  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 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.
