Hyperbolic neighbourhoods as organizers of finite-time exponential stretching

Abstract

Hyperbolic points and their unsteady generalization – hyperbolic trajectories – drive the exponential stretching that is the hallmark of nonlinear and chaotic flow. In infinite-time steady or periodic flows, the stable and unstable manifolds attached to each hyperbolic trajectory mark fluid elements that asymptote either towards or away from the hyperbolic trajectory, and which will therefore eventually experience exponential stretching. But typical experimental and observational velocity data are unsteady and available only over a finite time interval, and in such situations hyperbolic trajectories will move around in the flow, and may lose their hyperbolicity at times. Here we introduce a way to determine their region of influence, which we term a hyperbolic neighbourhood, that marks the portion of the domain that is instantaneously dominated by the hyperbolic trajectory. We establish, using both theoretical arguments and empirical verification from model and experimental data, that the hyperbolic neighbourhoods profoundly impact the Lagrangian stretching experienced by fluid elements. In particular, we show that fluid elements traversing a flow experience exponential boosts in stretching while within these time-varying regions, that greater residence time within hyperbolic neighbourhoods is directly correlated to larger finite-time Lyapunov exponent (FTLE) values, and that FTLE diagnostics are reliable only when the hyperbolic neighbourhoods have a geometrical structure that is ‘regular’ in a specific sense.