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Invariant tori of dissipatively perturbed Hamiltonian systems under symplectic discretization

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E. Hairer and Ch. Lubich

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Abstract. In a recent paper, Stoffer showed that, under a very weak restriction on the stepsize, weakly attractive invariant tori of dissipative perturbations of integrable Hamiltonian systems persist under symplectic numerical discretizations. Stoffer's proof works directly with the discrete scheme. Here, we show how such a result, together with approximation estimates, can be obtained by combining Hamiltonian perturbation theory and backward error analysis of numerical integrators. In addition, we extend Stoffer's result to dissipative perturbations of non-integrable Hamiltonian systems in the neighborhood of a KAM torus.

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Key Words. Weakly damped nonlinear oscillations, symplectic numerical integrator, Hamiltonian perturbation theory, backward error analysis, invariant manifold, long-time approximation

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