Generalized chronotaxic systems: time-dependent oscillatory dynamics stable under continuous perturbation

TL;DR

This paper develops a generalized deterministic theory of chronotaxic systems, which are resistant to continuous external perturbations, allowing for complex time-dependent oscillatory dynamics with varying parameters.

Contribution

The paper introduces a unified framework for analyzing chronotaxic systems without decoupling amplitude and phase, using contraction theory and time-dependent attractors.

Findings

Classified different types of chronotaxic dynamics.

Demonstrated the use of nonautonomous Poincaré oscillator as an example.

Discussed realizations including systems with temporal and interacting chronotaxicity.

Abstract

Chronotaxic systems represent deterministic nonautonomous oscillatory systems which are capable of resisting continuous external perturbations while having a complex time-dependent dynamics. Until their recent introduction in \emph{Phys. Rev. Lett.} \textbf{111}, 024101 (2013) chronotaxic systems had often been treated as stochastic, inappropriately, and the deterministic component had been ignored. While the previous work addressed the case of the decoupled amplitude and phase dynamics, in this paper we develop a generalized theory of chronotaxic systems where such decoupling is not required. The theory presented is based on the concept of a time-dependent point attractor or a driven steady state and on the contraction theory of dynamical systems. This simplifies the analysis of chronotaxic systems and makes possible the identification of chronotaxic systems with time-varying parameters. All types of chronotaxic dynamics are classified and their properties are discussed using the nonautonomous Poincar\'e oscillator as an example. We demonstrate that these types differ in their transient dynamics towards a driven steady state and according to their response to external perturbations. Various possible realizations of chronotaxic systems are discussed, including systems with temporal chronotaxicity and interacting chronotaxic systems.