Pendulum

Set up pendulum and power it, this should illuminate the leds and point them at the webcam, open exposure software and start. Set the pendulum in motion and the laptop should record the path. Can people notice a pattern.

Explain how a single pendulum forms a nice pattern, in fact we can predict its period and figure it out, however the double pendulum we can't predict. There's obviously an equation for its motion (we combine pendulum equations) it's just really complicated.

In fact we can observe flips and 360s of both pendulums, something which can't happen with single pendulums. It's really quite difficult to explain the chaos with out Lagrangian physics however by repeating the experiment and trying to recreate the same starting positions you should be able to demonstrate it.

There are actually some non-chaotic normal modes of the pendulum, the main two modes are demonstrable. The first one has both arms roughly inline and not too large an angle, this acts like a normal pendulum however the two arms have different frequencies. The second has the ends of the arm mirrored in the pivot and the arm move in the opposite direction.

Turbulent mixing

Butterfly effect

Fractals

If you want to take a more mathematical approach we can talk about fractals. A great way to introduce this is by playing the CHaOS game, start by drawing a triangle (any works) and colour the corners red, blue and green. Take a dice with equal probability of rolling red blue or green. Start by placing a dot inside the triangle (in fact this dot can be placed anywhere however convergence is slow), roll the dice and place a new dot at the mid point of the dot and the corner corresponding to the colour rolled. What happens if we keep doing this, try it again from the new dot, and the next one. Try it at least 5 times, do we think we'd see anything if we did a thousand times? Will we fill the entire triangle? Or will their be regions with very few points? In fact we end up with a fractal called the Sierpinski Triangle, mathematically the convergence is almost sure. We call the result of this process the 'orbit' of the start point ('seed') and each step an iteration, this is part of a whole range of mathematics called dynamical systems which also deals with things like population models.

We make a Sierpinski triangle by dividing the triangle into 4 pieces and deleting the middle section, then doing this again. We can then see how a point tends to the Sierpinski triangle, pick a point in the initial triangle removed, when we roll the dice it will move to one of the next three removed triangles, on the next iteration it will move to one removed in the next stage. However these triangles are getting smaller and smaller so these points are getting closer to the Sierpinski triangle. We say the orbit is attracted to the Sierpinski triangle and call the triangle a strange attractor.

With fractals we talk about self-similarity, looking at the triangle we see there are 3 copies of the triangle each half the size (magnification factor of 2). We can then talk about dimension, but we need to understand what this means. One thing people may say is about number of directions you can move, sizes of a linearly independent basis is one way to define dimension but we will look at another. Consider cutting a straight line down the middle, we get two pieces each with a magnification factor of 2. Cutting a square into 4 pieces each with magnification factor 2. So the dimension can then be given by the log(pieces)/log(mag). This agrees with lines, squares and cubes and gives the triangle dimension log(3)/log(2)~1.58.