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

Monday and Friday

2:00 - 3:25 pm

119 Downs Laboratory of Physics (DWN)

Sarthak Parikh

210-2 Math Building (Building 15)

Office hours: By appointment

Email:

Office hours: By appointment

Email:

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

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

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

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).