Introduction to Deterministic Chaos with Lorenz Model

The paradox of deterministic chaos is that physical systems can be subject to precise laws called deterministic and seemingly simple, yet exhibit behavior that appears complex and random. Such dynamic systems can be found in all fields of science, and here are just a few examples:

• in fundamental mathematics: prime numbers have a very simple definition (they are integers with no divisors other than themselves) but their distribution seems complex and random, associated with a famous conjecture: the Riemann hypothesis. Then there's the geodesic motion of a point in a compact space of negative curvature;

• Historically, among the first laws of physics to be discovered were the laws of gravitational attraction, which govern the solar system (Newton 1687) and explain the simple, almost periodic motion of planetary orbits. We now know that these movements are in fact highly chaotic on the scale of millions of years. Poincaré (1890) pioneered the study of deterministic chaos with his famous three-body problem;

• the motion of a fluid with an infinite number of degrees of freedom is more complex. In the so-called turbulent regime, it appears chaotic. These models are very important in meteorology for understanding the dynamics of flow around aircraft wings, rivers, oceans and the atmosphere. As explained in this article, Lorenz (1965) proposed a very schematic and simplified model, obtaining a model with three variables, but which is very chaotic;

• In musical acoustics, the hip movement of a clarinet or bassoon under too much airflow has a chaotic dynamic that produces an unpleasant sound called "canard" .

The study of such dynamical systems has been going on for a long time now, and many techniques have been used and are still being discovered: linear algebra for the stability of fixed points and trajectories, functional analysis and probabilities to describe statistical properties, etc. Numerical experiments are also very useful for observing new phenomena. Numerical experiments are also very useful for observing new phenomena.

Note to the reader: the links to some of the video animations illustrating each part of this article can be found in the Websites section of the Further reading section. In this article we don't put the detailed...