When simple things interact new behaviour emerges.

The Double Pendulum

A double pendulum is like the upper and lower parts of your arm: pinned at the shoulder and swinging freely with a hinge at the elbow. Each part of the arm swings like a simple pendulum on its own. But when they swing together, the interaction generates new, complicated, behaviour.

 

Newton and Chaos

The motion of the pendulum is described by mathematical formulas based on Newton’s laws of motion.

Unfortunately, these equations are too complicated to solve by hand. Computer simulations help us explain why the double pendulum moves in very complicated ways. The figure below shows how long it takes the lower arm to do a 360° ‘loop the loop’, starting from different initial arrangements of the arms.

The x axis is the starting angle between the upper arm and the vertical, and the y axis is the angle between the lower arm and the vertical. Colours show how long you have to wait before the lower arm ‘loops the loop’. In a green blob it happens quickly, red ones take longer, and in the yellow bit it never happens! The sharp boundaries between colours show that small changes to the initial arrangement have a big effect on the waiting time: this shows the double pendulum's ‘sensitivity to initial conditions’.

 

Coffee and Weather

Because the double pendulum and the weather obey similar kinds of mathematical equations, they have similar kinds of solutions – and that means chaos. The same ‘sensitivity to initial conditions’ that occurs in the double pendulum makes weather prediction a big challenge!

Looked at close up, atmospheric flows are irregular and turbulent. From space we can see a bigger picture: structures such as hurricanes, typhoons and zonal jets emerge.

 

Planets and Chaos

Stars and planets interact through gravity. A century ago, Henri Poincaré noticed that when three bodies (stars or planets) were attracted to each other under gravity, it was possible that their motion would be chaotic, and, as a result, unpredictable.

 

Although chaos in the solar system makes long term predictions (for example about whether asteroids will hit the Earth) difficult, chaos can also be useful, for example in planning trajectories for spacecraft. And there’s a long mathematical journey for the next century too, through strange and attractive new worlds. Come and enjoy the scenery!

A scenic view of a Lorenz Attractor

(Image: Hinke Osinga and Bernd Krauskopf, Dept of Engineering Mathematics, University of Bristol)