论文标题
部分可观测时空混沌系统的无模型预测
Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions
论文作者
论文摘要
可以通过多项式功能对物理，经济学，化学和生物学的许多系统的动力学进行建模。在本文中，我们提供了一种计算手段，通过使用半芬特编程来求解平方之和的程序（SOS）程序，以找到一组多项式动态系统。随着SOS程序的出现，可以有效地搜索Lyapunov功能，以保证多项式系统的稳定性。但是，SOS计算通常无法找到功能，因此条件在整个状态空间中的存在。我们在这里表明，将SOS优化限制为特定域，使我们能够获得积极的不变集，从而通过单独考虑每个正相不变的集合来促进动力学的分析。此外，我们超越了经典的Lyapunov稳定性分析，并使用SOS分解来计算实施足够的阳性条件，以保证限制周期的存在，独特性和指数稳定性。重要的是，这种方法适用于任何维度的系统，因此超出了仅限于二维相空间的经典方法。我们通过应用于经典系统的应用来说明我们的不同结果，例如Van der Pol振荡器，Fitzhugh-Nagumo神经元方程和Lorenz系统。
The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by using semidefinite programming to solve sum-of-squares (SOS) programmes. With the emergence of SOS programmes, it is possible to efficiently search for Lyapunov functions that guarantee stability of polynomial systems. Yet, SOS computations often fail to find functions, such that the conditions hold in the entire state space. We show here that restricting the SOS optimisation to specific domains enables us to obtain positively invariant sets, thus facilitating the analysis of the dynamics by considering separately each positively invariant set. In addition, we go beyond classical Lyapunov stability analysis and use SOS decompositions to computationally implement sufficient positivity conditions that guarantee existence, uniqueness, and exponential stability of a limit cycle. Importantly, this approach is applicable to systems of any dimension and, thus, goes beyond classical methods that are restricted to two-dimensional phase space. We illustrate our different results with applications to classical systems, such as the van der Pol oscillator, the Fitzhugh-Nagumo neuronal equation, and the Lorenz system.
