2017/02/07 by Francesca Di Patti, Duccio Fanelli, Filippo Miele +2
Computer Science · Mathematics · Physics and Astronomy · #Asymmetry #Bifurcation #Brusselator #Computer science #Geometry #Hopf bifurcation #Instability #Mathematical analysis #Mathematics #Mechanics #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Physics #Quantum mechanics #Reaction–diffusion system #Stability (learning theory) #Statistical physics #Symmetry (geometry) #Symmetry breaking #cond-mat.stat-mech #nlin.PS #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/j.cnsns.2017.08.012
arxiv created 2017/02/07 · openalex publication_date 2017/09/05 · arxiv updated 2017/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.