Theory of Turing Patterns on Time Varying Networks

Abstract

The process of pattern formation for a multispecies model anchored on a time varying network is studied. A nonhomogeneous perturbation superposed to an homogeneous stable fixed point can be amplified following the Turing mechanism of instability, solely instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation to derive a closed analytical prediction for the onset of the instability in the time dependent framework. Continuously and piecewise constant periodic time varying networks are analyzed, setting the framework for the proposed approach. The extension to nonperiodic settings is also discussed.

Turing patterns in a network-reduced FitzHugh-Nagumo model

“…[12,13], enabling to better capture geometries present in many real cases. Since then, a number of generalisations to various types of networks and dynamics has been made, including asymmetric, multiplex, time-varying, time-delayed, and non-normal networks [14][15][16][17][18][19], and intriguing relations between self-organisation and network topology have been revealed.…”

Classes of random walks on temporal networks with competing timescales

“…Remark 1. (On random walk time-dependent Laplacian) In previous works dealing with synchronisation [27,28], desynchronisation [29], instabilities in reaction-diffusion systems [30] and other works about dynamical systems on time-varying networks [14,31], a time-dependent Laplacian L(t ) has replaced the usual graph Laplacian L in the equations. Here we want to comment on (5) in this time-varying setting, that is,…”

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

“…For network (6), suppose Assumptions 1 and 2 hold, and linear matrices are bounded with ||Γ 1 || ≤ 1 , ||Γ 2 || ≤ 2 , where 1 and 2 are positive constants. Under the action of adaptive aperiodically intermittent controllers (4) and adaptive updating laws (5), if there exist positive constants , , , L f , , and 0 < < 1, where is defined in Definition 2, such that…”

“…[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.…”