Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems

摘要: Abstract We present three examples of delayed bifurcations for spike solutions reaction–diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analyzed in context ODE’s Mandel Erneux (1987). It found that instability would not be fully realized until had entered well into regime. bifurcation is said have been “delayed” relative threshold value computed directly linear stability analysis. In contrast study Erneux, we analyze systems partial differential equations (PDE’s). particular, singularly perturbed generalized Gierer–Meinhardt Gray–Scott models, resulting slow passage regimes oscillatory competition instability. first example, model on infinite real line, tuning control parameter through Hopf bifurcation. second consider finite one-dimensional domain. this scenario, opposed extrinsic value, triggered by intrinsic dynamics PDE system. third instabilities feed rate parameter. all cases, find must pass regime before onset observed, indicating delay. also has important eventual give analytic predictions magnitude delays obtained analysis certain explicitly solvable nonlocal eigenvalue problems (NLEP’s). theory confirmed numerical full