Advertisement

Journal of Dynamics and Differential Equations

, Volume 17, Issue 1, pp 175–215 | Cite as

Homoclinic Shadowing

  • Brian A. Coomes
  • Hüseyin KoçakEmail author
  • Kenneth J. Palmer
Article

Abstract

A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in R n is presented. It is a computer-assisted technique with two main components. First, a global Newton’s method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.

Keywords

Transversal homoclinic orbits pseudo orbits chaos shadowing Newton’s method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beyn, W.-J., Kleinkauf, J.-M. 1997Numerical approximation of homoclinic chaosNumer Algorithms142554CrossRefGoogle Scholar
  2. 2.
    Beyn, W.-J., Kleinkauf, J.-M. 1997The numerical computation of homoclinic orbits for mapsSIAM J. Numer Anal.20106120Google Scholar
  3. 3.
    Birkhoff, G. 1935Nouvelles recherches sur les systémes dynamiquesMem. Pon Acad. Sci. Novi. Lyncaei185216Google Scholar
  4. 4.
    Brown, R. 1995Horseshoes in the measure-preserving Hénon mapErgod. Th. Dyn. Sys.1510451059Google Scholar
  5. 5.
    Coomes, B.A., Koçak, H., Palmer, K.J. 1996Shadowing in discrete dynamical systems In Six Lectures on Dynamical SystemsWorld ScientificSingapore163217Google Scholar
  6. 6.
    Coomes, B.A., Koçak, H., Palmer, K.J. 1997Computation of long periodic orbits in chaotic dynamical systems.The Australian Math Soc Gazette24183190Google Scholar
  7. 7.
    Coppel, W.A. 1978Dichotomies in Stability Theory. Lecture Notes in MathematicsSpringer-VerlagBerlin629Google Scholar
  8. 8.
    Devaney, R.L. 1984Homoclinic bifurcations and the area-conserving Hénon mappingJ. Diff Eqns.51254266CrossRefGoogle Scholar
  9. 9.
    Devaney, R.L, Nitecki, Z. 1979Shift automorphisms in the Hénon mappingComm. Math. Phys.67137146CrossRefGoogle Scholar
  10. 10.
    Fontich, E. 1990Transversal homoclinic points of a class of conservative diffeomorphismsJ. Diff. Eqns.87127CrossRefGoogle Scholar
  11. 11.
    Glasser, M.L., Papageorgiou, V.G., Bountis, T.C. 1989Melnikov’s function for two-dimensional mappingsSIAM J. Appl. Math.49692703CrossRefGoogle Scholar
  12. 12.
    Hénon, M. 1969Numerical study of quadratic area-preserving mappingsQuart. Appl. Math.27291312Google Scholar
  13. 13.
    Hénon, M. 1976A two-dimensional mapping with a strange attractorCommun. Math. Phys.506977CrossRefGoogle Scholar
  14. 14.
    Henry, D. 1981Geometric Theory of Semilinear Parabolic Equations Lecture. Notes in Mathematics. 840Springer-VerlagNew YorkGoogle Scholar
  15. 15.
    Kantorovich, L.V., Akilov, G.P. 1959Functional Analysis in Normed Spaces. FIZMATGIZ. Functional Analysis2Pergamon PressMoscow1982Google Scholar
  16. 16.
    Marotto, F.R. 1979Perturbation of stable and chaotic difference equationsJ. Math. Anal. Appl.72715729CrossRefGoogle Scholar
  17. 17.
    Marotto, F.R. 1979Chaotic behaviour in the Hénon attractorComm. Math. Phys.68187194CrossRefGoogle Scholar
  18. 18.
    McGehee, R., Meyer, K.R. 1974Homoclinic points of area preserving diffeo- morphismsAmer. J. Math.96409421Google Scholar
  19. 19.
    Melnikov, V.K. 1963On the stability of the center for time periodic perturbationsTrans. Moscow Math. Soc.12157Google Scholar
  20. 20.
    Mischaikow, K., Mrozek, M. 1995Isolating neighborhoods and chaosJapan J. Indust. Appl. Math.12205236Google Scholar
  21. 21.
    Misiurewicz, M., Szewc, B. 1980Existence of a homoclinic orbit for the Hénon mapComm. Math. Phys.75285291CrossRefGoogle Scholar
  22. 22.
    Morinaka, M. 1983On the existence of transversal homoclinic points of some real analytic plane transformationsJ. Math. Kyoto Univ.23707714Google Scholar
  23. 23.
    Neumaier, A., Rage, T. 1993Rigorous chaos verification in discrete dynamical systemsPhysica D.67327346Google Scholar
  24. 24.
    Neumaier, A., Schlier, C. 1994Rigorous verification of chaos in a molecular modelPhys. Rev. E.5026822688CrossRefGoogle Scholar
  25. 25.
    Newhouse, S. 1980Lectures on dynamical systems, Dynamical Systems. Dynamical Systems, C.I.M.E. LecturesBirkhauserBostonGoogle Scholar
  26. 26.
    Palis J., Takens F. (1993). Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press.Google Scholar
  27. 27.
    Palmer, K.J. 1988Exponential dichotomies, the shadowing lemma and transversal homoclinic pointsDynamics Reported1265306Google Scholar
  28. 28.
    Palmer, K.J. 2000Shadowing in Dynamical SystemsKluwer Academic PublishersDordrechtGoogle Scholar
  29. 29.
    Poincaré, H. 1890Surle probléme des trois corps et les équations de la dynamiqueActa Math.131270Google Scholar
  30. 30.
    Robinson, C. 1995Dynamical Systems: Stability, Symbolic Dynamics and ChaosCRC PressBoca Raton, FloridaGoogle Scholar
  31. 31.
    Rogers, T.D., Marotto, F.R. 1983Perturbations of mappings with periodic repellersNonlin. Anal. TMA.797100CrossRefGoogle Scholar
  32. 32.
    Slyusarchuk, V.E. 1983Exponential dichotomy for solutions of discrete systemsUkrain. Math. J.3598103CrossRefGoogle Scholar
  33. 33.
    Smale S. (1965). Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology, Princeton University Press.Google Scholar
  34. 34.
    Stoffer, D., Palmer, K.J. 1999Rigorous verification of chaotic behaviour of maps using validated shadowingNonlinearity.1216831698CrossRefGoogle Scholar
  35. 35.
    Wilkinson, J.H. 1963Rounding Errors in Algebraic ProcessesPrentice-HallEnglewood Cliffs, New JerseyGoogle Scholar
  36. 36.
    Wilkinson, J.H. 1965The Algebraic Eigenvalue ProblemClarendon PressOxfordGoogle Scholar
  37. 37.
    Zgliczyński, P. 1997Computer assisted proof of the horseshoe dynamics in the Hénon mapRand. Comp. Dyn.5117Google Scholar
  38. 38.
    Zgliczyński, P. 1997Computer assisted proof of chaos in the Roessler equations and the Hénon mapNonlinearity.10243252CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Brian A. Coomes
    • 1
  • Hüseyin Koçak
    • 1
    Email author
  • Kenneth J. Palmer
    • 2
  1. 1.Departments of Mathematics and Computer ScienceUniversity of MiamiCoral GablesUSA
  2. 2.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan

Personalised recommendations