Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction—diffusion medium
Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction—diffusion medium
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可激发反应-扩散介质中孤立脉冲的数值分岔和稳定性分析
DOI：
10.1016/s0045-7825(98)00198-4
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发表时间：
1999
影响因子：
7.2
通讯作者：
Bär Markus
中科院分区：
文献类型：
--
作者：
J. Krishnan;I. Kevrekidis;M. Or;Martin G. Zimmerman;Bär Markus
我们提出了一个系统的，计算机辅助的研究孤立脉冲的分叉和不稳定性的可激发介质能够显示稳定的脉冲传播和时空混沌动力学的参数空间的间隔。所用的反应扩散模型是活化剂-抑制剂型的;只有活化剂在这种介质中扩散。控制参数是激活剂和抑制剂动力学的时间尺度的比率和激发阈值。这项研究的重点是旅行脉冲，它们的存在域和分叉，使他们不稳定。这些脉冲被近似为：（a）同宿轨道在行波ODE框架;和（B）作为解决方案的全偏微分方程（PDE）周期性边界条件在大的域。观察到行波ODE框架中的各种分叉（包括异宿环，所谓的T点[A.R. Champneys和Y.A.余维2同宿分支的数值检测与延拓，国际学术期刊，Bif。Chaos 4（1994）785; H. Kokobu，向量场的同宿和异宿分支，日本应用数学杂志5（1988）455]）。完整PDE框架中的不稳定性包括调制行波的霍普夫分叉（涉及离散脉冲谱）以及涉及连续谱的跃迁（诸如所谓的“回火”跃迁[M. Bär，M. Hildebrand，M. M. Falcke，H. Engel和M. Neufeld，表面反应模型中的化学湍流和驻波：全球耦合和波不稳定性的影响，混沌4（1994）499]）。调制脉冲的稳定性计算通过数值Floquet分析和级联的倍周期分岔观察，以及某些全球性的分岔。这些结果，证实了直接数值积分的观测，提供了一个“骨架”周围的PDE的整体复杂的时空动力学的许多功能组织。
We present a systematic, computer-assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space. The reaction—diffusion model used is of the activator-inhibitor type; only the activator diffuses in this medium. The control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold. This study focuses on travelling pulses, their domain of existence and the bifurcations that render them unstable. These pulses are approximated as: (a) homoclinic orbits in a travelling wave ODE frame; and (b) as solutions of the full partial differential equation (PDE) with periodic boundary conditions in large domains. A variety of bifurcations in the travelling wave ODE frame are observed (including heteroclinic loops, so-called T-points [A.R. Champneys and Y.A. Kuznetsov, Numerical detection and continuation of codimension-2 homoclinic bifurcations, Int. J. Bif. Chaos 4 (1994) 785; H. Kokobu, Homoclinic and heteroclinic bifurcations of vectorfields, Japan J. Appl. Math. 5 (1988) 455]). Instabilities in the full PDE frame include both Hopf bifurcations to modulated travelling waves (involving the discrete pulse spectrum) as well as transitions involving the continuous spectrum (such as the so-called ‘backfiring’ transition [M. Bär, M. Hildebrand, M. Eiswirth, M. Falcke, H. Engel and M. Neufeld, Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities, Chaos 4 (1994) 499]). The stability of modulated pulses is computed through numerical Floquet analysis and a cascade of period doubling bifurcations is observed, as well as certain global bifurcations. These results, corroborated by observations from direct numerical integration, provide a ‘skeleton’ around which many features of the overall complex spatiotemporal dynamics of the PDE are organized.