Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method
Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method
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DOI：
10.3934/jcd.2022006
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发表时间：
2021-11
影响因子：
1
通讯作者：
H. Dankowicz;J. Sieber
中科院分区：
文献类型：
--
作者：
H. Dankowicz;J. Sieber
本文提出了一个严格的框架，用于连续求解非线性约束，并使用基于伴随的方法同时分析测试函数对每个解点的约束违规的敏感性。通过相关拉格朗日乘子中问题拉格朗日的线性，该形式表明可以直接使用 COCO 软件包进行分析，特别是其分阶段问题构建的范例。在代数方程和边值问题的背景下阐述了一般理论，重点是光滑和混合动力系统中的周期轨道以及流的准周期不变环面。在后一种情况下，正态双曲性用于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，即使控制边值问题的线性化缺乏一般理论所要求的有界逆。横向稳定性的假设意味着这些解基于远离环面扰动的初始条件来预测轨迹的渐近相。示例 COCO 代码用于说明将敏感性分析附加到常规参数延续所需的设置成本的最小额外投资。 200字。
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.