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Homoclinic Classes and Finitude of Attractors For Vector Fields On n ‐Manifolds
Author(s) -
Carballo C. M.,
Morales C. A.
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001601
Subject(s) - homoclinic orbit , mathematics , vector field , attractor , orbit (dynamics) , manifold (fluid mechanics) , pure mathematics , heteroclinic orbit , conjecture , closure (psychology) , mathematical analysis , geometry , bifurcation , physics , mechanical engineering , quantum mechanics , nonlinear system , economics , engineering , market economy , aerospace engineering
A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor is a transitive set to which every positive nearby orbit converges; likewise, every negative nearby orbit converges to a repeller . It is shown in this paper that a generic C 1 vector field on a closed n ‐manifold has either infinitely many homoclinic classes, or a finite collection of attractors (or, respectively, repellers) with basins that form an open‐dense set. This result gives an approach to use in proving a conjecture by Palis. A proof is also given of the existence of a locally residual subset of C 1 vector fields on a 5‐manifold having finitely many attractors and repellers, but infinitely many homoclinic classes.