Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear flow
File(s)
Accepted version
Author(s)
Lakshmi, Mayur Venkatram
Fantuzzi, Giovanni
Fernández-Caballero, Jesús David
Yongyun, Hwang
Chernyshenko, Sergei
Type
Journal Article
Abstract
Tobasco et al. [Phys. Lett. A, 382:382–386, 2018] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Netwon–Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.
Date Issued
2020-04-08
Date Acceptance
2020-02-18
Citation
SIAM Journal on Applied Dynamical Systems, 2020, 19 (2), pp.763-787
ISSN
1536-0040
Publisher
Society for Industrial and Applied Mathematics
Start Page
763
End Page
787
Journal / Book Title
SIAM Journal on Applied Dynamical Systems
Volume
19
Issue
2
Replaces
10044/1/72578
Copyright Statement
© 2020, Society for Industrial and Applied Mathematics
Sponsor
Engineering and Physical Sciences Research Council
Engineering & Physical Science Research Council (EPSRC)
Identifier
https://arxiv.org/abs/1906.04001v1/
Grant Number
2092930
EP/J011126/1
Subjects
periodic orbits
polynomial optimization
ergodic optimization
Publication Status
Published
Date Publish Online
2020-04-08