Universal description of stochastic oscillators - higher dimensional examples, extraction of the mapping from data, and networks of oscillators

随机振荡器的通用描述 - 高维示例、从数据中提取映射以及振荡器网络

• 批准号: 539274742
• 负责人: Professor Dr. Benjamin Lindner
• 金额: --
• 依托单位: Institut für Physik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/539274742?language=en
• 关键词: Universal; description; stochastic; oscillators; higher

Many important non-equilibrium systems display oscillatory behavior that is shaped by pronounced fluctuations in phase and amplitude. Different mathematical models with distinct mechanisms have been suggested for such stochastic oscillators: (i) dynamics with a focus fixed point endowed with noise (the damped harmonic oscillator with thermal fluctuations would be a prominent example); (ii) limit cycles and relaxation oscillators perturbed by white noise; (iii) excitable systems driven by fluctuations; (iv) heteroclinic systems driven by noise. Most of these models are strongly nonlinear and are thus difficult to treat theoretically. A recent theory, put forward by my collaborators and me, provides a universal framework for the description of stochastic oscillators in terms of one of the eigenfunctions of the backward Kolmogorov operator. After projecting the system's variables to a new complex-valued variable, the spontaneous fluctuation statistics, the linear response of the oscillator, and its cross-correlation with another stochastic oscillator to which it is coupled, are all captured by simple and exact analytical expressions. This description is independent of the mechanism for the stochastic oscillation. In this project I want to apply and generalize this theory in three respects: (i) it should be applied to oscillators that have higher dimensions than two going beyond what has been studied before; (ii) the nonlinear transformation at the core of the procedure should be extracted from (simulation or experimental) data; (iii) the theory for coupled stochastic oscillators will be more thoroughly grounded and, moreover, extended to the case of networks of stochastic oscillators (more than two oscillators, the case previously investigated). These explorations and extensions of the framework have potential applications for the theoretical description of many systems outside of thermodynamic equilibrium, in particular in biology.

许多重要的非平衡系统显示出由相位和振幅的显著波动所形成的振荡行为。对于这样的随机振子，已经提出了具有不同机制的不同数学模型：（i）具有赋予噪声的焦点不动点的动力学（具有热涨落的阻尼谐振子将是一个突出的例子）;（ii）受白色噪声扰动的极限环和弛豫振子;（iii）由涨落驱动的可激发系统;（iv）由噪声驱动的异宿系统。这些模型大多是强非线性的，因此很难在理论上处理。最近的一个理论，提出了我的合作者和我，提供了一个通用的框架来描述随机振荡器的一个特征函数的向后柯尔莫哥洛夫运营商。在将系统的变量投影到一个新的复值变量后，自发波动统计，振荡器的线性响应，以及它与它所耦合的另一个随机振荡器的互相关，都被简单而精确的解析表达式所捕获。这种描述与随机振荡的机制无关。 在这个项目中，我想在三个方面应用和推广这个理论：（i）它应该被应用到比以前研究的更高维度的振子;（ii）在过程的核心的非线性变换应该被提取出来。（模拟或实验）数据;（iii）耦合随机振子的理论将更彻底地扎根，而且，将扩展到随机振子网络的情况。（两个以上的振荡器，以前调查的情况）。这些探索和扩展的框架有潜在的应用程序的热力学平衡以外的许多系统的理论描述，特别是在生物学。