Global bifurcation map of the homogeneus states in the Gray-Scott model

Control of spatiotemporal patterns in the Gray-Scott model

This paper studies the effects of a time-delayed feedback control on the appearance and development of spatiotemporal patterns in a reaction-diffusion system. Different types of control schemes are investigated, including single-species, diagonal, and mixed control. This approach helps to unveil different dynamical regimes, which arise from chaotic state or from traveling waves. In the case of spatiotemporal chaos, the control can either stabilize uniform steady states or lead to bistability between a trivial steady state and a propagating traveling wave. Furthermore, when the basic state is a…

The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model

The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity $ε$ of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue problems is used to predict the dynamical behavior and instability mechanisms of multi-spot quasi-equilibrium patterns for the GS model in the limit $ε\to 0$. A differential algebraic ODE system for the coll…

非线性反应扩散系统中的奇异图灵分岔与空间鸭解

Singular Turing bifurcations and spatial canard solutions in nonlinear reaction-diffusion systems

本研究针对一类非线性反应扩散方程，解析证明了奇异图灵分岔产生的空间周期解为鸭解，具有快慢结构和更快的振幅增长，并揭示了其由可逆折叠鞍结奇性的真/假鸭解机制驱动。