Kolmogorov-Arnold-Moser theorem

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

The above situation is the same as for the celebrated theorem of Kolmogorov-Arnold-Moser (KAM)—that is, the problem of small denominators. The convergence can be proved for sufficiently nonresonant combinations of the vibrational frequencies [31]. In other words, when tori of the vibrational motions on the NHIM Mq are sufficiently nonresonant, they survive under small perturbations. 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

In such a case the iterates of the two maps will also be conjugate and, if they are numerical methods, they will have similar stability properties and performance (e.g. the same effective order). It is difficult to separate the relevance of the two properties in cases where symplectic and reversible maps are conjugate. This is however rarely the case and certainly does not hold generically for discrete maps in many dimensions [209]. There is no direct correspondence between reversible and symplectic maps, however each class of maps admits certain theorems of dynamical systems which are in many ways analogous (for example, the Kolmogorov-Arnold-Moser, or KAM, theory for symplectic maps near elliptic fixed points [386] has an analogue for reversible maps [97]). 

As already noted, the strength of the interaction is a crucial parameter the KAM theorem (after Kolmogorov, Arnold and Moser [523, 524, 525] expresses the fact that the invariant tori on which the EBK quantisation (see above) is based are, in the classical system, replaced by very complex fractal structures in which only residues of the tori can survive. Thus, a progressively increasing volume of phase space is lost to chaotic orbits as the strength of the dynamical coupling between the modes is increased. 

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