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

Francesca Di Patti, Duccio Fanelli, Filippo Miele, Timoteo Carletti

Research output: Contribution to journalArticle

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Abstract

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.
Original languageEnglish
Pages (from-to)447-456
JournalCommunication in Nonlinear Science and Numerical Simulation
Volume56
DOIs
Publication statusPublished - 16 Aug 2017

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Complex Ginzburg-Landau Equation
Hopf bifurcation
Ginzburg-Landau
Directed graphs
Directed Graph
Reaction-diffusion System
Hopf Bifurcation
Approximation
Oscillating Solutions
Symmetry Breaking
Normal Form
Probe
Adjacent
Coefficient
Vertex of a graph
Model

Keywords

  • Reaction-diffusion model
  • Complex Ginzburg–Landau equation
  • Pattern formation
  • Synchronization

Cite this

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title = "Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs",
abstract = "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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Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs. / Di Patti, Francesca ; Fanelli, Duccio; Miele, Filippo; Carletti, Timoteo.

In: Communication in Nonlinear Science and Numerical Simulation, Vol. 56, 16.08.2017, p. 447-456.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

AU - Di Patti, Francesca

AU - Fanelli, Duccio

AU - Miele, Filippo

AU - Carletti, Timoteo

PY - 2017/8/16

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N2 - 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.

AB - 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.

KW - Reaction-diffusion model

KW - Complex Ginzburg–Landau equation

KW - Pattern formation

KW - Synchronization

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DO - https://doi.org/10.1016/j.cnsns.2017.08.012

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VL - 56

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EP - 456

JO - Communication in Nonlinear Science and Numerical Simulation

JF - Communication in Nonlinear Science and Numerical Simulation

SN - 1007-5704

ER -