 00411 A. Jorba
 Numerical computation of the normal behaviour of invariant curves of
$n$dimensional maps
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Oct 19, 00

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Abstract. We describe a numerical method for computing the linearized normal
behaviour of an invariant curve of a diffeomorphism of $\RR^n$, $n\ge
2$. In the reducible case, the method computes not only the normal
eigenvalues either elliptic or hyperbolic but also the
corresponding eigendirections, that are the first order approximation
to the invariant manifolds (stable, unstable and central) around the
curve. Moreover, the method seems to be able to detect the
nonreducibility if this is the case of the linearized system.
The input of the method is the invariant curve including its
rotation number as well as a numerical procedure for computing the
map and its differential. Hence, this method can be easily used on
Poincar\'e sections of ODE. Due to the spectral character of the
approximations used, the convergence of the process is very fast for
sufficiently smooth cases. We note that the method is also valid for
computing the normal behaviour of tori of higher dimensions. Finally,
as examples, we study the stability of the invariant curves that
appear in some concrete problems. In particular, we compute the
unstable manifold for a given invariant curve of a 6D symplectic map.
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