Nonchaotic Stagnant Motion in a Marginal Quasiperiodic Gradient System

TL;DR

This paper introduces a one-dimensional dynamical system with a marginal quasiperiodic gradient that exhibits nonchaotic stagnant motion, revealing unique quasiperiodic alternations and long-term behaviors distinct from traditional intermittency.

Contribution

It presents a novel mathematical model demonstrating nonchaotic stagnant motion with quasiperiodic alternations and derives its long-time asymptotic behavior, expanding understanding of nonchaotic dynamics.

Findings

Residence time density follows an inverse-square law.

Alternation between stagnant and moving phases is quasiperiodic.

Long-time behavior involves nested logarithmic decay.

Abstract

A one-dimensional dynamical system with a marginal quasiperiodic gradient is presented as a mathematical extension of a nonuniform oscillator. The system exhibits a nonchaotic stagnant motion, which is reminiscent of intermittent chaos. In fact, the density function of residence times near stagnation points obeys an inverse-square law, due to a mechanism similar to type-I intermittency. However, unlike intermittent chaos, in which the alternation between long stagnant phases and rapid moving phases occurs in a random manner, here the alternation occurs in a quasiperiodic manner. In particular, in case of a gradient with the golden ratio, the renewal of the largest residence time occurs at positions corresponding to the Fibonacci sequence. Finally, the asymptotic long-time behavior, in the form of a nested logarithm, is theoretically derived. Compared with the Pomeau-Manneville intermittency, a significant difference in the relaxation property of the long-time average of the dynamical variable is found.