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Growing 2D manifolds computation

M. Jia

Abstract


This paper mainly studies the computation of 2D manifolds of maps. An algorithm for computing 1D manifold, which uses a prediction and correction scheme to add new mesh points, is proposed at first. 2D manifold is computed by covering it with 1D sub-manifolds. A generalized Foliation Condition is proposed and used to make sure that the 2D manifold grows uniformly along the 1D sub-manifolds of different directions. The weak point of our algorithm is too much mesh points are generated at the inner part of the 2D manifold, and it is a promising key point where the algorithm can be revised in the future. But inserting 1D sub-manifold has advantages over inserting mesh points, because it avoids searching the preimage of the inserted mesh points on the surface of the 2D manifold.

Keywords


Dynamical system, map, stable manifold, unstable manifold, derivative transportation, 3D Hénon map, Lorenz system, chaotic attractor.

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