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On the sharpness of the Rüssmann estimates. (English) Zbl 1510.37093

Summary: Estimating the norm of the solution of the linear difference equation \(u(\theta)-u(\theta+\omega)=v(\theta)\) plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by Rüssmann in the mid 70’s. The aim of this paper is to compare the sharpness of these classical estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
39A30 Stability theory for difference equations

Software:

MPFI
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References:

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