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

作者：Ma Tian; Bollt Erik M^*

来源：SIAM Journal on Applied Dynamical Systems, 2014, 13(3): 1106-1136.

DOI：10.1137/130940633

• shape coherent set
• curvature evolution
• finite-time stable and unstable foliations
• implicit function theorem
• continuation
• mixing
• transport

Mixing and coherence are essential topics for understanding and describing transport in fluid dynamics and other nonautonomous dynamical systems. Only recently has the idea of coherence gained more serious footing, particularly with the recent advances of finite-time studies of nonautonomous dynamical systems. Here we define shape coherent sets as a means to emphasize the intuitive notion of ensembles which %26quot;hold together%26quot; for some period of time, and we contrast this notion to other recent perspectives of coherence, notably %26quot;coherent pairs,%26quot; 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 foliations locally correspond to points where curvature will evolve only slowly in time. Then we develop curves of points with zero-angle, meaning nonhyperbolic 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 method with popular benchmark examples and further investigate the intricate structure of foliation geometry.

• 出版日期2014