Methods of continuation and their implementation in the COCO software platform with application to delay differential equations
Methods of continuation and their implementation in the COCO software platform with application to delay differential equations
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DOI：
10.1007/s11071-021-06841-1
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发表时间：
2021-04
影响因子：
5.6
通讯作者：
Zaid Ahsan;H. Dankowicz;Mingwu Li;J. Sieber
中科院分区：
文献类型：
--
作者：
Zaid Ahsan;H. Dankowicz;Mingwu Li;J. Sieber
本文全面讨论了适用于参数延拓技术的非线性动力学和约束优化问题的构造，并特别强调了具有延迟的多段边值问题。讨论的背景是生态软件包及其对社区驱动的发展的明确支持。为此，本文首先形式化的协同构造范式增广连续问题兼容的同时分析隐式定义的流形的解决方案的非线性方程和相应的伴随变量与优化的标量目标函数沿着这样的流形。本文使用应用程序的数据同化从有限的时间历史和相位响应分析的周期性轨道，以确定一个普遍的范式建设，允许抽象和概括。然后，它详细的理论框架acoco兼容的工具箱能够支持延迟耦合多段边值问题，包括周期轨道，准周期轨道，连接轨道，初值问题和最优控制问题的分析，如一套数值例子所示。本文的目的是提出一种教学治疗，是访问的新手和启发专家呼吁应用非线性动力学的许多感官。散布在一个系统的讨论问题的建设，图形表示的延迟耦合问题，和矢量化公式的问题离散化，本文包括一个原始的推导使用拉格朗日灵敏度分析的相位响应泛函的轨道问题在抽象的Banach空间，以及使用真实的-时间实验数据。
This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of thecocosoftware package and its explicit support for community-driven development. To this end, the paper first formalizes thecococonstruction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corresponding adjoint variables associated with optimization of scalar objective functions along such manifolds. The paper uses applications to data assimilation from finite time histories and phase response analysis of periodic orbits to identify a universal paradigm of construction that permits abstraction and generalization. It then details the theoretical framework for acoco-compatible toolbox able to support the analysis of a large family of delay-coupled multi-segment boundary-value problems, including periodic orbits, quasiperiodic orbits, connecting orbits, initial-value problems, and optimal control problems, as illustrated in a suite of numerical examples. The paper aims to present a pedagogical treatment that is accessible to the novice and inspiring to the expert by appealing to the many senses of the applied nonlinear dynamicist. Sprinkled among a systematic discussion of problem construction, graph representations of delay-coupled problems, and vectorized formulas for problem discretization, the paper includes an original derivation using Lagrangian sensitivity analysis of phase response functionals for periodic-orbit problems in abstract Banach spaces, as well as a demonstration of the regularizing benefits of multi-dimensional manifold continuation for near-singular problems analyzed using real-time experimental data.