On planar curves with position-dependent curvature

2021/07/16 by Berger, Arno
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.07680

Abstract

Motivated by homothetic solutions to curvature-driven flows of planar curves, as well as their many physical applications, this work carries out a systematic study of oriented curves whose curvature κ is a given function of position or direction. The analysis is informed by a dynamical systems point of view. Though focussed on situations where the prescribed curvature depends only on the distance r from a distinguished point, the basic dynamical concepts are seen to apply in other situations as well. As an application, a complete classification of all simple closed solutions of κ= ar^b, with real constants a,b, is established.

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