Chronotaxic systems with separable amplitude and phase dynamics

Yevhen F Suprunenko,Aneta Stefanovska,Philip T Clemson

doi:10.1103/physreve.89.012922

Yevhen F Suprunenko, Aneta Stefanovska + Show 1 more

Open Access

https://doi.org/10.1103/physreve.89.012922

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Abstract

Until recently, deterministic nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies were not recognized as such and have often been mistreated as stochastic. These systems, named chronotaxic, were introduced in Phys. Rev. Lett. 111, 024101 (2013). In contrast to conventional limit cycle models of self-sustained oscillators, these systems posses a time-dependent point attractor or steady state. This allows oscillations with time-varying frequencies to resist perturbations, a phenomenon which is ubiquitous in living systems. In this work a detailed theory of chronotaxic systems is presented, specifically in the case of separable amplitude and phase dynamics. The theory is extended by the introduction of chronotaxic amplitude dynamics. The wide applicability of chronotaxic systems to a range of fields from biological and condensed matter systems to robotics and control theory is discussed.

Highlights

Real life dynamical systems are often explicitly time dependent: a power-grid network depends on supply and demand [1], the area of a rainforest depends on logging and reforestation [2], a cell must continuously generate ATP to cope with external perturbations [3,4] and the brain continuously matches oxygen and ion supply to its actual function [5] with the recently discovered [6] involvement of astrocytes
In this paper we extend the theory of chronotaxic systems [18] by studying the time-dependent and stable dynamics of both the amplitude and phase of oscillations in the case when these dynamics are separated
In Ref. [18] we identified which nonautonomous systems with time-dependent steady states represent an adequate description of time-dependent and stable oscillatory dynamics

Summary

INTRODUCTION

Real life dynamical systems are often explicitly time dependent: a power-grid network depends on supply and demand [1], the area of a rainforest depends on logging and reforestation [2], a cell must continuously generate ATP to cope with external perturbations [3,4] and the brain continuously matches oxygen and ion supply to its actual function [5] with the recently discovered [6] involvement of astrocytes. The description of oscillatory nonautonomous systems is still often based on conventional autonomous limit cycle models of self-sustained oscillators. In this paper we extend the theory of chronotaxic systems [18] by studying the time-dependent and stable dynamics of both the amplitude and phase of oscillations in the case when these dynamics are separated. This extension is achieved by introducing a time-dependent (driven) steady state or point attractor into the amplitude and phase dynamics. We provide an analytical calculation of the time-dependent steady state as a pullback point attractor in a chronotaxic system

FROM AUTONOMOUS TO NONAUTONOMOUS DYNAMICS AND TIME-DEPENDENT ATTRACTORS

Model of separable chronotaxic phase and amplitude dynamics

Conditions of existence

APPLICATIONS AND DISCUSSIONS

SUMMARY