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Multiscale time averaging, reloaded. (English) Zbl 1310.34051

The paper deals with a general dynamical system of the form \[ i\,\frac{\partial}{\partial t}\, \vec c = \beta A(t)\vec c, \] where \(\beta\) is a small parameter and \(\vec c\) has finite dimension. The authors introduce the mean \[ \bar A_0^{(n)} = \frac{1}{T_0} \int\limits_{nT_0}^{(n+1)T_0} A(s)ds \] and develop a rigorously controlled multi-time scale averaging technique. The averaging is done on a finite time interval, properly chosen, and then, via iterations and normal form transformations, the time intervals are scaled to arbitrary order. The authors consider as an example the problem of a finite dimensional conservative dynamical system, which is quasi periodic and dominated by slow frequencies, leading to small divisor problems in perturbative schemes. The obtained estimates hold for arbitrary long time intervals similar to the Nekhoroshev-type results.

MSC:

34C29 Averaging method for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34D15 Singular perturbations of ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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