Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources

Abstract

We study the pattern generating mechanism of a generalized Gierer-Meinhardt model with diffusions. We show the existence and stability of the Hopf bifurcation for the corresponding kinetic system under certain conditions. With spatial uneven diffusions, the obtained stable Hopf periodic solution may become unstable, which results in Turing instability. We derive conditions for the existence of Turing instability. Numerical simulations reveal that the Turing patterns are of stripe and spot shapes. In the analysis, we use bifurcation analysis, center manifold reduction for ordinary differential equations and partial differential equations. Though the Gierer-Meinhardt system is classical, our system with more general settings has yet to be analyzed in the literature.

Reaction-diffusion effects and spatiotemporal oscillations under SEM, STM and AFM-assisted charging in fiber-like and wire-like systems: From molecular and quantum wires to cooperative ferroelectric nanofibers and microfibers

“…Consequently, SEM technique is more optimal for the analysis of cooperatively driven ferroelectric fibers then AFM. It is quite obvious that oscillations observed in such systems are oscillations in distributed systems, and the corresponding models of the dynamics of reaction-diffusion processes in such systems should be interpreted as the models of 3D (4D) processes in distributed systems with spatiotemporal reactions under the electron beam, which is a control agent for the wave or pulse propagation with different charges and polarities (as Turing activators and inhibitors in the classical approaches [110][111][112][113][114][115][116][117][118][119][120][121] ).…”

Stability, Discretization, and Bifurcation Analysis for a Chemical Reaction System

“…Wilhelm and Heinrich [18] proposed a bimolecular chemical reaction model with the smallest possible reaction system for appearance of Hopf bifurcation. Wang et al [19] studied Gierer-Meinhardt type chemical reaction system incorporating pattern formation mechanism and emergence of Hopf bifurcation. Dutt [20] reported oscillatory behavior for a diffusion type chemical reaction model.…”