Stochastic Logistic Maps and Invariant Distributions

Abstract: Abstract: The logistic map, given by the mapping $f(x)=\lambda x(1-x)$, maps the interval $[0,1]$ to itself when $\lambda$ takes values between 0 and 4. It's a famous example of a map that displays chaotic behavior - behavior that is seemingly without pattern or predictability. One hallmark of chaos is what's known as ``sensitivity to initial conditions" - two points that start arbitrarily close to each other will eventually have orbits that diverge from each other. The deterministic logistic map has been well studied. We are interested in the \emph{stochastic} logistic map - the map given when the $\lambda$ values take independent random values according to a distribution on $[0,4]$. Since there is a stochastic element to this map, we can no longer study the sequence of points given by taking iterates of the logistic maps; the value of $f(x)$ is now a random variable, and we study its distribution under successive iterates. In this talk, we'll explore what the pattern of distributions can tell us about the map, and look for \emph{invariant} distributions - distributions that remain fixed under successive iterates of the stochastic logistic map.

Mathematics

Audience: advanced learners

Graduate Online Seminar Series (GOSS)

Series comments: Meeting Password: MATHGOSS

Announcement mailing list: groups.google.com/g/goss2021

Website: dzackgarza.com/GOSS/2021/

Recordings: www.youtube.com/watch?v=n3xhHlOzFPM&list=PLkscP0p2V2U5J-Gc4foDjQxVD-4c0dVU4