Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method

Abstract

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatio-temporal systems including fluid turbulence is supported by non-chaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatio-temporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification; or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop-dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatio-temporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

Variational methods for finding periodic orbits in the incompressible Navier–Stokes equations

“…Due to the line search requiring more evaluations, each SV iteration is on average around 40 % slower than the two PV methods, which leads to fewer iterations in 72 h. More importantly, SV shows a significantly slower convergence rate. All three methods show an initial fast improvement of the objective function, followed by a much slower period, consistent with Azimi et al (2022); the convergence rate in this latter region is much faster with the PV methods, as much smaller steps were required to be taken for SV. The precise reason for this is unclear, and we make no claim that our conjugate-gradient algorithm uses optimal parameters, but this result was robust after significant trial-and-error tweaking of them.…”

“…The simplest method is gradient descent with a fixed step size at each iteration. Previous authors (Farazmand 2016;Azimi et al 2022) have considered a dynamical system…”

“…Previous authors (Farazmand 2016; Azimi et al. 2022) have considered a dynamical system by the introduction of a fictitious time such that Simple gradient descent is equivalent to discretising this system with a forward-Euler scheme, and some improvements may be found by considering terms implicitly or using higher-order schemes, or adaptive time steppers. This differs from the method of Lan & Cvitanović (2004), whose fictitious system evolves in a direction equivalent to an infinitesimal step of a Newton–Raphson method, as opposed to in the steepest descent direction for the cost function.…”

“…Boghosian et al (2011) applied this method to a lattice Boltzmann formulation of Navier-Stokes, but noted the prohibitively large computational requirements. Azimi, Ashtari & Schneider (2022) used a matrix-free method inspired by Farazmand (2016) to find periodic orbits in the Kuramoto-Sivashinsky equation. Though they successfully demonstrated the method, challenges remain towards the ultimate goal of applying this method to find unstable periodic orbits in fluid dynamics, as the studied one-dimensional equation differs from the twoand three-dimensional incompressible Navier-Stokes equations in important aspects.…”

“…In this paper we study methods which follow ideas of Azimi et al. (2022), but which are formulated for 2-D Kolmogorov flow and explicitly address the incompressibility constraints of Navier–Stokes. In § 2 we present the governing equations of the flow, formulate objective functionals that are minimised at periodic orbits, and derive the necessary gradients of these.…”