Spring 2016
  • MATH 614: Dynamical Systems and Chaos
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    Time and venue:  MWF 1:50-2:40 p.m., BLOC 160

    First day hand-out

    Office hours (BLOC 223b):
  • Monday, 3:00-4:00 p.m.
  • by appointment


    • Additional office hours (BLOC 223b):
  • Monday, May 9, 3:00-4:00 p.m.
    •

    Homework assignments ##1-4


    Topics for the projects

    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.

    Lecture 4: Logistic map. Itineraries.

    Lecture 5: Cantor sets. Metric and topological spaces.

    Lecture 6: Symbolic dynamics.

    Lecture 7: Symbolic dynamics (continued).

    Lecture 8: Topological conjugacy.

    Lecture 9: Compact sets. Definition of chaos.

    Lecture 10: Chaos. Structural stability.

    Lecture 11: 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). Inverse limit space extension.

    Lecture 23: Attractors. Hyperbolic periodic points.

    Lecture 24: Bifurcation theory in higher dimensions. The Hopf bifurcation.

    Lecture 25: Chain recurrence.

    Lecture 26: Morse-Smale diffeomorphisms. Hyperbolic dynamics.

    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).

    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). More on holomorphic dynamics.

    Part IV: Brief introduction to ergodic theory

  • Invariant measure
  • Ergodic theorem
  • Ergodicity and mixing
  • Spectral properties of a dynamical system
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    Lecture 36: Invariant measure.

    Lecture 37: Ergodic theorems. Ergodicity.

    Lecture 38: Ergodicity (continued). Mixing.