Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations

Abstract

The coupling of spatial and temporal symmetry breaking instabilities is studied in a two-variable reactiondiffusion model describing semiconductor transport. A variety of spatiotemporal patterns corresponding to pure Hopf and Turing modes, localized patterns, and mixed Turing-Hopf modes including subharmonic spatiotemporal spiking are found. By organizing the results in a time-scale versus space-scale diagram, and by comparing them with a chemical reaction-diffusion model, it is shown that such behavior is generic for a class of extended nonlinear dynamic systems near codimension-two Turing-Hopf bifurcations.

Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations

“…In this kind of bifurcation the formation of inhomogeneous stationary patterns caused by Turing instabilities "interacts" with the appearance of oscillations due to a Hopf bifurcation. This mechanism of the emergence of spatio-temporal patterns has been studied intensively during the last years in physical and chemical system (De Wit et al (1996);Meixner et al (1997)). …”

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

“…In the neighbourhood of a codimension-two TH point, chaotic 2D dynamics has also been observed both experimentally [5] and numerically [49] although a theoretical characterization of the mechanisms leading to chaos has only rarely been provided [1]. In this respect, within the ecological context, a fascinating debate is currently going on: see e.g.…”

“…The coupling between these two kinds of instabilities was first theoretically investigated in the 90's in the framework of the Lengyel-Epstein model of the CIMA reaction [56]. Since then it has become a key topic for pattern formation in the reaction-diffusion framework and has been extensively investigated in chemistry, physics and biology [2,3,24,33,48,49,55].…”

Temperature control of pattern formation in the Ru(bpy)32+-catalyzed BZ-AOT system

“…The Turing-Hopf interaction has been the subject of active theoretical research in recent years. 28,[32][33][34][35][36][37][38][39][40][41] The results presented here show what kinds of patterns can be found in actual experimental systems as a result of this interaction. One of these patterns, the reversed oscillatory Turing pattern, has, to the best of our knowledge, been seen here for the first time.…”