A construction of patterns with many critical points on topological tori

TL;DR

This paper constructs specific closed surfaces of genus 1 with reaction-diffusion patterns exhibiting an arbitrarily large number of critical points, advancing understanding of pattern complexity on topological tori.

Contribution

It introduces a method to design genus 1 surfaces with reaction-diffusion patterns that have any desired number of critical points, demonstrating high pattern complexity.

Findings

Constructed surfaces with arbitrarily many critical points

Established existence of stable nonconstant stationary solutions

Enhanced understanding of pattern formation on tori

Abstract

We consider reaction-diffusion equations on closed surfaces in $\mathbb R^3$ having genus $1$. Stable nonconstant stationary solutions are often called patterns. The purpose of this paper is to construct closed surfaces together with patterns having as many critical points as one wants.