Reverse-engineering invariant manifolds with asymptotic phase

2016/08/30 by Matthew Kvalheim, Kvalheim, Matthew, Shai Revzen +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Dynamical Systems (math.DS) #FOS: Biological sciences #FOS: Mathematics #Optimization and Control (math.OC) #Quantitative Methods (q-bio.QM) #math.DS #math.OC #q-bio.QM

paper · pdf · doi:10.48550/arxiv.1608.08442

arxiv created 2016/08/30 · arxiv updated 2016/08/31

Abstract

We present a recipe for rendering a submanifold normally hyperbolic and invariant within a stability basin. The construction includes the ability to choose the asymptotic phase map. We are motivated by the notion of "templates and anchors" -- the biomechanical observation that animal motions are often governed by low dimensional dynamics -- and the growing applications in robotics which desire means for making such biologically derived templates govern the dynamics of robots. Our approach is fairly universal, in the sense that a broad range of model reduction constructions must be normally hyperbolic if they are robust, and a broad range of such normally hyperbolic systems can be produces from our construction.

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