Ultradiscretization of solvable one-dimensional chaotic maps

TL;DR

This paper explores the ultradiscretization of a solvable one-dimensional chaotic map derived from elliptic functions, revealing its connection to the tent map and providing a geometric interpretation via tropical geometry.

Contribution

It introduces a novel ultradiscretization process for elliptic function-based maps and links the resulting dynamics to tropical geometry, extending to higher multiplication formulas.

Findings

Ultradiscrete limit yields the tent map and its solution.

Provides a geometric interpretation using tropical Jacobian.

Discusses generalization to higher multiplication formulas.

Abstract

We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the $m$-th multiplication formula of the elliptic functions is also discussed.