Differential Geometry Perspective of Shape Coherence and Curvature Evolution by Finite-Time Non-hyperbolic Splitting

Mixing, and coherence are fundamental issues at the heart of understanding transport in fluid dynamics and other non-autonomous dynamical systems. Recently, the notion of coherence has come to a more rigorous footing, and particularly within the recent advances of finite-time studies of non-autonomous dynamical systems. Here we define shape coherent sets as a means to emphasize the intuitive notion of ensembles which “hold together” for some period of time, and we contrast this notion to other recent perspectives of coherence, notably “coherent pairs”, and likewise also to the geodesic theory of material lines. We will relate shape coherence to the differential geometry concept of curve congruence through matching curvatures. We show that points in phase space where there is a zero-splitting between stable and unstable manifolds locally correspond to points where curvature will evolve only slowly in time. Then we develop curves of points with zero-angle, meaning non-hyperbolic splitting, by continuation methods in terms of the implicit function theorem. From this follows a simple ODE description of the boundaries of shape coherent sets. We will illustrate our methods with popular benchmark examples, and further investigate the intricate structure of foliations geometry.