Angular dynamical spectra. A new method for determining frequencies, weak chaos and cantori

摘要

We define the invariant spectra of rotation angles and twist angles (angular dynamical spectra) and study the properties of their moments (angular moments) in a model two-dimensional map. The angular moments give the main frequencies of the orbits. A main frequency is defined both for a regular and chaotic orbit. For KAM curves around a centre this frequency corresponds to the rotation number. Inside islands of stability, we obtain both the main frequency (rotation number) and the 'epicyclic frequency'. A fast detection of thin chaotic layers is obtained on the basis of the behaviour of the frequency curve. We explore the resonant structure near a last KAM boundary. The secondary islands of various multiplicities, forming Farey sequences, and the noble tori between them are located. We find a criterion to determine whether various resonant chaotic zones communicate, or are separated by invariant tori. [References: 19]