Joint Reduced Model for the Laminar and Chaotic Attractors in Plane Couette Flow

TL;DR

This paper develops a low-dimensional nonlinear model for plane Couette flow's chaotic and laminar regimes using spectral submanifolds, accurately capturing trajectories and enabling efficient periodic orbit computation.

Contribution

It introduces a novel three-dimensional reduced-order model based on spectral submanifolds for chaotic plane Couette flow, unifying laminar and chaotic dynamics.

Findings

Model accurately reconstructs individual trajectories.

Enables rapid computation of embedded periodic orbits.

Captures both laminar and chaotic attractors.

Abstract

We use the theory of spectral submanifolds (SSMs) to develop a low-dimensional reduced-order model for plane Couette flow in the permanently chaotic regime studied by Kreilos & Eckhardt (2012). Our three-dimensional model is obtained by restricting the dynamics to the slowest mixed-mode SSM of the edge state. We show that this results in a nonlinear model that accurately reconstructs individual trajectories, representing the entire chaotic attractor and the laminar dynamics simultaneously. In addition, we derive a two-dimensional Poincar\'e map that enables the rapid computation of the periodic orbits embedded in the chaotic attractor.