Low-Dimensional Galerkin Approximations of Nonlinear Delay Differential Equations
Low-Dimensional Galerkin Approximations of Nonlinear Delay Differential Equations
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非线性时滞微分方程的低维伽辽金近似
DOI：
10.3934/dcds.2016.36.4133
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Shouhong Wang
中科院分区：
文献类型：
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作者：
M. Chekroun;M. Ghil;Honghu Liu;Shouhong Wang
本文重新讨论了非线性时滞微分方程组的常微分方程逼近问题。我们的工作在Hilbert空间赋予了一个自然的内积，包括一个点质量，并引入多项式正交相对于这样的内积，生活在域中的线性算子与相关的基础DDE。然后，这些多项式被用来设计一个一般的Galerkin计划，我们推导出严格的收敛结果，并表明它可以通过简单的解析公式数值实现。这样得到的计划适用于三个非线性DDE，两个自治和一个强制：（i）一个简单的DDE与分布延迟的解决方案回忆布朗运动;（ii）一个DDE与离散延迟，表现出双峰和混沌动力学;和（iii）一个周期性强制DDE与两个离散延迟产生的气候动力学。在所有这三种情况下，本文介绍的Galerkin计划提供了一个很好的逼近DDE的奇怪吸引子的低维ODE系统，以及其非线性动力学的统计特征。
This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.