Methods of continuation and their implementation in the COCO software platform with application to delay differential equations

Zaid Ahsan, Harry Dankowicz, Mingwu Li, Jan Sieber

Research output: Contribution to journalArticlepeer-review


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 analyzed using real-time experimental data.

Original languageEnglish (US)
Pages (from-to)3181-3243
Number of pages63
JournalNonlinear Dynamics
Issue number4
StatePublished - Mar 2022


  • Adjoint equations
  • Constrained optimization
  • Delay differential equations
  • Graph representations
  • Implicitly defined manifolds
  • Lagrange multipliers
  • Multi-segment boundary-value problems
  • Phase response curves
  • Problem regularization

ASJC Scopus subject areas

  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Control and Systems Engineering


Dive into the research topics of 'Methods of continuation and their implementation in the COCO software platform with application to delay differential equations'. Together they form a unique fingerprint.

Cite this