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Coupled nonautonomous oscillators

Research output: Contribution in Book/Report/ProceedingsChapter (peer-reviewed)

Published

Publication date2013
Host publicationNonautonomous dynamical systems in the life sciences
EditorsPeter E. Kloeden, Christian Pötzsche
PublisherSpringer
Pages163-197
Number of pages35
ISBN (Electronic)9783319030807
ISBN (Print)9783319030791
Original languageEnglish

Publication series

NameLecture Notes in Mathematics
PublisherSpringer
Volume2102
ISSN (Print)0075-8434

Abstract

First, we introduce nonautonomous oscillator—a self-sustained oscillator subject to external perturbation and then expand our formalism to two and many coupled oscillators. Then, we elaborate the Kuramoto model of ensembles of coupled oscillators and generalise it for time-varying couplings. Using the recently introduced Ott-Antonsen ansatz we show that such ensembles of oscillators can be solved analytically. This opens up a whole new area where one can model a virtual physiological human by networks of networks of nonautonomous oscillators. We then briefly discuss current methods to treat the coupled nonautonomous oscillators in an inverse problem and argue that they are usually considered as stochastic processes rather than deterministic. We now point to novel methods suitable for reconstructing nonautonomous dynamics and the recently expanded Bayesian method in particular.We illustrate our new results by presenting data from a real living system by studying time-dependent coupling functions between the cardiac and respiratory rhythms and their change with age. We show that the well known reduction of the variability of cardiac instantaneous frequency is mainly on account of reduced influence of the respiration to the heart and moreover the reduced variability of this influence. In other words, we have shown that the cardiac function becomes more autonomous with age, pointing out that nonautonomicity and the ability to maintain stability far from thermodynamic equilibrium are essential for life.