Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX1 2JD, UK
Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AGs, UK.
J R Soc Interface. 2017 Dec;14(137). doi: 10.1098/rsif.2017.0447.
Oscillations in dynamical systems are widely reported in multiple branches of applied mathematics. Critically, even a non-oscillatory deterministic system can produce cyclic trajectories when it is in a low copy number, stochastic regime. Common methods of finding parameter ranges for stochastically driven resonances, such as direct calculation, are cumbersome for any but the smallest networks. In this paper, we provide a systematic framework to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations relying on real root counting algorithms and graph theoretic methods. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix , unlike deterministic oscillations which are determined by By using graph theoretic tools, analysis of stochastic behaviour for larger interaction networks is simplified and stochastic dynamical systems with multiple resonant modes can be identified easily.
动力学系统中的振荡在应用数学的多个分支中都有广泛的报道。关键的是，即使是一个非振荡的确定性系统，当处于低拷贝数、随机状态时，也可以产生循环轨迹。寻找随机驱动共振参数范围的常用方法，如直接计算，对于任何除了最小的网络之外的网络都很麻烦。在本文中，我们提供了一个系统的框架，通过使用实根计数算法和图论方法，有效地确定随机振荡的共振模式数量和参数范围。我们认为随机共振是一个网络属性，通过表明共振模式只取决于平方雅可比矩阵，而不同于由确定的振荡，这是由决定的。通过使用图论工具，简化了对更大交互网络的随机行为的分析，并且可以很容易地识别具有多个共振模式的随机动力系统。
