Symbolic dynamics and chaos in plane Couette flow

Abstract

According to a recent theory \cite{Li14}, when the Reynolds number is large, fully developed turbulence is caused by short term unpredictability (rough dependence upon initial data); when the Reynolds number is moderate, often transient turbulence is caused by chaos (long term unpredictability). This article aims at studying chaos in plane Couette flow at moderate Reynolds number. Based upon the work of L. van Veen and G. Kawahara \cite{VK11} on a transversal homoclinic orbit asymptotic to a limit cycle in plane Couette flow, we explore symbolic dynamics and chaos near the homoclinic orbit. Mathematical analysis shows that there is a collection of orbits in the neighborhood of the homoclinic orbit, which is in one-to-one correspondence with the collection of binary sequences. The Bernoulli shift on the binary sequences corresponds to a chaotic dynamics of a properly defined return map.