Numerical methods for control-based continuation of relaxation oscillations
Numerical methods for control-based continuation of relaxation oscillations
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DOI：
10.1007/s11071-023-08288-y
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发表时间：
2023-02-15
影响因子：
5.6
通讯作者：
Renson，Ludovic
中科院分区：
文献类型：
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作者：
Blyth，Mark;Tsaneva-Atanasova，Krasimira;Renson，Ludovic
基于控制的延拓（CBC）是一种实验方法，可以揭示物理系统的稳定和不稳定动力学。它将数值延拓的路径跟踪原理扩展到实验，并提供系统的动力学分析，而无需数学建模。CBC在研究机械系统的分叉结构方面取得了相当大的成功。然而，该方法对于研究弛豫振荡是不实用的。需要大量的傅立叶模式来描述它们，并且当使用许多傅立叶模式时，实验的长度显著增加，因为系统必须运行多次才能收敛。此外，弛豫振荡经常出现在自治系统中，需要适当的相位约束。为了克服这些挑战，我们引入了自适应B样条离散化，可以产生一个简约的描述，否则需要许多傅立叶模式的响应。我们耦合到一个新的相位约束，锁相控制目标和解决方案的阶段。结果表明，一个慢-快的合成基因网络和俄勒冈模型的模拟。我们的方法将CBC扩展到比迄今为止所研究的更广泛的系统，在慢-快系统上开辟了一系列新的实验机会。
Control-based continuation (CBC) is an experimental method that can reveal stable and unstable dynamics of physical systems. It extends the path-following principles of numerical continuation to experiments and provides systematic dynamical analyses without the need for mathematical modelling. CBC has seen considerable success in studying the bifurcation structure of mechanical systems. Nevertheless, the method is not practical for studying relaxation oscillations. Large numbers of Fourier modes are required to describe them, and the length of the experiment significantly increases when many Fourier modes are used, as the system must be run to convergence many times. Furthermore, relaxation oscillations often arise in autonomous systems, for which an appropriate phase constraint is required. To overcome these challenges, we introduce an adaptive B-spline discretisation that can produce a parsimonious description of responses that would otherwise require many Fourier modes. We couple this to a novel phase constraint that phase-locks control target and solution phase. Results are demonstrated on simulations of a slow-fast synthetic gene network and an Oregonator model. Our methods extend CBC to a much broader range of systems than have been studied so far, opening up a range of novel experimental opportunities on slow-fast systems.