Brøns M, Sturis J
Department of Mathematics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Aug;64(2 Pt 2):026209. doi: 10.1103/PhysRevE.64.026209. Epub 2001 Jul 18.
We consider a simple model of an autocatalytic chemical reaction where a limit cycle rapidly increases to infinite period and amplitude, and disappears under variation of a parameter. We show that this bifurcation can be understood from seeing the system as a singular perturbation problem, and we find the bifurcation point by an asymptotic analysis. Scaling laws for period and amplitude are derived. The unphysical bifurcation to infinity disappears under generic modifications of the model, and for a simple example we show is replaced by a canard explosion, that is, a narrow parameter interval with an explosive growth of the amplitude. The bifurcation to infinity introduces a strong sensitivity that may result in chaotic dynamics if diffusion is added. We show that this behavior persists even if the kinetics is modified to preclude the bifurcation to infinity.
我们考虑一个自催化化学反应的简单模型，其中一个极限环迅速增加到无限周期和振幅，并在一个参数变化时消失。我们表明，这种分岔可以通过将系统视为奇异摄动问题来理解，并且我们通过渐近分析找到分岔点。推导了周期和振幅的标度律。在对模型进行一般修改后，到无穷大的非物理分岔消失了，并且对于一个简单的例子，我们表明它被一个鸭式爆炸所取代，即一个振幅呈爆炸式增长的狭窄参数区间。到无穷大的分岔引入了强烈的敏感性，如果加入扩散，可能会导致混沌动力学。我们表明，即使修改动力学以排除到无穷大的分岔，这种行为仍然存在。
