Olsen模型的多尺度几何与非经典弛豫振荡
Multiscale Geometry of the Olsen Model and Non-Classical Relaxation Oscillations
摘要：
研究了过氧化物酶-氧化酶反应的Olsen模型。使用基于多个时间尺度的几何分解来分析动力学。Olsen模型是四维的，不是几何奇异扰动理论所要求的标准形式，并且包含多个小参数。这三个障碍是我们通过分析解决的主要挑战。标度和爆破方法被用来识别几个子系统。这里给出的结果提供了一个严格的分析两个振荡模式。特别地，我们证明了非经典松弛振荡在两种情况下的存在。分析的基础上，结合可积松弛阶段的跨临界线和子流形的折叠奇点的desingularization。在这种情况下，我们的分析也解释了一个假设，已被利用，纯粹基于数值推理，在以前的分歧分析由Desroches，Krauskopf和Osinga [{离散。}【继续】【动态】{系统} S，2（4），p.807--827，2009]。此外，我们开发的几何分解的基础上，证明存在的混合模式和混沌振荡的奥尔森模型，这将在未来的工作中更详细地讨论。
英文摘要：
We study the Olsen model for the peroxidase-oxidase reaction. The dynamics is analyzed using a geometric decomposition based upon multiple time scales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. These three obstacles are the main challenges we resolve by our analysis. Scaling and the blow-up method are used to identify several subsystems. The results presented here provide a rigorous analysis for two oscillatory modes. In particular, we prove the existence of non-classical relaxation oscillations in two cases. The analysis is based upon desingularization of lines of transcritical and submanifolds of fold singularities in combination with an integrable relaxation phase. In this context our analysis also explains an assumption that has been utilized, based purely on numerical reasoning, in a previous bifurcation analysis by Desroches, Krauskopf and Osinga [{Discret.}{Contin.}{Dyn.}{Syst.}S, 2(4), p.807--827, 2009]. Furthermore, the geometric decomposition we develop forms the basis to prove the existence of mixed-mode and chaotic oscillations in the Olsen model, which will be discussed in more detail in future work.
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