The numerical approximation of center manifolds in Hamiltonian systems |
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Authors: | Wei-Hua Du,Wolf-Jü rgen Beyn |
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Affiliation: | a Mathematical Department of Harbin Engineering University, Harbin 150001, PR China b Department of Mathematics, University of Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany |
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Abstract: | In this paper we develop a numerical method for computing higher order local approximations of center manifolds near steady states in Hamiltonian systems. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear solver and a low-dimensional invariant subspace is available. Our method combines this restriction from linear algebra with the requirement that the center manifold is parametrized by a symplectic mapping and that the reduced equation preserves the Hamiltonian form. Our approach can be considered as a special adaptation of a general method from Numer. Math. 80 (1998) 1-38 to the Hamiltonian case such that approximations of the reduced Hamiltonian are obtained simultaneously. As an application we treat a finite difference system for an elliptic problem on an infinite strip. |
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Keywords: | Center manifolds Hamiltonian systems Numerical methods Bordered linear systems |
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