Shadowing in Dynamical Systems pp 225-239 | Cite as

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

Chapter

## Abstract

We consider the autonomous system of ordinary differential equations
where
then we know from Theorem 8.2 that the set
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

$$\mathop x\limits^.= F(x),$$

(1)

*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$${T_{{p_0}}}{W^s}(u) \cap {T_{{p_0}}}{W^u}(u) = span\{ F({p_0})\} ,$$

$$S = \{ u(t): - \infty< t < \infty \}\cup \{ {\phi ^t}({p_0}): - \infty< t < \infty \} $$

*S.*## Keywords

Periodic Orbit Homoclinic Orbit Transversality Condition Tubular Neighbourhood Symbolic Dynamics
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media Dordrecht 2000