The Kolmogorov–Arnold–Moser theorem

Quantum analogue of the Kolmogorov–Arnold–Moser transition in field-induced barrier penetration in a quartic potential

Quantum signatures of the Kolmogorov–Arnold– Moser (KAM) transition from the regular to chaotic classical dynamics of a double-well oscillator in the presence of an external monochromatic field of different amplitudes are analysed in terms of the corresponding Bohmian trajectories. It is observed that the classical chaos generally enhances the quantum fluctuations, while the quantum nonclassica...

Controllability for a class of area-preserving twist maps

In this paper, we study controllability of two-dimensional integrable twist maps with bounded area-preserving time-dependent (control) perturbations. In contrast to the time-independent perturbation case of the Kolmogorov–Arnold–Moser theorem, there are no invariant sets other than the whole phase space if the perturbation is made a function of time. We give necessary and sufficient conditions ...

Quantum KAM Technique and Yang-Mills Quantum Mechanics

We study a quantum analogue of the iterative perturbation theory by Kolmogorov used in the proof of the Kolmogorov-Arnold-Moser (KAM) theorem. The method is based on sequent canonical transformations with a "running" coupling constant ; ; , etc. The proposed scheme, as its classical predecessor, is "superconvergent" in the sense that after the nth step, a theory is solved to the accuracy of ord...

Kolmogorov-Arnold-Moser (KAM) Theory

Kam Theory: the Legacy of Kolmogorov’s 1954 Paper

Kolmogorov-Arnold-Moser (or kam) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and kam theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov’s pioneering work.