## Chaos IX: Chaotic or not?

There are many kinds of dynamics. Some are complicated, others are not. To try and understand this better, we can take a vector field that depends on just one parameter, and let this parameter change slowly. This shows that the dynamics, under influence of this parameter, is sometimes simple and then, without warning, becomes very complicated. We see bifurcations happening.

Chaos IX: Chaotic or not?

## Chaos VII: Strange Attractors

In 1963, Edward Lorenz (1917-2008), studied convection in the Earth’s atmosphere. As the Navier-Stokes equations that describe fluid dynamics are very difficult to solve, he simplified them drastically. The model he obtained probably has little to do with what really happens in the atmosphere. Read More

## Chaos VI: Chaos and the horseshoe

First, an old idea by Henri Poincaré (1854-1912): when studying a vector field in space, we can sometimes find a small disc that the trajectories hit repeatedly. Studying the points on the disc where the trajectories pass through is often a lot simpler than studying the vector field as a whole. We go from dynamics in continuous time to dynamics in discrete time.

Chaos VI: Chaos and the horseshoe

## Chaos V: Billiards

Studying the motion of celestial bodies is quite complicated, so lets’s take a simpler example. The motion of a ball rolling in a bowl does not seem too difficult to understand, but if their are a few bulges in the bowl, then the motion becomes very complicated!…
Chaos V: Billiards

## Chaos IV : Oscillations

We need two numbers to describe a swinging pendulum: one is its position, the angle versus a vertical line, and the other is its speed, the sign of which indicating that it moves to the right or to the left. Read More

## Chaos II: Vector fields

At the end of the 17th century, Gottfried Wilhelm Leibniz (1646-1716) and Isaac Newton (1643-1727), independently one from the other, invented a brilliant mathematical tool: infinitesimal calculus or differential and integral calculus. Read More

## Chaos I: Motion and Determinism

Chaos I: Motion and Determinism
The start of Chaos, with one of the foremost ideas of philosopher Heraclitus of Ephesis, who lived in the sixth century B.C. Creatures develop eternally, things have no substance and everything is always on the move: everything becomes everything, everything is everything. The first minutes of the film illustrate this idea with some everyday examples, as well as some mathematical ones.