Spring 2018
MATH 614: Dynamical Systems and Chaos
Time and venue: MWF 11:30 a.m.-12:20 p.m., BLOC 161
Office hours (BLOC 223b):
Monday, 1:00-2:00 p.m.
Friday, 1:00-2:00 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 periodic points.
Lecture 4: Logistic map. Itineraries.
Lecture 5: Cantor sets. Fractal dimension. Metric spaces.
Lecture 6: Symbolic dynamics.
Lecture 7: Symbolic dynamics (continued).
Lecture 8: Topological conjugacy.
Lecture 9: Compact sets. Definition of chaos.
Lecture 10: Chaos in unimodal maps. Structural stability.
Lecture 11: Structural stability (continued). Sharkovskii's theorem.
Lecture 12: Sharkovskii's theorem (continued).
Lecture 13: Bifurcation theory.
Lecture 14: Orbit diagram for the logistic map. Topological Markov chains.
Lecture 15: Maps of the circle.
Lecture 16: Rotation number. The standard family.
Lecture 17a: Morse-Smale diffeomorphisms.
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 17b: Dynamics of linear maps.
Lecture 18: Dynamics of linear maps (continued). Toral endomorphisms.
Lecture 19: The horseshoe map. Invertible symbolic dynamics.
Lecture 20: Stable and unstable sets. Hyperbolic toral automorphisms.
Lecture 21: Markov partitions. Solenoid.
Lecture 22: Solenoid (continued). Attractors.
Lecture 23: Hyperbolic dynamics.
Lecture 24: Bifurcation theory in higher dimensions. The Hopf bifurcation.
Lecture 25: Chain recurrence.
Lecture 26: More on hyperbolic dynamics. Morse-Smale diffeomorphisms.
Part III: Complex analytic dynamics
Complex quadratic maps
Classification of periodic points
The Julia set
The Mandelbrot set
Devaney's book: Chapter Three
Lecture 27: Holomorphic dynamics.
Lecture 28: Periodic points of holomorphic maps. Möbius transformations.
Lecture 29: Local holomorphic dynamics at fixed points.
Lecture 30: Neutral fixed points (continued). The Julia set.
Lecture 31: The quadratic maps. The Mandelbrot set.
Lecture 32: The Julia and Fatou sets.
Lecture 33: The Julia and Fatou sets (continued).
Lecture 34: The Fatou components. The filled Julia set.
Lecture 35: The filled Julia set (continued).
Part IV: Brief introduction to ergodic theory
Invariant measure
Ergodic theorem
Ergodicity and mixing
Spectral properties of a dynamical system
Lecture 36: Invariant measure.
Lecture 37: Ergodic theorems. Ergodicity.