Transition to Chaos: The Orbit Diagram and the Mandelbrot Set

In this video-lecture Prof. Robert L. Devaney explores and explains two of the more fascinating images that arise in the study of Dynamical Systems, namely the orbit (or bifurcation) diagram and the Mandelbrot Set. Both of these images arise when a quadratic function is iterated, so the level of mathematical sophistication necessary to understand the lecture is minimal. The main goal of the lecture is to describe the relationship between these two images, how they are generated as well as what they mean mathematically. Several important related concepts in dynamical systems theory are also described, including period doubling bifurcations, Feigenbaum’s number, Sarkovskii’s Theorem and the role of critical orbits. The mathematical concepts are illustrated with colorful slides, films and computer experiments done in real time.