Methods of continuation and their implementation in the COCO software platform with application to delay differential equations
Ahsan, Z; Dankowicz, H; Li, M; et al.Sieber, J
Date: 27 January 2022
Journal
Nonlinear Dynamics
Publisher
Springer
Publisher DOI
Abstract
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 the COCO software package and its explicit ...
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 the COCO software package and its explicit support for community-driven development. To this end, the paper first formalizes the COCO construction 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 a COCO-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 delaycoupled 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.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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