiltonian systems. Math. Annalen 268, 387–403.CrossRefGoogle Scholar
Hofer, H. & Wysocki, K. 1990 First order elliptic systems and the existence of homoclinic orbits in Hamilionian systems. Math. Annalen 288, 483–503.CrossRefGoogle Scholar
Hunt, G. W., Bolt, H. M. & Thompson, J. M. T. 1989 Localisation and the dynamic phase-space analogy. Proc. Roy. Soc. Lond. A 425, 245–267.Google Scholar
Hunt, G. W. & Wadee, M. K. 1991 Comparative Lagrangian formulations for localized buckling. Proc. Roy. Soc. Lond. A 434, 485–502.Google Scholar
Hunt, G. W. & Lucena, Neto E. 1991 Localized buckling in long axially-loaded cylindrical shells. J. Math. Phys. Solids 39, 881–894.CrossRefGoogle Scholar
Iooss, G. & Kirchgässner, K. 1990 Bifurcation d'ondes solitaires en presence d'une faible superficielle. C.R. Acad. Sci. Paris, Ser. I 311, 265–268 (in French).Google Scholar
Mielke, A. 1991 Personal communication, University of Bath, UK.Google Scholar
Rabinowitz, P. H. 1971 Some global results for nonlinear eigenvalue problems. Journal Fnal. Anal. 7, 487–513.Google Scholar
Stuart, C. A. 1985 A global branch of solutions to a semilinear equation on an unbounded interval. Proc. Roy. Soc. Edin. 101A, 273–282.CrossRefGoogle Scholar
Toland, J. F. 1986 Existence of symmetric homoclinic orbits for systems of Euler–Lagrange equations. A.M.S. Symp. Pure Math. 45, 447–459.CrossRefGoogle Scholar