Advertisement

Constructing dynamical systems having homoclinic bifurcation points of codimension two

  • Björn Sandstede
Article

Abstract

A procedure is derived which allows for a systematic construction of three-dimensional ordinary differential equations having homoclinic solutions. The equations are proved to exhibit codimension-two homoclinic bifurcation points. Examples include the non-orientable resonant bifurcation, the inclination-flip, and the orbit-flip. In addition, an equation is constructed which has a homoclinic orbit converging to a saddle-focus satisfying Shilnikov's condition. The vector fields are polynomial and non-stiff in that the eigenvalues are of moderate size.

Key words

Homoclinic orbits bifurcations 

AMS subject classifications

34C37 58F14 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bel80]
    L. A. Belyakov. The bifurcation set in a system with a homoclinic saddle curve.Mat. Zam. 28 (1980), 910–916.Google Scholar
  2. [Bey90]
    W.-J. Beyn. The numerical computation of connecting orbits in dynamical systems.IMA J. Numer. Anal. 9 (1990), 379–405.Google Scholar
  3. [CDF90]
    S.-N. Chow, B. Deng, and B. Fiedler. Homoclinic bifurcation at resonant eigenvalues.J. Dyn. Diff. Eq. 2 (1990), 177–244.CrossRefGoogle Scholar
  4. [CHS96]
    A. R. Champneys, J. HÄrterich, and B. Sandstede. A non-transverse homoclinic orbit to a saddle-node equilibrium.Ergod. Theory Dyn. Syst. 16 (1996), 431–450.Google Scholar
  5. [CK94]
    A. R. Champneys and Yu. A. Kuznetsov. Numerical detection and continuation of codimension-two homoclinic bifurcations.Int. J. Bifurc. Chaos 4 (1994), 795–822.Google Scholar
  6. [CKS95]
    A. R. Champneys, Yu. A. Kuznetsov, and B. Sandstede.HomCont: An AUTO86 Driver for Homoclinic Bifurcation Analysis, Version 2.0, Technical report, CWI, Amsterdam, 1995.Google Scholar
  7. [CKS96]
    A. R. Champneys, Yu. A. Kuznetsov, and B. Sandstede. A numerical toolbox for homoclinic bifurcation analysis.Int. J. Bifurc. Chaos 6 (1996), 867–887.CrossRefGoogle Scholar
  8. [Den94]
    B. Deng. Constructing homoclinic orbits and chaotic attractors.Int. J. Bifurc. Chaos 4 (1994), 823–841.CrossRefGoogle Scholar
  9. [DKO95]
    F. Dumortier, H. Kokubu, and H. Oka. A degenerate singularity generating geometric Lorenz attractors.Ergod. Theory Dyn. Syst. 15 (1995), 833–856.Google Scholar
  10. [FD91]
    M. J. Friedman and E. J. Doedel. Numerical computation and continuation of invariant manifolds connecting fixed points.SIAM J. Numer. Anal. 28 (1991), 789–808.CrossRefGoogle Scholar
  11. [HKK94]
    A. J. Homburg, H. Kokubu, and M. Krupa. The cusp horseshoe and its bifurcations from inclination-flip homoclinic orbits.Ergod. Theory Dyn. Syst. 14 (1994), 667–693.Google Scholar
  12. [KKO93]
    M. Kisaka, H. Kokubu, and H. Oka. Bifurcation toN-homoclinic orbits andN-periodic orbits in vector fields.J. Dyn. Diff. Eq. 5 (1993), 305–358.CrossRefGoogle Scholar
  13. [Pal84]
    K. J. Palmer. Exponential dichotomies and transversal homoclinic points.J. Diff. Eq. 55 (1984), 225–256.CrossRefGoogle Scholar
  14. [San93]
    B. Sandstede.Verzweigungstheorie homokliner Verdopplungen, Doctoral thesis, University of Stuttgart, Stuttgart, 1993.Google Scholar
  15. [San95]
    B. Sandstede. Convergence estimates for the numerical approximation of homoclinic solutions.IMA J. Numer. Anal. (1997), to appear.Google Scholar
  16. [San96]
    B. Sandstede. A unified approach to homoclinic bifurcations with codimension two. II. Applications, in preparation (1996).Google Scholar
  17. [Shi70]
    L. P. Shilnikov. A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type.Mat. USSR Sb. 10 (1970), 91–102.Google Scholar
  18. [SS95]
    B. Sandstede and A. Scheel. Forced symmetry breaking of heteroclinic cycles.Nonlinearity 8 (1995), 333–365.CrossRefGoogle Scholar
  19. [Ter92]
    D. Terman. The transition from bursting to continuous spiking in excitable membrane models.J. Nonl. Sci. 2 (1992), 135–182.Google Scholar
  20. [Yan87]
    E. Yanagida. Branching of double pulse solutions from single solutions in nerve axon equations.J. Diff. Eq. 66 (1987), 243–262.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Björn Sandstede
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidence

Personalised recommendations