Sensitivity Analysis for Periodic Orbits and Quasiperiodic Invariant Tori Using the Adjoint Method

TL;DR

This paper develops an adjoint-based framework for analyzing the sensitivities of periodic orbits and quasiperiodic tori in dynamical systems, facilitating continuation and stability analysis with minimal setup effort.

Contribution

It introduces a rigorous adjoint-based method for continuation and sensitivity analysis of invariant tori and periodic orbits, applicable to smooth and hybrid systems, with implementation in COCO software.

Findings

Normal hyperbolicity ensures existence of sensitivity solutions.

Sensitivity analysis predicts asymptotic phases of trajectories.

Minimal additional setup needed for COCO-based continuation.

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.