Ma 4/104 - Introduction to Mathematical Chaos (Spring 2018)

Course Description

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


Course Meeting Time and Location
Monday and Friday
2:00 - 3:25 pm
119 Downs Laboratory of Physics (DWN)

Course Instructor Contact Information and Office Hours
Sarthak Parikh
210-2 Math Building (Building 15)
Office hours: by appointment

Grading

Class participation: 40%. Student presentations: 60%.


Suggested books

1. Chaos: An Introduction to Dynamical Systems, by Alligood, Sauer and Yorke.

2. Lectures on Fractal Geometry and Dynamical Systems, by Pesin and Climenhage.


Lecture notes

1. April 2, Monday: Dynamical systems, representations (differential equations, maps), one-dimensional systems, fixed points, stability, phase portraits (notes I  pp. 1-7).

2. April 4, Wednesday: Nonlinear systems, linearization, stability (notes I pp. 7-14); Logistic map, cobweb diagrams, bifurcation diagram (notes II pp. 1-3).

3. April 6, Friday: Conditions for sources and sinks (notes I pp. 15-18); periodic orbits, stability of periodic orbits (notes II pp. 5-8).

4. April 9, Monday: Logistic map, period-doubling bifurcations, stability condition for period-k orbits revisited (notes II pp. 1-10).

5. April 11, Wednesday: Saddle-node bifurcation, period-doubling cascade, universality, Feigenbaum constant, onset of chaos (notes II pp. 10-18).

6. April 13, Friday: Sensitive dependence on initial conditions (notes II pp. 19-22); higher-dimensional systems (differential equations), phase space diagrams (notes III pp. 1-10).

7. April 16, Monday: Stable/unstable manifolds, Jacobian matrix, types of phase space behaviors (notes III pp. 1-8, 18, 20-21).

8. April 20, Friday: Complex eigenvalues and eigenvectors, matrix times circle equals ellipse, nonlinear differential equations, stability of nonlinear systems, examples (notes III pp. 8-17, 19-20, 28-35).

9. April 23, Monday: More examples: including (damped) simple pendulum, (damped) Duffing oscillator, limit cycles (notes III pp. 35-42).

10. April 27, Friday: Existence and uniqueness theorem, Lipschitz functions, Gronwall's inequality, continuous dependence on initial conditions, Lyapunov functions (notes III pp. 22-27, 43-46).

11. April 30, Monday: Poincare map, bifurcations in differential equations: period-doubling bifurcation (notes III pp. 47-56).

12. May 4, Friday: Bifurcations in differential equations: saddle-node and Hopf bifurcations (notes III pp. 56-63).

13. May 7, Monday: Bifurcations in differential equations: Hopf bifurcations - supercritical, subcritical (notes III pp. 63-70).

14. May 11, Friday: Student presentation: Hodgkin-Huxley model; Limit sets and their properties (notes IV pp. 1-6).

15. May 14, Monday: Poincare-Bendixson theorem, Lorenz equations: Hopf bifurcation & chaotic attactor, Lyapunov exponents: maps & flows (notes IV pp. 7-13).

16. May 21, Monday: Lyapunov exponents for flows, chaos in differential equations, Lyapunov dimension (notes IV pp. 14-21).

17. May 25, Friday: Ditch Day holiday.

18. May 28, Monday: Memorial Day holiday.

19. June 1, Friday: Lyapunov exponents -ctd- dissipative, chaotic flows, maps (notes IV pp. 21-27).

20. June 4, Monday: Fractals, Cantor set and its properties, Sierpinski gasket/carpet, box-counting dimension (notes V pp 1-8).

21. June 8, Friday: Higher dimensional maps: Henon map, stable/unstable manifolds and fractals, Mandelbrot and Julia sets (notes V pp 8-25).