Synchronization and intermittency of the two point motion.

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In the study of dynamic systems involving particles or states over time, a fundamental inquiry emerges in the examination of the "two-point motion". This phenomenon focuses on understanding the evolution of two particles within the same system over time.Consider, for instance, the scenario of population modeling, where two distinct populations of equal size are observed to discern whether they converge to a shared random state (synchronization) or exhibit divergence indicative of chaotic behavior. Alternatively, a fascinating phenomenon known as intermittency in the two-point motion can occur. This entails a peculiar behavior wherein two points spend extended periods in close proximity, only to inevitably diverge again, regardless of their initial closeness.The intricacies of intermittency in the two-point motion remain elusive, constituting a gap in our understanding. This research project seeks to address this gap by identifying specific classes of systems in which intermittency is prevalent. Through comprehensive investigation, the aim is to shed light on the underlying dynamics that give rise to this intriguing behavior and contribute to a deeper comprehension of complex systems and their temporal evolution.This project falls within the EPSRC statistics and applied probability, and non-linear systems research areas.