Archive for Rational Mechanics and Analysis

, Volume 76, Issue 2, pp 135–165 | Cite as

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

  • Philip Holmes
  • Jerrold Marsden


This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=fo(X)+εf1(X,t), where fo(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. R. Abraham & J. Marsden [1978] “Foundations of Mechanics,” 2nd Edition, Addison-Wesley.Google Scholar
  2. V. Arnold [1964] Instability of dynamical systems with several degrees of freedom. Dokl. Akad. Nauk. SSSR 156, 9–12.Google Scholar
  3. V. Arnold [1966] Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynomique des fluides parfaits. Ann. Inst. Fourier, Grenoble 16, 319–361.Google Scholar
  4. J. Carr [1980] Application of Center Manifolds, Springer Applied Math. Sciences (to appear).Google Scholar
  5. J. Carr & M. Z. H. Malhardeen [1979] Beck's problem, SIAM J. Appl. Math. 37, 261–262.Google Scholar
  6. P. Chernoff & J. Marsden [1974] “Properties of Infinite Dimensional Hamiltonian Systems,” Springer Lecture Notes in Math., no. 425.Google Scholar
  7. S. N. Chow, J. Hale & J. Mallet-Paret [1980] An example of bifurcation to homoclinic orbits, J. Diff. Eqn's. 37, 351–373.Google Scholar
  8. C. Conley & J. Smoller [1974] On the structure of magneto hydrodynamic shock waves, Comm. Pure Appl. Math. 27, 367–375, J. Math. of Pures et. Appl. 54 (1975) 429–444.Google Scholar
  9. D. Ebin & J. Marsden [1970] Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92, 102–163.Google Scholar
  10. J. Gollub [1980] The onset of turbulence: convection, surface waves, and oscillators, Springer Lecture Notes in Physics.Google Scholar
  11. J. Guckenheimer [1979] On a codimension two bifurcation (preprint).Google Scholar
  12. O. Gurel & O. Rössler (eds) [1979] “Bifurcation theory and applications in scientific disciplines,” Ann. of N.Y. Acad. Sciences, vol. 316. Google Scholar
  13. B. Hassard [1980] Computation of invariant manifolds, in Proc. “New Approaches to Nonlinear Problems in Dynamics”, ed. P. Holmes, SIAM publications.Google Scholar
  14. E. Hille & R. Phillips [1957] “Functional Analysis and Semigroups”, A.M.S. Colloq. Publ.Google Scholar
  15. M. Hirsch, C. Pugh & M. Shub [1977] “Invariant Manifolds”, Springer Lecture Notes in Math. no. 583.Google Scholar
  16. P. Holmes [1979a] Global bifurcations and chaos in the forced oscillations of buckled structures, Proc. 1978 IEEE Conf. on Decision and Control, San Diego, CA, 181–185.Google Scholar
  17. P. Holmes [1979b] A nonlinear oscillator with a strange attractor, Phil. Trans. Roy. Soc. A 292, 419–448.Google Scholar
  18. P. Holmes [1980a] Averaging and chaotic motions in forced oscillations, SIAM J. on Appl. Math. 38, 65–80.Google Scholar
  19. P. Holmes [1980b] Space and time-periodic perturbations of the sine-Gordon equation (preprint).Google Scholar
  20. P. Holmes & J. Marsden [1978] Bifurcation to divergence and flutter in flow induced oscillations; an infinite dimensional analysis, Automatica 14, 367–384.Google Scholar
  21. P. Holmes & J. Marsden [1979] Qualitative techniques for bifurcation analysis of complex systems, in Gurel & Rössler [1979], 608–622.Google Scholar
  22. T. Kato [1977] “Perturbation Theory for Linear Operators” 2nd Ed., Springer.Google Scholar
  23. N. Kopell & L. N. Howard [1976] Bifurcations and trajectories joining critical points, Adv. Math. 18, 306–358.Google Scholar
  24. M. Levi, F. C. Hoppenstadt & W. L. Miranker [1978] Dynamics of the Josephson junction, Quart. of Appl. Math. 36, 167–198.Google Scholar
  25. V. K. Melnikov [1963] On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12, 1–57.Google Scholar
  26. F. Moon & P. Holmes [1979] A magneto-elastic strange attractor, J. Sound and Vibrations 65, 275–296.Google Scholar
  27. J. Moser [1973] “Stable and Random Motions in Dynamical Systems,” Ann. of Math. Studies no. 77, Princeton Univ. Press.Google Scholar
  28. S. Newhouse [1974] Diffeomorphisms with infinitely many sinks, Topology 12, 9–18.Google Scholar
  29. S. Newhouse [1979] The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. I.H.E.S. 50, 100–151.Google Scholar
  30. J. Rauch [1979] Qualitative behavior of dissipative wave equations. Arch. Rational Mech. An. 62, 77–91.Google Scholar
  31. J. Robbin [1971] A structural stability theorem, Ann. of Math. 94, 447–493.Google Scholar
  32. I. Segal [1962] Nonlinear Semigroups, Ann. of Math. 78, 334–362.Google Scholar
  33. Y. Shizuta [1980] On the classical solutions of the Boltzmann equations, Comm. Pure. Appl. Math. (to appear).Google Scholar
  34. S. Smale [1963] Diffeomorphisms with many periodic points, in “Differential and Combinatorial Topology” (ed. S. S. Cairns), Princeton Univ. Press, 63–80.Google Scholar
  35. S. Smale [1967] Differentiable dynamical systems, Bull. Am. Math. Soc. 73, 747–817.Google Scholar
  36. W. Y. Tseng & J. Dugundji [1971] Nonlinear vibrations of a buckled beam under harmonic excitation. J. Appl. Mech. 38, 467–476.Google Scholar
  37. I. Vidav [1970] Spectra of perturbed semigroups, J. Math. An. Appl. 30, 264–279.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1981

Authors and Affiliations

  • Philip Holmes
    • 1
  • Jerrold Marsden
    • 2
  1. 1.Theoretical and Applied MechanicsCornell UniversityIthaca
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeley

Personalised recommendations