Stationary patterns in star networks of bistable units: Theory and application to chemical reactions

Abstract

We present theoretical and experimental studies on pattern formation with bistable dynamical units coupled in a star network configuration. By applying a localized perturbation to the central or the peripheral elements, we demonstrate the subsequent spreading, pinning, or retraction of the activations; such analysis enables the characterization of the formation of stationary patterns of localized activity. The results are interpreted with a theoretical analysis of a simplified bistable reaction-diffusion model. Weak coupling results in trivial pinned states where the activation cannot propagate. At strong coupling, a uniform state is expected with active or inactive elements at small or large degree networks, respectively. A nontrivial stationary spatial pattern, corresponding to an activation pinning, is predicted to occur at an intermediate number of peripheral elements and at intermediate coupling strengths, where the central activation of the network is pinned, but the peripheral activation propagates toward the center. The results are confirmed in experiments with star networks of bistable electrochemical reactions. The experiments confirm the existence of the stationary spatial patterns and the dependence of coupling strength on the number of peripheral elements for transitions between pinned and retreating or spreading fronts in forced network configurations (where the central or periphery elements are forced to maintain their states).

Localized patterns in star networks of class-B lasers with optoelectronic feedback

“…In close analogy to previous findings in electrochemical bistable networks [19,20] Fig. 4 shows that the coupling strength required for a transition to occur to the orange line, drops with η (see Fig.…”

“…Figure 3 As a result of the above stability analysis, when the system starts at the passive-active state, both lasers jump to the active state (green branch) through a SN bifurcation at rather low coupling strengths η > 0.008819. This resembles the chemical bistable media where an activation front can propagate thus activating the passive nodes [19,20]. On the other hand, when the system is prepared in the passive-passive state, higher values η > 0.04922 are required for the lasers to jump to the active-active state, and this transition takes place through a PB bifurcation.…”

“…Previous studies with electrochemical systems [19,20,27] theoretical and experimental studies, it was demonstrated that such a reduced system could produce all the rich dynamics of the original network despite its simpler form.…”

Synchronization in nonlinear complex networks with multiple time‐varying delays via adaptive aperiodically intermittent control

“…[3][4][5] In recent years, many novel achievements of complex networks such as the universal resilience patterns, stationary patterns, Turing patterns, feedback-induced stationary localized patterns, stability and control synchronization, and so on have been proposed and further improved our understanding of the properties and mechanism of network structure. [6][7][8][9][10][11][12][13][14] The deep exploration of network structure features and functions and the scientific understanding and applications of network dynamics have become the frontier subject of the current network science.…”

Intertangled stochastic motifs in networks of excitatory-inhibitory units

“…One can then assemble large networks with designated topology, by replicating the aforementioned module on each node of the collection and incorporating the specific nature of the coupling [17]. Stationary patterns [18] of asynchronous activity for the scrutinized species can eventually emerge, following symmetry breaking instabilities that necessitate a fine tuning of the parameters involved. These patterns could define the basic architectural units for natural systems to perform efficient computations [19][20][21].…”