# Causal Stability and Synchronization

## TL;DR

This paper introduces the concept of causal stability as a new spatial perspective on chaos synchronization, providing a theorem and empirical criteria for identifying synchronizing variables in coupled chaotic systems.

## Contribution

It proposes the novel idea of causal stability, proves a synchronization theorem, and offers empirical criteria for causality-based identification of synchronizing variables.

## Key findings

- Causal stability is a necessary and sufficient condition for synchronization.
- A causal stability synchronization theorem is established.
- Empirical criteria for identifying synchronizing variables are provided.

## Abstract

Synchronization of chaos arises between coupled dynamical systems and is very well understood as a temporal phenomena which leads the coupled systems to converge or develop a dependence with time. In this work, we provide a complementary spatial perspective to this phenomenon by introducing the novel idea of causal stability. We then propose and prove a causal stability synchronization theorem as a necessary and sufficient condition for synchronization. We also provide an empirical criteria to identify synchronizing variables in coupled identical chaotic dynamical systems based on causality analysis on time series data of the driving system alone.