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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Bibliographic Details
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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Summary: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.
ISSN:0003-486X
1939-8980
DOI:10.2307/1971313