Semi-discrete linear hyperbolic polyharmonic flows of closed polygons

James McCoy; Jahne Meyer; James McCoy; Jahne Meyer

doi:10.3934/mine.2025013

Mathematics in Engineering

2025, Volume 7, Issue 3: 281-315. doi: 10.3934/mine.2025013

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Research article

Semi-discrete linear hyperbolic polyharmonic flows of closed polygons

James McCoy ,
Jahne Meyer ^,

School of Information and Physical Sciences, University of Newcastle, NSW, Australia

Received: 15 April 2024 Revised: 14 May 2025 Accepted: 15 May 2025 Published: 23 May 2025

We consider the damped hyperbolic motion of polygons by a linear semi-discrete analogue of polyharmonic curve diffusion. We show that such flows may transition any polygon to any other polygon, reminiscent of the Yau problem of evolving one curve to another by a curvature flow, before converging exponentially to a point that, under appropriate rescaling, is a planar basis polygon. We also consider a hyperbolic linear semi-discrete flow of the Yau curvature difference flow, where a polygonal curve is able to flow to any other such that we get convergence to the target polygon in infinite time.
- second order hyperbolic system of linear ordinary equation,
- linear hyperbolic geometric flow,
- evolving polygons
Citation: James McCoy, Jahne Meyer. Semi-discrete linear hyperbolic polyharmonic flows of closed polygons[J]. Mathematics in Engineering, 2025, 7(3): 281-315. doi: 10.3934/mine.2025013

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Abstract

We consider the damped hyperbolic motion of polygons by a linear semi-discrete analogue of polyharmonic curve diffusion. We show that such flows may transition any polygon to any other polygon, reminiscent of the Yau problem of evolving one curve to another by a curvature flow, before converging exponentially to a point that, under appropriate rescaling, is a planar basis polygon. We also consider a hyperbolic linear semi-discrete flow of the Yau curvature difference flow, where a polygonal curve is able to flow to any other such that we get convergence to the target polygon in infinite time.

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