Extended networks as a route of stabilization of divergent dynamics

This work examines the dynamical states of coupled Hénon maps that arise due to the network's coupling configuration in a ring topology. The parameters for each individual node are selected in a way that ensures the absence of stable attractors in phase space, such that in the event of synchronization across the network, all maps exhibit divergence after a brief transient period. However, contrary to what one would expect, we find that the coupled network demonstrates the ability to stabilize and produce non-divergent dynamics, depending on the coupling strength and radius. Thus, the dynamical states observed following the transient phase are exclusively a consequence of the network's coupling. Using spatial recurrence matrix, the study correlates nondivergent dynamics with parameter regions prone to chimera and incoherent states, demonstrating multistability for certain coupling strengths and showing that individual nodes’ dynamics remain close to the chaotic saddle of the uncoupled maps. The paper is organized to discuss the Hénon map, coupling mechanisms, characterization of nondivergent states and dynamical switch states transitions.