Synchronization in slowly switching networks of coupled oscillators

Abstract

Networks whose structure of connections evolves in time constitute a big challenge in the study of synchronization, in particular when the time scales for the evolution of the graph topology are comparable with (or even longer than) those pertinent to the units’ dynamics. We here focus on networks with a slow-switching structure, and show that the necessary conditions for synchronization, i.e. the conditions for which synchronization is locally stable, are determined by the time average of the largest Lyapunov exponents of transverse modes of the switching topologies. Comparison between fast- and slow-switching networks allows elucidating that slow-switching processes prompt synchronization in the cases where the Master Stability Function is concave, whereas fast-switching schemes facilitate synchronization for convex curves. Moreover, the condition of slow-switching enables the introduction of a control strategy for inducing synchronization in networks with arbitrary structure and coupling strength, which is of evident relevance for broad applications in real world systems.

Designing temporal networks that synchronize under resource constraints

“…( 3 ) and ( 4 ) is the periodic λ ( t ) vs. the fixed α , it is natural to expect the stability of the temporal network to be related to the master stability function averaged over a suitable range of α . Specifically, one might reasonably associate Γ with the averaged master stability function 18 , 22 , 26 , 27 , 30 where W ( λ ) is the probability distribution of λ (it follows that ). However, it is clear that cannot be used to predict Γ in general.…”

“…It is worth noting that the equivalence between Γ and at the slow-switching limit is not specific to Stuart–Landau oscillators and can be expected for generic oscillator models 26 , 30 . As a result, Λ ″ ( α 0 ) < 0 is a robust indicator that synchronization in a system can benefit from temporal networks.…”

“…(3) and ( 4) is the periodic λ(t) vs. the fixed α, it is natural to expect the stability of the temporal network to be related to the master stability function averaged over a suitable range of α. Specifically, one might reasonably associate Γ with the averaged master stability function 18,22,26,27,30 Λ ¼…”

“…An insight provided by our theory is that the effectiveness of introducing time-varying coupling depends critically on the curvature of the master stability function 29 at its first zero, which extends the results presented in ref. 30 . Moreover, we demonstrate analytically that the condition for improved synchronizability in temporal networks is universally satisfied by coupled one-dimensional maps.…”

Connection adaption for control of networked mobile chaotic agents

“… 30 , while a study for slow-switching networks is presented in ref. 31 . Further, the compound effect of different time scales between structure dynamics and unit dynamics has been studied in ref.…”