Emergence of chaos and fractality in the basin boundary of subcritical shear flow

Abstract

From a dynamical systems perspective of subcritical transition in shear flows, the basin boundary separating the laminar and turbulent attractors, along with the edge state that governs the long-term dynamics on that boundary, are of fundamental interest. As the Reynolds number is increased from small values, a multiplicity of simple exact coherent structures (ECS) appear in phase space, of which one often undertakes the role of the edge state. At higher values of the Reynolds number, however, the dynamics on the basin boundary and the edge state are sensitive to initial conditions, as shown by a wealth of numerical and experimental studies. The mechanism behind this transition remains unclear. To address this, we examine the subcritical regime of Taylor-Couette flow using a minimal computational box and reveal a generic mechanism whereby the edge state becomes chaotic. The first step in this process involves the formation of a heteroclinic tangle between a travelling-wave-type ECS and an independently engendered chaotic saddle. The interaction causes the basin boundary to incorporate the saddle, thus inheriting its fractal structure, while the edge state itself remains the simple ECS. The transition of the edge state to chaos requires a second step: the formation of a heteroclinic cycle involving the ECS and the chaotic saddle. In consequence, the observation of a simple non-chaotic edge state at given values of the parameters is no guarantee that the same situation will hold at nearby values.