Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear flow

Lakshmi M, Fantuzzi G, Fernandez-Caballero JD, Hwang Y, Chernyshenko S (2020)

Publication Type: Journal article

Publication year: 2020

Journal

SIAM Journal on Applied Dynamical Systems Society for Industrial and Applied Mathematics

Book Volume: 19

Pages Range: 763-787

Journal Issue: 2

DOI: 10.1137/19M1267647

Abstract

Tobasco, Goluskin, and Doering [Phys. Lett. A, 382 (2018), pp. 382–386] 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 Newton–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.

Authors with CRIS profile

Giovanni Fantuzzi

Involved external institutions

Imperial College London / The Imperial College of Science, Technology and Medicine GB United Kingdom (GB) University of Warwick GB United Kingdom (GB)

How to cite

APA:

Lakshmi, M., Fantuzzi, G., Fernandez-Caballero, J.D., Hwang, Y., & Chernyshenko, S. (2020). Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear flow. SIAM Journal on Applied Dynamical Systems, 19(2), 763-787. https://dx.doi.org/10.1137/19M1267647

MLA:

Lakshmi, Mayur, et al. "Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear flow." SIAM Journal on Applied Dynamical Systems 19.2 (2020): 763-787.

BibTeX: Download