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Introduction to mathematical physics, fall 2012

Introduction to dynamical systems and chaotic systems


Carlos Mejia-Monasterio
Paolo Muratore-Ginanneschi


10 cu.


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 non-linear 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.


Linear systems, fixed points and cycles in 2-D, simple bifurcations in 2-D, flows in 3-D chaos and Ljapunov exponents. Methods for simplifying dynamical systems: central manifold, normal forms and multi-scale perturbation theory.


The course is intended for undergraduate students of mathematics, physics. Prior courses in advanced calculus and linear algebra are required (Diff.Int. 1-2 and Lineaarialgebra 1, or Mapu 1-2). Background material will be on request discussed during the course.


Weeks 37-42 and 44-50, Tuesday 14-16 and Thursday 14-16 in room C123.

From 12.11 exercises on Mondays 14-16 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 1-10

Lectures 10-20

Lecture_01 Existence and uniqueness of solutions (17.09)

Lecture_11 Periodic orbits (27.11)

Lecture_02 Time-autonomous linear systems (18.09)


Lecture_03 Time-non-autonomous 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




The main references for the course are

References for numerical exercises

Octave packages can be retrieved also from


Did you forget to register? What to do.

Exercise groups











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