2005/02/24 by Rui Dilão, Rui Dilao, Dilao, Rui · 2 citations
Computer Science · Mathematics · Medicine · Physics and Astronomy · #Chaos control and synchronization #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0502047
27 pages, 5 figures
arxiv created 2005/02/24 · openalex publication_date 2005/02/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from vector fields with one fixed point and a supercritical Hopf bifurcation. For the Brusselator and the Ginzburg-Landau reaction-diffusion equations, we obtain the bifurcation diagrams associated with the transition between time periodic solutions and asymptotically stable solutions (Turing patterns). In two-component systems of reaction-diffusion equations, we show that the existence of Turing instabilities is neither necessary nor sufficient for the existence of Turing pattern type solutions. Turing patterns can exist on both sides of the Hopf bifurcation associated to the local vector field, and, depending on the initial conditions, time periodic and stable solutions can coexist.