Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci
Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci
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DOI：
10.3934/dcdss.2022036
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
P. Carter;A. Champneys
中科院分区：
文献类型：
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作者：
P. Carter;A. Champneys
研究了一类两快一慢多时间尺度动力系统，它包含一维Fitzhugh-Nagumo方程的行波解所得到的常微分方程组。所讨论的问题是在Hopf分叉中产生的小幅度周期轨道在小参数区间内经历快速幅度增长的机制，类似于鸭子爆炸。慢流形周围的鞍焦点结构意味着，随着振幅的增加，单个周期轨道会经历一系列的折叠。利用鸭子爆炸理论和希尔尼科夫分析相结合的思想，在一些一般假设下进行了分析。得到了褶皱的参数位置对奇异参数和控制鞍焦本征值的参数的依赖关系的渐近公式。结果表明，无论是对合成正规形的例子，还是对Fitzhugh-Nagumo系统的数值结果，分析结果都是一致的。
A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a canard explosion. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.