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A survey of methods for computing (un)stable manifolds of vector fields. (English) Zbl 1086.34002
The article presents a survey of different methods for computing stable or unstable manifolds of vector fields. The discussed approaches include: approximation by geodesic level sets by the first two authors [Chaos 9, No. 3, 768–774 (1999; Zbl 0983.37110)]; BVP continuation of trajectories by Doedel; computation of fat trajectories by M. E. Henderson [Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, No. 3, 451–476 (2002; Zbl 1044.37053)]; the PDE approach by J. Guckenheimer and A. Vladimirsky [SIAM J. Appl. Dyn. Syst. 3, No. 3, 232–260 (2004; Zbl 1059.37019)]; and the box covering algorithm by M. Dellnitz and A. Hohmann [Numer. Math. 75, No. 3, 293–317 (1997; Zbl 0883.65060)].
The authors concentrate on the case of two-dimensional manifolds. All methods are illustrated by means of the two-dimensional stable manifold of the origin in the Lorenz system.

MSC:
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
65L99 Numerical methods for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
37C10 Dynamics induced by flows and semiflows
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DSTool; GAIO; PITCON; psSchur
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