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Eigenvalues and delay differential equations: periodic coefficients, impulses and rigorous numerics
Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2020-10-08 , DOI: 10.1007/s10884-020-09900-0
Kevin E. M. Church

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We consider systems with a single delay where the period is at most equal to the delay, and the latter two are commensurate. We first represent the monodromy operator (period map) as an operator acting on a product of sequence spaces that represent the Chebyshev coefficients of the state-space vectors. Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation error in addition to some other technical operator norms, this leads to the method being suitable to computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs on two example problems. We also test our discretization scheme in floating point arithmetic on a gamut of randomly-generated high-dimensional examples with both periodic and constant coefficients to inspect the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check for periodic systems) for increasing numbers of Chebyshev modes.



中文翻译:

特征值和时滞微分方程:周期系数,脉冲和严格的数值

我们开发了经过验证的数值方法来计算Floquet均衡器乘积和时滞微分方程以及脉冲时滞微分方程的周期解。使用我们的方法,人们可以精确地计算以零为中心的闭盘外的Floquet乘法器的数量,或者紧凑集合中以零为界的乘法器的数量。我们考虑具有单个延迟的系统,其中周期最多等于延迟,而后两个延迟是相当的。我们首先将单峰算子(周期图)表示为对表示状态空间向量的切比雪夫系数的序列空间乘积起作用的算子。截断模式数量产生了数值方法,并通过仔细限制除其他一些技术操作员规范之外的截断误差,这导致该方法适用于Floquet乘数位置的计算机辅助证明。我们演示了两个示例问题的计算机辅助证明。我们还使用浮点算法对离散生成的离散化方案进行了测试,该离散化方案是在随机产生的具有周期系数和常数系数的高维实例的整个范围上进行的,以检查单峰算子的谱半径估计的精度(即对周期系统的稳定性/不稳定性检查)用于增加Chebyshev模式的数量。

更新日期:2020-10-08
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