Abstract
Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase- synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable.This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.
Highlights
Many dynamical systems, both natural and man-made, are composed of interacting parts
Examples include Josephson junctions [1,2], neuronal networks [3,4,5], the cardiorespiratory system [6,7,8], cardiorespiratory–brain interactions [9,10,11,12], and systems occurring in social sciences [13,14], communications [15,16] and chemistry [17,18,19]
In a differential equation or stochastic differential equation describing a system of interacting components, the terms on the right-hand side arising from the interactions between the components are referred to as coupling functions
Summary
Both natural and man-made, are composed of interacting parts. Many recent studies of interactions are designed for, and focus exclusively on, the effect of the net coupling strength of interacting systems This approach is often found in informationtheoretic methods for the detection of directionality and causality of influence between time-series including, for example, methods for Granger causality, transfer entropy, mutual information and symbolic transfer entropy [34,35,36,37]. We extend [23] by studying theoretically the effects of time-varying coupling functions that induce a transition to synchronization, while keeping the net coupling strength constant.
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