Introduction to mathematical physics, fall 2012
Introduction to dynamical systems and chaotic systems
Lecturer
Carlos MejiaMonasterio
Paolo MuratoreGinanneschi
Scope
10 cu.
Type
Advanced studies
Course description
Dynamical systems are mathematical models describing the evolution of systems in terms of equation of motion and initial values. Examples are mechanics in physics, population dynamics in biology and chemical kinetics in chemistry. Dynamical systems theory is also finding an increasing number of applications in social sciences as mathematical economy and finance.
The course will focus on models based on ordinary differential equations (ODE). Under rather general conditions, ODE are known to have unique solutions for any complete set of initial data. In the general case of nonlinear dynamics, fully deterministic solutions may become unpredictable for practical purposes due a to sensitive dependence on the initial conditions. This striking phenomenon is called chaos .
The purpose of the course is to illustrate tools and techniques to characterize qualitative properties of the solutions like dependence on initial data, large time behavior and sensitivity to variation of the parameters. Reduction and perturbative methods for quantitative construction of the solutions will be also illustrated.
The emphasis of the course will be on concrete examples and geometric thinking. Theorems will be stated but the main interest will be to show their meaning and relevance in the treatment of concrete examples.
Contents
Linear systems, fixed points and cycles in 2D, simple bifurcations in 2D, flows in 3D chaos and Ljapunov exponents. Methods for simplifying dynamical systems: central manifold, normal forms and multiscale perturbation theory.
Prerequisites
The course is intended for undergraduate students of mathematics, physics. Prior courses in advanced calculus and linear algebra are required (Diff.Int. 12 and Lineaarialgebra 1, or Mapu 12). Background material will be on request discussed during the course.
Lectures
Weeks 3742 and 4450, Tuesday 1416 and Thursday 1416 in room C123.
From 12.11 exercises on Mondays 1416 in room CK108. Last exercise session 25.11.
First lecture: Tuesday 11.09
The lecture notes cover and sometimes integrate the material expounded in the lessons.
They also give bibliographic references for the same topics. Note that, however, the subdivision of the lecture notes
does not necessarily reflects the number of taught lessons.
Lectures 110 
Lectures 1020 

Lecture_01 Existence and uniqueness of solutions (17.09) 
Lecture_11 Periodic orbits (27.11) 
Lecture_02 Timeautonomous linear systems (18.09) 

Lecture_03 Timenonautonomous linear systems (26.11) 

Lecture_04 Hamiltonian systems (27.09) 

Lecture_05 Fixed point stability (02.10) 

Lecture_06 Invariant manifolds I (02.10) 

Lecture_07 Invariant manifolds II (12.10) 

Lecture_08 Normal forms theory (20.10) 

Lecture_09 Lyapunov exponents (19.11) 

Lecture_10 Billiard tables (19.11) 

Attachment to lectures 


pendulum A Mathematica notebook for the forced pendulum 

Exams
Bibliography
The main references for the course are
 Chaos: From Simple Models to Complex Systems by Massimo Cencini, Fabio Cecconi and Angelo Vulpiani (2009),
 Concepts and Results in Chaotic Dynamics by Pierre Collet and JeanPierre Eckmann (2006),
 Geometrical theory of dynamical systems, Nils Berglund (2001),
 Perturbation theory of dynamical systems, Nils Berglund (2001)
References for numerical exercises
 GNU Octave documentation
 Lecture Notes for the Octave class at BFHTI by Andreas Stahel
 Octave wiki tutorial
 Introduction to Octave by P.J.G. Long (2005)
Octave packages can be retrieved also from
Registration
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Exercise groups
Group 
Day 
Time 
Place 
Instructor 

1. 



