Coexistence of regularity and irregularity in a nonlinear dynamical system

Abstract

A bounded trajectory of a continuous or discrete dynamical system is decomposed into two components. One component, termed as regular, is periodic with a given period and the transformation of the trajectory into the regular component is based on a variational principle. The other component, termed as irregular, is not periodic with the given period. The regular and irregular components have a property similar to orthogonality. The decomposition amounts to representation of the vector space of bounded trajectories as a product of two vector spaces comprising regular and irregular components. The decomposition leads to a new classification of bounded trajectories of a one-dimensional discrete dynamical system into asymptotically regular, mixed and irregular types. It also leads to new measures called periodic mean, periodic covariance and to two measures of irregularity. The potential of these diagnostic tools is explored by detecting windows of period seven, and by elucidating the structure of one such window of the benchmark system governed by logistic map. A window of period k begins with the period-doubling regime having a basic cycle of period k. It is followed by a mixed regime which contains windows of periods of multiples of k. The windows have a similarity property and they are nested.