Topological stabilization for synchronized dynamics on networks

Abstract

A general scheme is proposed and tested to control the symmetry breaking instability of a homogeneous solution of a spatially extended multispecies model, defined on a network. The inherent discreteness of the space makes it possible to act on the topology of the inter-nodes contacts to achieve the desired degree of stabilization, without altering the dynamical parameters of the model. Both symmetric and asymmetric couplings are considered. In this latter setting the web of contacts is assumed to be balanced, for the homogeneous equilibrium to exist. The performance of the proposed method are assessed, assuming the Complex Ginzburg-Landau equation as a reference model. In this case, the implemented control allows one to stabilize the synchronous limit cycle, hence time-dependent, uniform solution. A system of coupled real Ginzburg-Landau equations is also investigated to obtain the topological stabilization of a homogeneous and constant fixed point. PACS numbers: 89.75.-k 89.75.Kd 89.75.Fb 02.30.Yy 05.45.Xt

Spectral control for ecological stability

“…Notably, the corrections δΛ can be chosen such that matrix D is also zero-row-sum, as D is. This is rigorously proven in Appendix C, building on the derivation reported in [13]. Moreover, D is real: this property is also inherited from D, as shown again in Appendix C. We are thus brought back to the linear problem: (6) or, equivalently tov i = j C ij v j , where:…”

“…Starting from these premises, in this paper we provide an alternative approach to ecological stability by developing a self-consistent mathematical strategy which implements a spectral control algorithm. The method builds on the technique developed in [13] and extends its domain of applicability, beyond diffusion mediated (linear) processes to the interesting setting where pairwise, hence non-linear, non-local interactions are considered. In doing so, we will contribute to identifying the key topological features that should be possessed by a stable (resilient) ecological network.…”

Stability of synchronization in simplicial complexes

“…In particular, synchronous behaviors have been observed and characterized in small-world 23 , weighted 24 , multilayer 25 , and adaptive networks 26 , 27 . Outside complete synchronization, moreover, other types of synchronization have been revealed to emerge in networked systems, including remote synchronization 28 , 29 , cluster states 30 and synchronization of group of nodes 31 , chimera 32 , 33 , Bellerophon states 34 , 35 , and Benjamin–Feir instabilities 36 – 38 . Finally, the transition to synchronization has been shown to be either smooth and reversible, or abrupt and irreversible (as in the case of explosive synchronization, resembling a first-order like phase transition 39 ).…”

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.…”