Spring 2014
  • MATH 614: Dynamical Systems and Chaos
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    Time and venue:  TR 9:35-10:50 a.m., BLOC 161

    First day hand-out

    Office hours (BLOC 223b):
  • Tuesday, 1:00-3:00 p.m.
  • by appointment
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    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. 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
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    Lecture 24: Invariant measure. Recurrence.

    Lecture 25: Ergodic theorems. Ergodicity and mixing.