# Winter 2014 - 2015

## Classical Complex Systems

(MSc. Advanced Theory, Lecturer: David Gross, Johan Aberg, Jens Timmer)

**Announcements**

Due to a traffic accident, the original lecturer will unfortunately not be able to teach this term. Additional information is available from the original course website.

**Time & Venue**

Lecture:

Tuesdays 10 - 12, SR 1, physics high rise

Thursdays 12 - 2, SR 1, physics high rise

Exercise:

Wednesdays 2 - 4 CIP pool

**Contents**

- Deterministic Dynamical Systems
- Logistics map, Bernoulli shift, tent map
- Orbits, stability, Lyapunov exponents
- Bifurcation mechanisms
- Invariant measures
- Feigenbaum universality
- Fractal dimension: box counting dimension, Koch curve, Cantor sets

- Stochastic Processes
- Probability theory recap: random variables, conditional probabilities, distribution functions
- Markov chains
- Transition matrices
- Stationary and limit distributions
- Time averages
- Recurrence times
- Reversibility, detailed balance
- Markov chain Monte Carlo simulations
- Random walks on graphs
- Continuous time, discrete space Markov chains: Gillespie algorithm

**Prerequisites**

Classical mechanics, differential equations, statistical physics

**Literature**

E. Ott, *Chaos in Dynamical Systems*

More to be added.

**Lecture Notes**

Inofficial lecture notes. All topics relevant for the exam are included. The last lectures will not be included. Some solutions of the last exercise are included to show an application of the Gillespie algorithm. If relevant errors are found they will be corrected. (last update: 10.02.15)

**Computer Code**

The Mathematica Notebook used to explore the logistics map.