Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

Harry Dankowicz; Jan Sieber

doi:10.3934/jcd.2022006

Article Contents

2022, Volume 9, Issue 3: 329-369. Doi: 10.3934/jcd.2022006

This issue Previous Article Next Article Continuation methods for principal foliations of embedded surfaces

Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

Harry Dankowicz^1, , and
Jan Sieber^2,

1.
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 68101, USA
2.
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, United Kingdom

^* Corresponding author: danko@illinois.edu
^* Corresponding author: danko@illinois.edu

Received: October 31, 2021

Revised: February 28, 2022

Published: July 2022

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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.

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Mathematics Subject Classification: Primary: 37C55, 37M21; Secondary: 49K40, 37E10, 37E45.

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Figure 1. Illustration of sensitivity analysis along an invariant curve: (top left) Bifurcation diagram, (bottom left) spectrum of $ \Gamma_\rho $ and $ \hat{\Gamma}_\rho $, (middle column) phase plots of invariant curves at points labeled A and B, and tangents to stable fibers at selected points, (right column) differences between trajectories of initial conditions near a point on the invariant curve (black dot) and forward iterates of the corresponding stable fiber projection

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Figure 2. At each stage of construction of the two-segment periodic orbit problem in (112)-(114), one obtains an extended continuation problem (left column) in terms of a subset of the complete vector of continuation variables and an associated set of adjoint conditions (right column) in terms of a subset of the complete vector of adjoint variables

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Figure 3. A logical matrix representation of the variable dependencies of zero and monitor functions introduced at different stages of problem construction. At the stage of construction illustrated in the figure, two zero/monitor functions are added that depend on three newly introduced continuation variables. Each function also depends on previously introduced continuation variables identified by ones in the corresponding row

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Figure 4. Fields of a function data structure may be used to store context-independent integer indices for subsets of continuation variables associated with a newly constructed function for later reference by subsequent constructor calls. In the figure, the fields $\mathtt{{fcndata.x1_idx}}$ and $\mathtt{{fcndata.x2_idx}}$ reference constructor-dependent details that are unknown to COCO and independent of the order of construction

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Figure 5. A logical matrix representation of the dependencies of the adjoint conditions on adjoint variables introduced at different stages of problem construction. At the stage of construction illustrated in the figure, three adjoint conditions are added that include terms linear in two newly introduced adjoint variables. Each variable also appears in contributions to previously introduced adjoint conditions identified by ones in the corresponding rowspan="$[0[]0]"

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