Benjamin–Feir instabilities on directed networks

Patti, Francesca Di; Fanelli, Duccio; Miele, Filippo; Carletti, Timotéo

doi:10.1016/j.chaos.2016.11.018

Cited by 18 publications

(15 citation statements)

References 33 publications

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“…for respectively the diagonal and off-diagonal (l = m) moments. The stationary values of the moments can be analytically computed by setting to zero the time derivatives on the left hand side of equations (17)- (18) and solving the linear system that is consequently obtained. Particularly relevant for our purposes is the quantity δ i = ζ 2 i , the variance of the fluctuations displayed, around the deterministic equilibrium, on node i.…”

Section: Reaction-diffusion Dynamics On a Directed Latticementioning

confidence: 99%

Resilience for stochastic systems interacting via a quasi-degenerate network

Nicoletti

Fanelli

Zagli

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasi degenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, may turn unstable, yielding seemingly regular patterns in the concentration amount. Non-normality and quasi-degenerate networks may therefore amplify the inherent stochasticity, and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.Models of interacting populations are of paramount importance in a broad range of applications of interdisciplinary breath. Beyond the simplified arena of deterministic approaches, stochastic effects play a role of paramount importance and might yield a large of plethora of non trivial behaviors. Furthermore, to account for the inherent complexity of the existing interactions, the inspected model are embedded on a network architecture. In this paper, we show how a non-normal reaction model coupled to a directed, quasi-degenerate, network can drive a resonant amplification of the noisy component of the dynamics. This observation, that we here substantiate analytically, calls for a revised concept of resilience, the ability of a system to oppose external disturbances.

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Section: Reaction-diffusion Dynamics On a Directed Latticementioning

confidence: 99%

Resilience for stochastic systems interacting via a quasi-degenerate network

Nicoletti

Fanelli

Zagli

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Im ), have been worked out in [25], building on the recipe outlined in [21]. The conclusion of [25] are here briefly reviewed for for the sake of completeness. Indeed, λ (α)…”

Section: Diffusive Oscillators On Networkmentioning

confidence: 99%

Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

Patti

Fanelli

Miele

et al. 2018

Communications in Nonlinear Science and Numerical Simulation

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A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg-Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.

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“…In Ref. [54] the analysis has been extended to the setting where the unperturbed homogeneous solution is a LC and thus depends explicitly on time. In the following, for the sake of consistency, we will go through the analysis of Ref.…”

Section: Controlling the Instability On Balanced Directed Networkmentioning

confidence: 99%

“…In the following, for the sake of consistency, we will go through the analysis of Ref. [54] to eventually obtain the conditions that instigate the topological instability of a timedependent solution of the LC type.…”

Section: Controlling the Instability On Balanced Directed Networkmentioning

confidence: 99%

Topological stabilization for synchronized dynamics on networks

et al. 2017

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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

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Benjamin–Feir instabilities on directed networks

Cited by 18 publications

References 33 publications

Resilience for stochastic systems interacting via a quasi-degenerate network

Resilience for stochastic systems interacting via a quasi-degenerate network

Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

Topological stabilization for synchronized dynamics on networks

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