Spring 2014
MATH 614: Dynamical Systems and Chaos
Time and venue: TR 9:35-10:50 a.m., BLOC 161
Office hours (BLOC 223b):
Tuesday, 1:00-3:00 p.m.
by appointment
Course outline:
Part I: One-dimensional dynamics
Introduction and preliminaries
Hyperbolicity
Logistic map
Symbolic dynamics
Definition of chaos
Structural stability
Bifurcation theory
Maps of the circle
Period-doubling
Devaney's book: Chapter One
Lecture 1: Examples of dynamical systems.
Lecture 2: Periodic points. Hyperbolicity.
Lecture 3: Classification of fixed points. Logistic map.
Lecture 4: Itineraries. Cantor sets.
Lecture 5: Cantor sets (continued). Metric and topological spaces. Symbolic dynamics.
Lecture 6: Symbolic dynamics (continued). Topological conjugacy. Definition of chaos.
Lecture 7: Compact sets. Topological conjugacy (continued). Definition of chaos (revisited).
Lecture 8: Structural stability. Sharkovskii's theorem.
Lecture 9: Sharkovskii's theorem (continued).
Lecture 10: Bifurcation theory.
Lecture 11: Maps of the circle.
Lecture 12: Maps of the circle (continued). Subshifts of finite type (revisited).
Part II: Higher-dimensional dynamics
Dynamics of linear maps
The horseshoe map
Attractors
Stable and unstable manifolds
The Hopf bifurcation
Devaney's book: Chapter Two
Lecture 13: Dynamics of linear maps. Hyperbolic toral automorphisms.
Lecture 14: The horseshoe map. Invertible symbolic dynamics. Stable and unstable sets.
Lecture 15: Markov partitions. Solenoid.
Lecture 16: Bifurcation theory in higher dimensions. The Hopf bifurcation.
Lecture 17: Hyperbolic dynamics. Chain recurrence.
Lecture 18: Stable and unstable manifolds. Hyperbolic sets.
Part III: Complex analytic dynamics
Complex quadratic maps
Classification of periodic points
The Julia set
The Mandelbrot set
Devaney's book: Chapter Three
Lecture 19: Holomorphic dynamics. Classification of periodic points.
Lecture 20: Möbius transformations. Local holomorphic dynamics at fixed points.
Lecture 21: Neutral periodic points. The Julia and Fatou sets.
Lecture 22: The Julia and Fatou sets (continued).
Lecture 23: The filled Julia set. The Mandelbrot set.
Part IV: Brief introduction to ergodic theory
Invariant measure
Ergodic theorem
Ergodicity and mixing
Spectral properties of a dynamical system
Lecture 24: Invariant measure. Recurrence.
Lecture 25: Ergodic theorems. Ergodicity and mixing.