Resolvent-Based Optimisation for Approximating the Statistics of a Chaotic Lorenz System

TL;DR

This paper introduces a new method combining variational and resolvent analysis to efficiently approximate the statistical properties of chaotic systems like the Lorenz equations, reducing computational complexity.

Contribution

It presents a novel framework that uses resolvent modes and gradient-based optimisation to approximate chaotic attractors without requiring exact unstable periodic orbits.

Findings

Rapid convergence of statistical measures to long-term averages

Effective dimensionality reduction from three to two dimensions in Lorenz system

Approximate trajectories capture key statistical properties of chaos

Abstract

We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional approaches relying on identifying all short, fundamental unstable periodic orbits to compute ergodic averages via cycle expansion are computationally prohibitive for high-dimensional fluid systems. Our framework stems from the observation in Lasagna, Phys. Rev. E (2020), that a single unstable periodic orbit with a period sufficiently long to span a large fraction of the attractor captures the statistical properties of chaotic trajectories. Given the difficulty of identifying unstable periodic orbits for high-dimensional fluid systems, approximate trajectories residing in a low-dimensional subspace are instead constructed using resolvent modes, which inherently capture the temporal periodicity of unstable periodic orbits. The amplitude coefficients of these modes are adjusted iteratively with gradient-based optimisation to minimise the violation of the projected governing equations, producing trajectories that approximate, rather than exactly solve, the system dynamics. A first attempt at utilising this framework on a chaotic system is made here on the Lorenz 1963 equations, where resolvent analysis enables an exact dimensionality reduction from three to two dimensions. Key observables averaged over these trajectories produced by the approach as well as probability distributions and spectra rapidly converge to values obtained from long chaotic simulations, even with a limited number of iterations. This indicates that exact solutions may not be necessary to approximate the system's statistical behaviour, as the trajectories obtained from partial optimisation provide a sufficient ``sketch'' of the attractor in state space.