SNAPSHOT: A Mathematical Route to Chaos

In dynamical systems, “chaos” refers to something very complicated and without general rules, which makes chaotic systems very hard to understand. “However, we can understand the pattern that leads to the chaos; the path from something we know to something irregular,” says Professor Tao Chen (LaGuardia Community College).

Chen, along with Professors Yunping Jiang (The Graduate Center, CUNY, and Queens College) and Linda Keen (Lehman College, emerita at The Graduate Center), discovered a new path that leads systems in the family of tangent functions from order into chaos. The route involves two new phenomena, called cycle doubling and cycle merging. The discovery, along with the technique the researchers used to prove it, could lead to new research directions.

A row of white doors with one painted yellow

The article appears in Conformal Geometry and Dynamics.

Imagine you take an equation and iterate it: Every time you get an answer out for y, you plug that in as your new x, and repeat. If you find that after several iterations, the equation spits out the same number you first plugged in, you have completed a cycle.

In the tangent family, the new paper explains, two different cycles can merge into one, or one cycle can split into two.

Though the math may sound niche, these patterns can appear in nature. Cycle doubling can model the population of a species with a constant food supply, Keen explains, or the number of animals whose moods change from stable to unstable as the group moves from one environment to the next.

In their paper, the researchers applied a mathematical tool called “holomorphic motions,” which had never before been used in this type of research. “Mathematicians working in both complex analysis and dynamical systems will be interested in this approach,” Jiang says. “We also hope this powerful method can be applied to other branches of mathematics.”