Towards topological chaos theory

Title

Towards topological chaos theory

Speaker

Eran Igra (SIMIS, Shanghai)

Time and Location

Friday, March 28, 18:30, E2-102 SOUTH CAMPUS

Abstract

Assume we have a flow generated by a system of ordinary Differential Equations, whose dynamics are chaotic – i.e., there exists a strange, possibly fractal, attractor on which the dynamics obey the “Butterfly Effect”. In other words, if we choose two initial conditions on the attractor, no matter how close they are to one another, their trajectories will eventually diverge away from one another – only to come arbitrarily close to one another at some future time, after which they will eventually diverge again, only to repeat this behavior in a non periodic way infinitely many more times in the future. This strange phenomenon was intuitively described as follows by E.N. Lorenz – “the flap of a butterfly’s wings in Hawaii could set off a Tornado in Texas”, who originally observed it in his meteorological studies. And indeed, such complex patterns of behavior appear almost everywhere in nature – whether in meteorology, the motion of asteroids in our solar system, and the spread of diseases (to name a few). This leads us to ask the following – given a natural process that is modeled (or approximated) by some system of ODEs, when should we expect it to have chaotic behavior? And when it is chaotic, how can we describe it qualitatively?

It is precisely these questions we study in this talk. Inspired by the seminal ideas of Stephen Smale, Leonid P. Shilnikov, James A. Yorke and William Thurston, we show how one can use topology to predict chaotic behavior for three dimensional systems of ODEs – as well as to analyze the “topology of chaos” induced by these equations. We achieve both by studying the topology of the periodic solutions on the attractor, which will allow us to transform the problem of prediction and qualitative description of chaotic dynamics from analysis into the realm of topology. In particular, we will prove that in certain topological scenarios chaotic behavior for the flow is forced by the topology of the space in which the solution curves flow.

These ideas will not be too abstract, and we will show their power by applying them to study two famous examples of chaotic attractors – the Lorenz Butterfly and the Rössler attractor. In more detail, we will prove how these ideas can be used to analyze the topology of these strange attractors – as well as to study their bifurcations, and how they evolve from order into chaos. Finally (and time permitting), inspired by the Thurston-Nielsen Classification Theorem we discuss how our results could possibly be generalized in a way that ties the onset of chaos in three dimension to the theory of three-dimensional manifolds, and then conjecture how these ideas can possibly be generalized to higher dimensional systems.

Free pizza will be provided.