# Connecting Cycles for Concentric Circles

## TL;DR

This paper investigates the critical points of connecting cycles around concentric circles, showing that generically the perimeter function is Morse, and analyzes specific configurations and bifurcations.

## Contribution

It characterizes critical connecting cycles for concentric circles, proves Morse properties of the perimeter function, and computes indices for specific configurations.

## Key findings

- Diametrically aligned configurations are critical points.
- Perimeter function is generically Morse on the configuration space.
- Bifurcation analysis for four concentric circles reveals pitchfork bifurcations.

## Abstract

We study perimeters of connecting cycles for concentric circles. More precisely, we are interested in characterization of those connecting cycles which are critical points of perimeter considered as a function on the product of given circles. Specifically, we aim at showing that, generically, perimeter is a Morse function on the configuration space, and computing Morse indices of critical configurations. In particular, we prove that the diametrically aligned configurations are critical and their indices can be calculated from an explicitly given tridiagonal matrix. For four concentric circles, we give examples of non-generic collections of radii and describe a pitchfork type bifurcation of stationary connecting cycles.