Causal Properties of Synchronizing Systems

Abstrakt

The idea of ‘causal stability’ and the ‘causal stability synchronization theorem’ have recently been proposed and proved by us [1]. This work theoretically establishes the ‘causal’ or ‘spatial’ conditions for chaotic synchronization. Synchronization of coupled chaotic systems is a ubiquitous phenomenon in nature, of much interest to the dynamical systems community. However, from the time it was discovered, synchronization has always been considered as a temporal phenomenon with the ‘temporal conditions’ required for systems to synchronize thoroughly studied and well established. On the other hand, ‘causality estimators’ based on time-series are widely used to study coupled dynamical systems. Though the fate of many coupled systems is synchronization, no rigorous mathematical relationship has been established between ‘causal properties’ of systems and their ability to synchronize. By proving the causal stability synchronization theorem for identical chaotic systems coupled unidirectionally, our work opens up the possibility to study chaotic synchronization with the lens of causality. Alongside, we provide causality based empirical conditions to predict ‘synchronizing variables’ based on the time series of the master system alone. Synchronizing variables are the variables which when coupled lead the systems to synchronization. In this work, we test for the robustness of the proposed empirical conditions on different dynamical systems started with different initial conditions. This is important for future real word applications where prediction of synchronizing variables is essential, e.g., in the control of chaos and the control of synchronization (such as in epilepsy, communication or electrical circuits). We also compare our empirical criterion with the criterion of ‘negative conditional Lyapunov exponents’, an existing necessary temporal criterion for synchronization.