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Introduction to Mathematical Chaos (Spring 2018-19)

Ma 4/104Spring 2018-19Caltech

An introduction to chaos theory and fractals in the context of dynamical systems.

Overview

This class gives an introduction to chaos theory and fractals in the context of dynamical systems.

Meeting time and location

  • Tuesday and Thursday, 10:30 – 11:55 am
  • 187 Linde Hall

Instructor

  • Sarthak Parikh (he/him) · 202 Kellogg/Linde Hall · Office hours by appointment

Grading

  • Problem sets: 50%
  • Student presentations: 30%
  • Class participation: 20%

Problem sets

  • Problem set 1 (due 5pm, Thursday April 25)
  • Problem set 2 (due 5pm, Tuesday May 7)
  • Problem set 3 (due 5pm, Monday June 3)

Lecture outline

  1. Dynamical systems, representations (differential equations, maps), one-dimensional systems, fixed points, stability, phase portraits, nonlinear systems, linearization.
  2. Nonlinear maps, conditions of attracting/repelling fixed points, condition of stable/unstable fixed point; logistic map, bifurcation diagram, cobweb diagram.
  3. Cobweb diagrams in the logistic map, periodic orbits, period-doubling bifurcations, stability of periodic orbits, period-3 orbit.
  4. Saddle-node bifurcation, Sharkovskii’s theorem, Feigenbaum constant and universality, proof of condition of sink/source, sensitive dependence on initial conditions.
  5. Differential equations in 2d, examples: saddle point, source, sink, stable, unstable, degenerate eigenvalues, stable but not attracting, stable/unstable manifolds.
  6. More examples: lines of fixed points, complex eigenvalues: elliptical orbits, spiralling orbits, globally asymptotically stable equilibrium points.
  7. A 3d example, existence and uniqueness theorem, Lipschitz functions, Gronwall’s inequality, continuous dependence on initial conditions.
  8. Stability of nonlinear systems, Jacobian matrix, examples: damped/undamped pendulum, damped/undamped Duffing oscillator.
  9. Stability of centers, Hamiltonian systems, Lyapunov center theorem, Lyapunov functions, Lyapunov theorem.
  10. Applications of Lyapunov theorem, basin of attraction, trapping region, LaSalle corollary, Poincaré maps, bifurcations.
  11. Floquet multipliers, Poincaré sections, limit cycles, bifurcation in differential equations: period-doubling, saddle-node & Hopf bifurcations.
  12. Hopf bifurcation -ctd-: Andronov-Hopf bifurcation theorem, supercritical & subcritical bifurcations, examples of Hopf bifurcation.
  13. Lorenz equation; omega-limit sets and alpha-limit sets, examples, cycles.
  14. Properties of omega-limit sets, proof of LaSalle corollary, Poincaré-Bendixson theorem, Lorenz attractor and chaotic attractors.
  15. Lyapunov exponents, Lyapunov dimension, examples.
  16. More results about Lyapunov exponents, Lyapunov exponents and dimensions of the Lorenz attractor.
  17. Cantor set, properties, fractals, fractal dimensions and examples, tent map, chaos & fractal basins, numerical results & plots for Hénon map: chaotic attractors, fractal stable/unstable manifolds, bifurcations.
  18. Stable/unstable manifolds and homoclinic points, stable manifold theorem, Julia and Mandelbrot sets, filled Julia sets.
  19. Hausdorff dimension and its properties.
  20. Student presentations (Adrian Lopez: Limit cycle in Van der Pol equation; Alexandra Klipfel: Chaotic mixing in fluids & baker transformation; Logan Apple: Iterated Function Systems; Richard Chen: Collatz conjecture).

References

  • Alligood, Sauer, and Yorke, Chaos: An Introduction to Dynamical Systems
  • Pesin and Climenhaga, Lectures on Fractal Geometry and Dynamical Systems