Rüssmann, Helmut KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. (English) Zbl 1221.37114 Discrete Contin. Dyn. Syst., Ser. S 3, No. 4, 683-718 (2010). The present paper is devoted to a new variant of the KAM theory obtained by adding an artificial parameter \(q\in (0, 1)\), which makes the steps of the KAM iteration infinitely small in the limit \(q\rightarrow 1\). This KAM procedure can be compared for \(q<1\) with a Riemann sum which tends for \(q\rightarrow 1\) to the corresponding Riemann integral. As a consequence, this limit has all advantages of an integration process compared with its preliminary stages. But there is a difference from integrals: the KAM iteration itself works only for \(q<1\); however, \(q\) can be chosen as near to \(1\) as we want, and the limit \(q\rightarrow 1\) exists for all involved parameters. The new technique of estimation differs completely from all what has appeared about KAM theory in the literature up to date. Reviewer: Mircea Crâşmăreanu (Iaşi) Cited in 1 ReviewCited in 19 Documents MSC: 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 70H08 Nearly integrable Hamiltonian systems, KAM theory 70K43 Quasi-periodic motions and invariant tori for nonlinear problems in mechanics 41A10 Approximation by polynomials Keywords:KAM theory; artificial parameters; polynomial approximations; cut functions PDFBibTeX XMLCite \textit{H. Rüssmann}, Discrete Contin. Dyn. Syst., Ser. S 3, No. 4, 683--718 (2010; Zbl 1221.37114) Full Text: DOI