Dynamical Systems

Written by: Shlomo Sterberg (Harvard University)

There’s over 150 pages of information in this online textbook written by Shlomo Sterberg, professor of Mathematics at Harvard University and apparently presented to students attending Math 118 in the Spring of 2000.

Table of Contents for Dynamical Systems

  1. Iterations and fixed points
    1. Square roots
    2. Newton’s method
      1. The guts of the method
      2. A vector version
      3. Implementation
      4. The existence theorem
      5. Basins of attraction
    3. The implicit function theorem
    4. Attractors and repellers
    5. Renormalization group
  2. Bifurcations
    1. The logistic family
      1. 0 < μ ≤ 1
      2. 1 < μ ≤ 2
      3. 2 < μ < 3
      4. μ = 3
      5. 3 < μ < 1 + √6
      6. 3.449499…< μ < 3.569946....
      7. Reprise
    2. Local bifurcations
      1. The fold.
      2. Period doubling.
    3. Newton’s method and Feigenbaum’s constant
    4. Feigenbaum renormalization.
    5. Period 3 implies all periods
    6. Intermittency
  3. Conjugacy
    1. Affine equivalence
    2. Conjugacy of T and L4
    3. Chaos
    4. The saw-tooth transformation and the shift
    5. Sensitivity to initial conditions
    6. Conjugacy for monotone maps
    7. Sequence space and symbolic dynamics
  4. Space and time averages
    1. histograms and invariant densities
    2. the histogram of L4
    3. The mean ergodic theorem
    4. the arc sine law
    5. The Beta distributions
  5. The contraction fixed point theorem
    1. Metric spaces
    2. Completeness and completion
    3. The contraction fixed point theorem
    4. Dependence on a parameter
    5. The Lipschitz implicit function theorem
  6. Hutchinson’s theorem and fractal images.
    1. The Hausdorff metric and Hutchinson’s theorem
    2. Affine examples
      1. The classical Cantor set
      2. The Sierpinski Gasket
  7. Hyperbolicity
    1. C0 linearization near a hyperbolic point
    2. invariant manifolds
  8. Symbolic dynamics
    1. Symbolic dynamics
    2. Shifts of finite type
      1. One step shifts
      2. Graphs
      3. The adjacency matrix
      4. The number of fixed points
      5. The zeta function.
    3. Topological entropy
      1. The entropy of YG from A(G)
    4. The Perron-Frobenius Theorem
    5. Factors of finite shifts
   

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Dynamical Systems

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