Chaotic Waves

Tasks

Tilt the wave machine to make two simultaneous waves.

Are the two waves the same or different?

Can you make two waves the same?
Maths

Chaotic waves is an example of chaos theory. Even though the waves are created simultaneously, small differences will lead to very difference results.

Chaotic systems are not random. In fact, the underlying maths can be quite simple. It is their sensitivity to initial conditions that makes them so unpredictable.

This is sometimes known as the “butterfly effect” which is the poetic idea that a butterfly’s wings can change the course of a tornado halfway around the world.

Chaotic behaviour exists in many natural systems, including weather, climate and road traffic.
History

The idea that small changes can lead to very different results started to appear in the late 1800s.

French mathematician Henri Poincaré noticed that it was possible for some planets to have an orbit that never repeated. Later, another French mathematician, Jacques Hadamard, devised a game of billiards where two balls starting very close together would result in very different paths. These observations would later become important features of chaos theory.

In the 1960s, American mathematician Edward Lorenz noticed that small changes in weather simulations lead to very different results. This showed that even very precise measurements cannot be used to make long-term weather predictions.
People

Henri Poincaré 1854 – 1912
Poincare was a French mathematician and theoretical physicist. In the 1880s, Poincare was studying complex orbits of planets (the three body problem), when he found that there could be orbits that never repeated, an important feature of modern chaos theory.

Poincare’s conjecture was a question about spheres in four dimensions. It was one of the great unsolved problems of 20th century mathematics and was worth one million dollars. The problem was solved in 2003 by Russian mathematician Grigori Perelman.

Edward Norton Lorenz 1917 – 2008
Lorenz was an American mathematician and meteorologist. In the 1950s, Lorenz started using computers to forecast weather based on observational data. In 1961, noticed that small changes in initial conditions lead to large long-term changes. This observation was one of the foundations of Chaos Theory.
Applications

The effects of chaos theory are found in weather forecasting, so even with very accurate measurements it is impossible to predict weather patterns more than ten days in advance.

Chaos effects are very useful in cryptography, where repeated applications of simple ciphers on two similar messages end up giving very different results. This makes secret messages much harder to work out.

Chaos can also be found in ecology (where certain species exhibit chaotic behaviour in population growth), robotics (where robots are designed to explore their environment chaotically), and AI (where similar inputs can produce different results).
Maths at Home

The shape of a wave is called a trochoid, and it is the shape traced by a point on a circle as it rolls along the ground.

The point tracing the wave doesn’t have to be on the edge of the circle, it can be a point inside the circle as well. This will create smaller, flatter waves.

Try to make a wave shape yourself using this online Spirograph (called Inspirograph) https://nathanfriend.io/inspirograph/

What do you need to make a wave shape? How does the shape change when you use a point closer to the edge or closer to the centre?

From the circle, what is the distance between peaks? What is the height of the wave?