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

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

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Erik Bollt , Clarkson University - Dept. of Mathematics
Fine Hall 214

Mixing, and coherence are fundamental issues at the heart of understanding fluid dynamics and other non- autonomous dynamical systems. Only recently has the notion of coherence come to a more rigorous footing, and particularly within the recent advances of finite-time studies of nonautonomous dynamical systems.. Here we define shape coherent sets which we relate to measure of coherence in differentiable dynamical systems from which we will show that tangency of finite time stable foliations (related to a forward time perspective) and finite time unstable foliations (related to a “backwards time" perspective) serve a central role.  We develop zero-angle curves, meaning non-hyperbolic splitting curves, by continuation methods in terms of the implicit function theorem, from which follows a simple ODE description of the boundaries of shape coherent sets.