## Abstract

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. ^{1}The code included in this paper constitutes fully executable scripts. Complete code, including that used to generate the results in Fig. 1, is available at https://github.com/jansieber/adjoint-sensitivity2022-supp. 2The equation Φ = 0 on U is said to be regular with dimensional deficit d at a solution point u˜ if there exists a function Ψ: U → Rd such that the map F: u ›→ (Φ(u),Ψ(u)) is continuously Frech´et differentiable on a neighborhood of u˜ and DF (u˜) has a bounded inverse. ^{3}In the absence of explicit encodings of second derivatives, coco relies on a suitable finitedifference approximation of these derivatives, as necessary. 4Here, the dual space R_{Φ}∗ is the space of functions of bounded variation. We restrict attention to the subspace of continuously differentiable functions λ (·) to allow the use of integration of parts when evaluating variations of L. 5The indicator function 1r: S → { 0, 1} is nonzero on |φ|<r/ 2 (appropriately defined in the metric on S). ^{6}See the reviews [9, 17] for the use of generalized implicit function theorems and Diophantine conditions on ρ (i.e., | exp(2π ikρ)− 1| ≥ C_{Diop}|k|−ν for all k/= 0 and some constants C_{Diop} > 0 and ν > 0) to establish existence of invariant tori with parallel flows and irrational rotation numbers. A general treatment for when formal expansions (such as (259)) permit one to establish the existence of invariant manifolds with a certain degree of regularity is given in [4, 5, 6]. The more recent monograph [12] develops numerical algorithms with rigorous error bounds for computing invariant manifolds (such as quasiperiodic tori) in the presence of unbounded inverses in (242) and small divisors.

Original language | English (US) |
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Pages (from-to) | 329-369 |

Number of pages | 41 |

Journal | Journal of Computational Dynamics |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 2022 |

Externally published | Yes |

## Keywords

- Hybrid systems
- constraint Lagrangian
- numerical continuation
- persistence
- software implementation

## ASJC Scopus subject areas

- Computational Mechanics
- Computational Mathematics