Seminar| Institute of Mathematical Sciences
Time: Thursday, October 23th, 2025,10:30-11:30
Location: RS408, IMS
Speaker: Eran Igra from SIMIS
Abstract: It is well known that chaotic dynamics often arise by a stable, attracting periodic orbit which undergoes a cascade of period multiplying bifurcations that turn it into a chaotic attractor – for example, the Rössler attractor, the Logistic map, and many other famous examples of chaotic dynamics bifurcate from order into chaos this way. Of course, there are many ways a stable periodic orbit can bifurcate and become chaotic – with the most famous one being the Feigenbaum scenario, i.e., a period-doubling route to chaos. This leads us to ask the following question – how many such routes to chaos exist, and which one is the most common?
It is precisely these questions we will consider in this talk. In detail, we will study the possible bifurcation diagrams of periodic orbits by reducing them to graphs, which we then color using the Orbit Index Theory (originally developed by J.A. Yorke, K. Alligood, J.M. Paret and S.N. Chow). This will allow us to study the genericity of different routes to chaos, as well as to analyze their possible complexity. As we will prove, when the dimension of the phase space is sufficiently high there is no upper bound on the complexity of a generic routes to chaos. Time permitting, we will discuss what happens in low-dimensional phase spaces, and how our results possibly suggest a fermionic description of bifurcation theory via the virial expansion and universal description via the Rado graph. Based on a joint paper with Valerii Sopin (arXiv:2509.06852).