Fisher Information and synchronisation transitions: A case-study of a finite size multi-network Kuramoto–Sakaguchi system

We analyse a Kuramoto–Sakaguchi dynamics on a two-layer multi-network using the Fisher Information which has been demonstrated in a variety of complex dynamical and thermodynamic systems to provide a lens on critical behaviour and transitions to chaos. Here we use a case-study, introduced elsewhere and thus providing a baseline, of multi-networks consisting of tree and random graphs with couplings and frequencies set at values in the vicinity of thresholds for locking, metastable and chaotic states. We observe transitions in the two-dimensional space of the frustrations in the cross-network interactions of the multi-layer system. While the Shannon entropy consistently identifies a range of transitions, the Fisher Information detects additional signals corresponding to significant changes in the microscopic dynamics. We argue that Fisher Information provides a single measure to analyse rich coupled dynamics and to detect meaningful transitions in a finite-size system that otherwise require multiple measures to establish. We support this analysis using a novel semi-analytical steady-state ansatz incorporating splay phase parameters, where the stability analysis concurs with key changes in the Fisher Information.