General stability conditions and stability transfer from quotient to parent systems
Establish general conditions that guarantee the stability of cluster-synchronised solutions in multiplex and hypergraph dynamical systems, including criteria that ensure stability is preserved when lifting a synchronised solution from the quotient dynamics to the parent system.
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Our results formalise and clarify the relationship between cluster synchronisation and equitability, including a new concept of dynamical stability, on networks, and higher-order networks, but several important open questions remain. These include fast and exhaustive algorithms to find equitable partitions in arbitrary multiplexes and hypergraphs; the realisation and ordering problem, that is, which equitable partitions and in which order they synchronise as we increase the coupling strength parameters (see for the network case); the stability question, that is, finding general conditions that guarantee the stability and a synchronised solution, for instance from a quotient to a parent solution; and an extension to non-identical dynamical units such as general multi-layer networks and to other synchronisation types beyond identical synchronisation.