Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations

  • Ken Palmer
Part of the Mathematics and Its Applications book series (MAIA, volume 501)


We consider the autonomous system of ordinary differential equations
$$\mathop x\limits^.= F(x),$$
where F : U → ℝ n is a C 1 vectorfield, the set U being open and convex. Denote by the corresponding flow. Suppose u (t) is a hyperbolic periodic orbit for Eq.(l) with minimal period T < 0 and let p 0 be in the intersection of the stable manifold W s (u) and the unstable manifold W u (u). If p 0 satisfies the transversality condition
$${T_{{p_0}}}{W^s}(u) \cap {T_{{p_0}}}{W^u}(u) = span\{ F({p_0})\} ,$$
then we know from Theorem 8.2 that the set
$$S = \{ u(t): - \infty< t < \infty \}\cup \{ {\phi ^t}({p_0}): - \infty< t < \infty \} $$
is hyperbolic. Now we state our main theorem, which uses symbolic dynamics to describe the solutions of Eq.(l) which remain in a neighbourhood of the set S.


Periodic Orbit Homoclinic Orbit Transversality Condition Tubular Neighbourhood Symbolic Dynamics 
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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Ken Palmer
    • 1
  1. 1.School of Mathematical & Statistical SciencesLa Trobe UniversityBundooraAustralia

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