# American Institute of Mathematical Sciences

ISSN:
2158-2491

eISSN:
2158-2505

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## Journal of Computational Dynamics

October 2021 , Volume 8 , Issue 4

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2021, 8(4): 403-445 doi: 10.3934/jcd.2021016 +[Abstract](330) +[HTML](178) +[PDF](572.97KB)
Abstract:

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio \begin{document}$1$\end{document}.

2021, 8(4): 447-494 doi: 10.3934/jcd.2021017 +[Abstract](185) +[HTML](82) +[PDF](5208.39KB)
Abstract:

We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function \begin{document}$r(\varphi)$\end{document} measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function \begin{document}$r(\varphi)$\end{document} and of the curvature \begin{document}$\kappa(\varphi)$\end{document} (characterized by \begin{document}$dr/d\varphi = 0$\end{document} and \begin{document}$d\kappa /d\varphi = 0$\end{document}, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.

We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.

2021, 8(4): 495-520 doi: 10.3934/jcd.2021018 +[Abstract](169) +[HTML](50) +[PDF](5286.25KB)
Abstract:

In this paper we combine an approach based on Runge-Kutta Nets considered in [Benning et al., J. Comput. Dynamics, 9, 2019] and a technique on augmenting the input space in [Dupont et al., NeurIPS, 2019] to obtain network architectures which show a better numerical performance for deep neural networks in point and image classification problems. The approach is illustrated with several examples implemented in PyTorch.

2020 CiteScore: 1