# From Tangents To Chaos, via Holomorphic Motion

Professors Linda Keen and Yunping Jiang and GC alumnus Tao Chen (Ph.D. '13) discovered a new mathematical route to chaos, one that applies to natural phenomena such as changes in a species' food supply. |

**Tao Chen** (Ph.D. ’13), a professor at LaGuardia Community College, continues to collaborate with his Graduate Center doctoral advisers, Professor **Linda Keen** (GC/Lehman College, Mathematics) and Professor **Yunping Jiang** (GC/Queens College, Mathematics). The three have kept up their weekly research seminar at The Graduate Center, and those discussions have yielded a new paper in Conformal Geometry and Dynamics.

The professors discovered new mathematical phenomena, called cycle merging and cycle doubling, in the tangent family — a family of systems familiar to many from high school trigonometry. To prove their observations, the mathematicians applied a technique called “holomorphic motion,” which had never been used before in this context.

Both the phenomena and the tool they used in their proof, the researchers say, could open up new possibilities for research.

For Keen, this study is a continuation of her longtime work on the dynamics of tangent functions in complex dynamical systems, while Jiang’s work focuses on the dynamics of quadratic polynomials in real dynamical systems.

Chen used his advisers’ expertise in both these areas to craft his Ph.D. thesis. Discussions on his results and on earlier work by Keen and Jiang eventually led to their new collaboration.

Imagine a simple equation relating the variables x and y. Say when you plug zero in for x, the equation gives negative one for y. Next you take your answer of negative one and plug that in for x, and the equation spits out zero for y. You are back where you started, and have completed one cycle of two points. In the tangent family, the new paper explains, two different cycles can merge into one, or one cycle can split into two.

Put another way, the researchers discovered a new route that leads these systems from order into chaos. While chaos itself is hard to understand, Chen says, it is possible to understand how a system arrived there in the first place. These math-heavy concepts also appear in nature.

“The population of a species with constant food supply, as it changes over time, can be modeled by dynamical systems that exhibit cycle doubling,” Keen says.

The researchers are now investigating new maps, other than the tangent family, to see if they also exhibit cycle doubling and merging. Holomorphic motion, the technique behind their proof for these phenomena, could also generate as many new research directions as the discovery itself.

“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.”

Submitted on: JAN 31, 2019

Category: Faculty | General GC News | Mathematics