High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays

IF 2.5 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Foundations of Computational Mathematics Pub Date : 2023-06-09 DOI:10.1007/s10208-023-09614-x

Robert Szczelina, Piotr Zgliczyński

引用次数: 0

Abstract

Abstract We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).

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多时滞时滞微分方程系统poincar映射严格计算的高阶lohner型算法

摘要提出了多时滞时滞微分方程(DDEs)系统严格积分的lohner型算法，并将其应用于poincar映射的计算，研究了一类有界永恒解的动力学问题。该算法基于相空间中解的分段泰勒表示，并利用DDEs中解的平滑来产生高阶解的外壳。我们应用拓扑技术来证明各种动力学行为，例如，在麦基-格拉斯方程中(在数值上观察到混沌的参数区)存在(显然)不稳定的周期轨道，以及延迟摄动混沌ODE (Rössler系统)中符号动力学的持久性。

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来源期刊

Foundations of Computational Mathematics 数学-计算机：理论方法

CiteScore

6.90

自引率

3.30%

发文量

审稿时长

>12 weeks

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