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A non-transverse homoclinic orbit to a saddle-node equilibrium

Published online by Cambridge University Press:  19 September 2008

Alan R. Champneys ,
Jörg Härterich  and
Björn Sandstede
Alan R. Champneys
Affiliation:
Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, UK
Jörg Härterich
Affiliation:
Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2–6, 14195 Berlin, Germany
Björn Sandstede
Affiliation:
Weierstraβ-Institut für Angewandte Analysis und Stochastik, Mohrenstraβe 39, 10117 Berlin, Germany
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Abstract

A homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve defining two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the Hénon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996 , pp. 431 - 450
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[BC94]Bai, F. and Champneys, A. R.. Numerical detection and continuation of saddle-node homoclinic bifurcations of codimension one or two. Preprint, University of Bath, 1994.Google Scholar
[CK94]Champneys, A. and Kuznetsov, Yu. A.. Numerical detection and continuation of codimension-two homoclinic bifurcations. Int. J. Bifurcation & Chaos 4 (1994), 785–822.CrossRefGoogle Scholar
[CL90]Chow, S.-N. and Lin, X. B.. Bifurcation of a homoclinic orbit with a saddle-node equilibrium. Diff. lntegr. Eq. 3 (1990), 435–466.Google Scholar
[Den90]Deng, B.. Homoclinic bifurcations with nonhyperbolic equilibria. SIAMJ. Math. Anal. 21 (1990), 693–720.CrossRefGoogle Scholar
[DK86]Doedel, E. and Kernévez, J.. AUTO: Software for continuation problems in ordinary differential equations with applications. Technical Report, California Institute of Technology, 1986.Google Scholar
[Fie92]Fiedler, B.. Global pathfollowing of homoclinic orbits in two-parameter flows. Preprint, 1992.Google Scholar
[GH90]Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, 1990.Google Scholar
[Gle88]Glendinning, P.. Global bifurcation in flows. New directions in dynamical systems ed Bedford, T. and Swift, J., pages 120–149. Cambridge University Press, 1988.CrossRefGoogle Scholar
[HKK93]Homburg, A. J., Kokubu, H. and Krupa, M.. The cusp horseshoe and its bifurcations from inclination-flip homoclinic orbits. Ergod. Th. & Dynam. Sys. 14 (1994), 667–693.CrossRefGoogle Scholar
[IY91]Il'yanshenko, Yu. S. and Yakovenko, S. Yu.. Finitely-smooth normal forms of local families of diffeomorphisms and vector fields. Russian Math. Surveys 46 (1991), 1–43.CrossRefGoogle Scholar
[Luk82]Lukyanov, V. I.. Bifurcations of dynamical systems with a saddle-node separatrix loop. Diff. Eq. 18 (1982), 1049–1059.Google Scholar
[MV93]Mora, L. and Viana, M.. Abundance of strange attractors. Acta. Math. 171 (1993), 1–71.CrossRefGoogle Scholar
[PT93]Palis, J. and Takens, F.. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Cambridge University Press, 1993.Google Scholar
[San93]Sandstede, B.. Verzweigungstheorie homokliner Verdopplungen. Doctoral Thesis, University of Stuttgart, 1993.Google Scholar
[San94a]Sandstede, B.. Center manifolds for homoclinic solutions. In preparation, 1994.Google Scholar
[San94b]Sandstede, B.. Constructing dynamical systems possessing homoclinic bifurcation points of codimension two. In preparation, 1994.Google Scholar
[Sch87]Schecter, S.. The saddle-node separatrix-loop bifurcation. SIAM J. Math. Anal. 28 (1987), 1142–1156.CrossRefGoogle Scholar
[Shi69]Shilnikov, L. P.. On a new type of bifurcation of multidimensional dynamical systems. Soviet Math. Dokl. 10 (1969), 1368–1371.Google Scholar
[Shi70]Shilnikov, L. P.. A contribution to a problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type. Math. USSR Sbornik 10 (1970), 91–102.CrossRefGoogle Scholar
[Sot74]Sotomayor, J.. Generic one-parameter families of vector fields. Inst. Hautes Etudes Sci. Publ. Math. 43 (1974), 5–46.CrossRefGoogle Scholar
[Van89]Vanderbauwhede, A.. Centre manifolds, normal forms and elementary bifurcations. Dynamics Reported vol 2, pp 89–169. ed Kirchgraber, U. and Walther, H. O.. Wiley, New York, 1989.CrossRefGoogle Scholar
[YA85]Yorke, J. A. and Alligood, K. T.. Period doubling cascades of attractors: a pre-requisite for horseshoes. Comm. Math. Phys. 101 (1985), 305–321.CrossRefGoogle Scholar