Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-06-17T04:21:24.249Z Has data issue: false hasContentIssue false

Basins of attraction near homoclinic tangencies

Published online by Cambridge University Press:  19 September 2008

J.C. Tatjer
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain
C. Simó
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain

Abstract

We describe the behaviour of the basin of attraction of the attracting periodic points which appear near a non-degenerate tangential homoclinic point of a dissipative saddle fixed point for one-parameter families of planar diffeomorphisms. This behaviour depends on certain relations between the eigenvalues of the saddle point and on the geometry of the tangency.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1994

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

REFERENCES

[1]Benedicks, M. & Carlesson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73169.CrossRefGoogle Scholar
[2]Chow, S. N. & Hale, J. K.. Methods of Bifurcation Theory. Springer: New York, 1982.CrossRefGoogle Scholar
[3]Da Silva Ritter, G. L., Ozorio de Almeida, A. M & Douady, R.. Analytical determination of unstable periodic orbits in area preserving maps. Physica 29D (1987), 181190.Google Scholar
[4]Franceschini, V. & Russo, L.. Stable and unstable manifolds on the Hénon mapping. J. Stat. Phys. 25 (1981).CrossRefGoogle Scholar
[5]Gavrilov, N. K. & Sitnikov, L. P.. On the three-dimensional dynamical systems close to a system with a structurally unstable homoclinic curve. (I) Math. USSR Sbornik 17 (1972), 467485;CrossRefGoogle Scholar
Gavrilov, N. K. & Sitnikov, L. P.. On the three-dimensional dynamical systems close to a system with a structurally unstable homoclinic curve. (II) Math. USSR Sbornik 19 (1973), 138156.CrossRefGoogle Scholar
[6]Hartman, P.. Ordinary Differential Equations. Reprint of the 2nd ed. Birkhauser: Boston, 1982.Google Scholar
[7]Irwin, M. C.. On the smoothness of the composition map. Quart. J. Math. Oxford (2) 23 (1972), 113133.CrossRefGoogle Scholar
[8]Marsden, J. E. & McCracken, M.. The Hopf Bifurcation and its Applications. Springer: Berlin, 1976.CrossRefGoogle Scholar
[9]Mora, L. & Viana, M.. Abundance of strange attractors. Acta Math. 171 (1993), 171.CrossRefGoogle Scholar
[10]Newhouse, S.. Lectures on dynamical systems. Dynamical Systems. CIME Lectures Bressanone (Italy). Birkhauser: Boston, 1980.Google Scholar
[11]Newhouse, S.. The creation of non trivial recurrence in the dynamics of diffeomorphisms. In Comportement Chaotique des Systèmes Déterministes (Les Houches 1981). Iooss, G., Helleman, R. and Stora, R., eds. North-Holland: Amsterdam, 1983.Google Scholar
[12]Newhouse, S.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. IHES 50 (1979), 101151.CrossRefGoogle Scholar
[13]Palis, J. & Melo, W. de. Geometric Theory of Dynamical Systems. An Introduction. Springer: New York, 1982.CrossRefGoogle Scholar
[14]Palis, J. & Takens, F.. Homoclinic bifurcations and hyperbolic dynamics. XVI Colóquio Brasileiro de Matemática (1985). IMPA/CNPq Rio de Janeiro (1987).Google Scholar
[15]Robinson, C.. Bifurcations to infinitely many sinks. Commun. Math. Phys. 90 (1983), 433459.CrossRefGoogle Scholar
[16]Simó, C.. On the Hénon-Pomeau attractor. J. Stat. Phys. 21, (1979).CrossRefGoogle Scholar
[17]Sotomayor, J.. Liçoes sobre Equaçōes Diferenciais Ordinarias. IMPA, Projeto Euclides: Rio de Janeiro, 1979.Google Scholar
[18]Tatjer, J. C.. Estudi del fenòmen de Newhouse per a difeomorfismes dissipatius del pla. Master Thesis. Universitat de Barcelona (1984).Google Scholar
[19]Tatjer, J. C.. Invariant manifolds and bifurcations for one-dimensional and two-dimensional dissipative maps. PhD Thesis. Universitat de Barcelona (1990).Google Scholar
[20]Tatjer, J. C.. Codimension two bifurcations near homoclinic tangencies of second order. Proc. European Conf. on Iteration Theory (ECIT 91), (Lisboa (Portugal)). Lampreia, J.P., Llibre, J., Mira, C., Ramos, J. Sousa and Targonski, Gy., eds. World Scientific: Singapore, 1992. pp 306318.Google Scholar