00-411 A. Jorba

Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps (500K, PostScript, gzipped and uuencoded) 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 non-reducibility --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 6-D symplectic map.

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