Heteroclinic and Homoclinic Connections in a Kolmogorov-Like Flow

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.

arXiv commentSupplemental Videos Showing Heteroclinic and Homoclinic Connections Are Available

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• Observing a Dynamical Skeleton of Turbulence in Taylor-Couette Flow Experiments
• Predicting chaotic statistics with unstable invariant tori
• Jacobian-Free Variational Method for Constructing Connecting Orbits in Nonlinear Dynamical Systems
• Data-driven detection of drifting system parameters