These are the notes for an optional course on dynamical systems that I will give to 4th year mathematics students this semester.

The course is in French, but for convenience I will write the notes in English. (In order to become competitive, my French students will have to learn to use English anyway).

The format of the course is as follows:

12 weeks, each week consists of 2.5h of lecture and 1.5h+1.5h of excercise sessions.

The main reference for this course is:

**Brin and Stuck, Introduction to Dynamical Systems**, Cambridge University press, 2003, 253 pages.

This book is relatively elementary, so it can serve as a first introduction to dynamical systems. And it contains lots of exercises, which is especially good for our excercise sessions.

I’ll not completely follow the book, but most of the things that I teach will be taked from this book.

Another introductory book, which also contains many exercises, is:

**Hasselblatt and Katok, A first course in dynamics**, 2003.

I will take some material from this second book too. One can find electronic copies of these books of the web. (Don’t ask me how).

So, the first lecture is:

# Lecture 1. Basic Concepts and Examples

## 1.1. What is a dynamical system ?

Anything which moves, anything which changes with time, is a dynamical system. So, basically, everything is a dynamical system. For example:

* The solar system (motion of planets and asteroids around the sun)

* The universe (big bang, expansion, formation of black holes, etc.)

* The temperature on the earth

* Air traffic control

* Migration of fish, birds

* Stock market prices

and so on

What we want to do is to construct mathematical models for all such dynamical systems, in order to study them, dervie from them informations which can be useful in one way or another.

So how to present a dynamical system?

We need a few keywords, like:

- **state**: to say that something chages or moves is to say that its state changes.

- **observable**: in order to describe a state, we need observables. Each observable is a function, say from the space of all possible different states of our problem to the set of real numbers.

- **phase space** = **state space** = set of all possible states

- **configuration space** = **position space** = a quotient space of phase space (in mechanics, state is determined by position + velocity, and not position alone).

- **generator**: it can be a map, or an ODE, or a PDE, or any other thing which indicates how states change with time.

There are basically two different ways to describe a dynamical systems, which, for lack of better terminology, I will call phase-space representation and evolutionary representation:

**1.1.a. Phase space representation**:

In this representation, the dynamical system is given by a set X (the phase space) and either a map or a vector field (differential equation) on X. This map or vector field is the generator of the system. When it is generated by a map, then the time variable is discrete, and when it is generated by a vector field then the time varable is continuous.

Example from number theory:

X = N is the set of natural numbers, and f is defined as follows

f(n) = sum of all divisors of n which are smaller than n.

f(9) = 1 + 3 = 4, f(4) = 1+ 2 = 3, f(3) = 1, f(1) = 0, f(0) = 0.

0 is called a fixed point in this example because f(0) = 0. But besides 0 there are other fixed points, i.e. n such that f(n) = n. These numbers are called **perfect numbers** in number theory.

f (6) = 1+2+3 = 6, f(28) = 1+ 2 + 4 + 7 + 14 = 28, … (perfect numbers are related to Mersenne prime numbers)

If f(n) =m and f(m) = n then {m ,n} is a **periodic orbit** of period 2 for the dynamical system generated by f. In number theory, they are called amicable pairs. An example going back to Pythagoras is {220, 284}. There ae also amicable triples (periodic orbits of period 3) and so on.

This simple example shows that many problems in number theory can also be viewed as problems about paricular solutions of dynamical systems (which can be very hard problems)

Another example (open problem)

X = N, f(n) = n/2 if n even, f(n) = 3n+1 if n odd.

conjecture: any orbit will contain 1.

Example: Lotka-Volterra system (predator-prey)

Questions: Equilibrium? Periodic orbits (cycles) ? Stability ?

Example; spinning top

configuration space = SO(3), phase space = T*SO(3)

ALL examples from Brin-Stuck are of phase space representation type. However, it is important to know that that there is another representation:

**1.1.b Evolutionary representation**

In this repsentation, we need a total spatial space or space-time M. A state will be represented by a (generalized) function on M (a map from M to R, or maybe to C, or maybe to some other linear space if need be)

This representation is especially useful for quantum dynamics and complex systems, but not only: one can also turn a “phase-space system” into an evolutionary system:

Instead of looking at f: X \to X one looks at

L: C(X) \to C(X)

(where C is an algebra of functions on X) given by the formula

L(g) (x) = g(f(x))

A big advantage of the evolutionary representation is that (very often) the system is LINEAR! In particular, quantum mechanics is linear (so one can talk about quantum numbers = eigenvalues).

## 1.2 Examples and other notions

(shifts, cross-section, attractor, etc):

We will go through the remaining sections of Chapter 1 of Brin-Stuck, from 1.2 to 1.13 (if time permits)