Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
H. Dankowicz, J. Sieber
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引用次数: 0

Abstract

This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. 200 words.
用伴随法分析周期轨道和准周期不变环面灵敏度
本文提出了一个严格的框架,用于非线性约束解的延拓,并使用基于伴随的方法同时分析测试函数在每个解点对约束违反的敏感性。通过相关拉格朗日乘子中问题拉格朗日量的线性,形式主义被证明可以直接适用于使用COCO软件包进行分析,特别是它的阶段问题构建范式。一般理论是在代数方程和边值问题的背景下说明的,重点是光滑和混合动力系统中的周期轨道,以及流动的准周期不变环面。在后一种情况下,尽管控制边值问题的线性化缺乏一般理论所要求的有界逆,但使用正规双曲性来证明与轨道周期对参数扰动和约束违反的敏感性相关的伴随条件的连续解的存在性。横向稳定性的假设则意味着这些解预测了基于初始条件下远离环面扰动的轨迹的渐近相位。示例COCO代码用于说明将灵敏度分析附加到正则参数延拓所需的最小额外设置成本投资。200个单词。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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