On the relationship between the theory of cointegration and the theory of phase synchronization

TL;DR

This paper reveals a close connection between cointegration in econometrics and phase synchronization in physics, enabling cross-disciplinary methods for analyzing dynamic systems and their equilibrium states.

Contribution

It establishes a theoretical link between cointegration and phase synchronization, allowing statistical inference techniques from econometrics to be applied to physical oscillatory systems.

Findings

Cointegration approximates stochastic Kuramoto equations.

Statistical tests for phase synchronization can be derived from cointegration theory.

Applications demonstrated on coupled Rössler-Lorenz and electrochemical oscillators.

Abstract

The theory of cointegration has been a leading theory in econometrics with powerful applications to macroeconomics during the last decades. On the other hand the theory of phase synchronization for weakly coupled complex oscillators has been one of the leading theories in physics for many years with many applications to different areas of science. For example, in neuroscience phase synchronization is regarded as essential for functional coupling of different brain regions. In an abstract sense both theories describe the dynamic fluctuation around some equilibrium. In this paper, we point out that there exists a very close connection between both theories. Apart from phase jumps, a stochastic version of the Kuramoto equations can be approximated by a cointegrated system of difference equations. As one consequence, the rich theory on statistical inference for cointegrated systems can immediately be applied for statistical inference on phase synchronization based on empirical data. This includes tests for phase synchronization, tests for unidirectional coupling and the identification of the equilibrium from data including phase shifts. We study two examples on a unidirectionally coupled R\"ossler-Lorenz system and on electrochemical oscillators. The methods from cointegration may also be used to investigate phase synchronization in complex networks. Conversely, there are many interesting results on phase synchronization which may inspire new research on cointegration.