A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

Bulcsú Sándor,Bence Schneider,Mária Ercsey-Ravasz,Zsolt I Lázár

doi:10.3390/e23010103

Abstract

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

Highlights

Complex network theory has had many interdisciplinary applications in different domains of social sciences, epidemiology, economy, neuroscience, biology etc. [1]
Inspired by the theory of dynamical systems we look at the trajectory length of a system evolving over discrete time according to the transition probabilities defining its State-transition networks (STN)
Mapping a dynamical system into an STN requires us to assign the different states of the dynamics to certain nodes of the network [15,16]

Summary

Introduction

Complex network theory has had many interdisciplinary applications in different domains of social sciences, epidemiology, economy, neuroscience, biology etc. [1]. Complex network theory has had many interdisciplinary applications in different domains of social sciences, epidemiology, economy, neuroscience, biology etc. In recent years different network approaches have been developed for nonlinear time series analysis. Proper mapping between a discrete time series and a complex network in order to apply the tools of network theory in an efficient manner is not a trivial question. There are several approaches to this problem, here we mention three large categories [2]: (1) Proximity networks are created based on the statistical or metric proximity of two time series segments. The most studied variant of proximity networks are recurrence networks [3]. These have found many applications, in the characterization of discrete [4]

Methods

Results

Conclusion