The aim of this course is to introduce the student to basic techniques and results in the broad field of "Dynamical Systems" by developing the theory for two settings: discrete time dynamical systems (generated by maps) and continuous time dynamical systems (generated by differential equations).
In the first part, this course explains basic ideas of the field in low dimensional settings of iterated maps on the line and in the plane. In particular, we develop theory of topological dynamics, symbolic dynamics and hyperbolic dynamics. Important results and ideas in this context that are explained include "period three implies chaos", period doubling route to chaos and the Smale horseshoe map. The second part develops dynamical systems analysis for differential equations and compares techniques and results to the discrete setting with focus on the occurrence of chaotic dynamics.
The key strength of dynamical systems analysis is the departure from explicit formulas and solution recipes which are replaced by more abstract qualitative techniques from geometry and topology to discuss global properties of solutions. While this modern viewpoint originates in the work of Poincare, it was, in fact, the study of chaotic dynamical systems from the 1960s that lead to a breakthrough in science and an explosion of interest in the field of dynamical systems. Its basic techniques have by now been used in various related settings such partial differential differential equations (which lead to infinite-dimensional dynamical systems) and control theory.
This course will serve as an introduction for the more advanced course "Ergodic Theory and Fractals" and other courses in dynamical systems. The part on continuous dynamical systems complements the modern canon of topics displayed in other courses such as "Introduction to dynamical systems" which has a different focus.
Mode of instruction
Weekly two hour lecture and two questions hours every other week
Ordinary differential equations and Topology
The grade for this course is determined by the four homeworksets, the midterm and the exam.
Chaotic Dynamics van Geoffrey Goodson, ISBN 9781316285572.
Nonlinear Dynamics and Chaos van Steven H. Strogatz. ISBN-13: 978-0738204536
Ordinary Differential Equations and Dynamical Systems van Gerald Teschl. ISBN-13: 978-0821883280