A signature of chaotic behavior in dynamical systems is sensitive dependence on initial conditions, and Lyapunov exponents measure the rates at which nearby orbits diverge. One might expect that geometric expansion or stretching in a map would lead to positive Lyapunov exponents. This, however, is very difficult to prove - except for maps with invariant cones (or a priori separation of expanding and contracting directions). An example that has come to symbolize the challenge is the standard map: there is a parameter that corresponds to the amount of stretching, and for no value of this parameter, no matter how large, has anyone been able to prove the positivity of Lyapunov exponents. In this talk, I would like to explain the reasons behind this challenge, and to show that a tiny amount of noise can make the problem much more tractable.
Sun, 10/04/2016 - 16:00 to 17:00
Lecture hall 2