A Brief Introduction to Chaotic Dynamical Systems

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Our speaker, Nicholas Reitter, is a member of Math for Math's Sake. He writes:

Chaotic dynamical systems, characterized by sensitive dependence on initial conditions and a few other properties, forms a rich topic-area at the intersection of physics and mathematics. Though such systems can be described in a number of ways, including as systems of differential equations, we will use the more accessible approach of iterative mappings to make a very introductory foray into studying their behavior.

More specifically, we will use tools like “graphical analysis” and “orbit diagrams” (or, “bifurcation plots”) – to be defined and described in some detail – to make a brief study of one paradigmatic family of such mappings: the simple quadratic family Q_(c ) of mappings under iteration.

Q_(c ): x ↦ x^2 + c

(Q_(c ) under iteration has its most interesting dynamics occurring in the region [-2 ≤ c ≤ (1/4)] .)

No prior exposure to dynamical systems will be assumed. Only basic undergraduate calculus should be considered a prerequisite. Some exposure to complex analysis may be helpful, but it is not essential. An astonishing amount of mathematical structure appears for certain such mappings on the real line alone.

During this workshop, you will be asked to do some simple exercises designed to familiarize those new to the topic with the abovementioned tools, after which we will go through the behavior of Q_(c ) under iteration. We’ll cover key results like Feigenbaum’s constant and Sarkovskii’s Theorem, and perhaps one or two simpler proofs, and we’ll view a sampling of the fascinating images that arise from these systems. Suggestions for further reading and study will also be provided.

Recommended Reading

Excerpt [with nice illustrations!] from Robert L. Devaney, "A First Course in Chaotic Dynamical Systems" (an undergraduate textbook): https://app.box.com/s/2j9zwmx84yb2l4ty14xacuwdgrrzr18m

Bio

Nicholas Reitter works in applied financial math for a large accounting firm, and has had a parallel college teaching career as an adjunct. This presentation will be essentially highlights of material from a one-semester course on this topic he gave to undergraduate engineering students at Cooper Union.