# Heteroclinic and Homoclinic Connections in a Kolmogorov-Like Flow

## TL;DR

This paper investigates the complex network of dynamical connections among unstable solutions in a weakly turbulent Kolmogorov flow, revealing heteroclinic and homoclinic structures that underpin turbulence.

## Contribution

It provides the first extensive computation of heteroclinic and homoclinic connections in a Kolmogorov-like flow, linking various types of solutions and elucidating the chaotic dynamics.

## Key findings

- Numerous heteroclinic connections between equilibria, periodic, and quasi-periodic orbits.
- Identification of a homoclinic connection forming a chaotic repeller.
- Evidence that the homoclinic tangle underpins transient turbulence.

## Abstract

Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions -- equilibria, periodic, and quasi-periodic orbits -- as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.