Hyperbolicity and the Creation of Homoclinic Orbits

We consider one-parameter families$\lbrace \varphi _\mu; \mu \epsilon R \rbrace$of diffeomorphisms on surfaces which display a homoclinic tangency for$\mu < 0$and are hyperbolic for$\mu < 0$(i.e., φμhas a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for...

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Published in:Annals of mathematics 1987-03, Vol.125 (2), p.337-374
Main Authors: Palis, J., Takens, F.
Format: Article
Language:English
Subjects:
Coordinate systems
Eigenvalues
Exact sciences and technology
Function theory, analysis
Horseshoes
Leaves
Mathematical methods in physics
Periodic orbits
Physics
Quadrants
Saddle points
Tangents
Zero
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container_issue 2
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container_title Annals of mathematics
container_volume 125
creator Palis, J.
Takens, F.
description We consider one-parameter families$\lbrace \varphi _\mu; \mu \epsilon R \rbrace$of diffeomorphisms on surfaces which display a homoclinic tangency for$\mu < 0$and are hyperbolic for$\mu < 0$(i.e., φμhas a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for μ positive. For many of these families, we prove than φμis also hyperbolic for most small positive values or μ (which implies much regularity of the dynamical structure). A main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency.
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ispartof Annals of mathematics, 1987-03, Vol.125 (2), p.337-374
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source JSTOR Archival Journals and Primary Sources Collection
subjects Coordinate systems
Eigenvalues
Exact sciences and technology
Function theory, analysis
Horseshoes
Leaves
Mathematical methods in physics
Periodic orbits
Physics
Quadrants
Saddle points
Tangents
Zero
title Hyperbolicity and the Creation of Homoclinic Orbits
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