Chaos Part III: Visualising Chaos

How is one system more chaotic than another? To answer this question we need a way to measure chaos. This can be challenging, with quantitative measures requiring some serious mathematics. There is however, a rather beautiful visual tool that can offer assistance; the ‘flip plot’.

This is typically used to analyse the double pendulum, which is a classic example of a chaotic system. Two double pendulum systems with almost identical initial conditions will diverge significantly over time, thus exhibiting the hallmark of chaos. This is in contrast to two single pendulums, that will maintain the same distance apart in phase space when released from two very similar initial conditions.

Employing the flip plot to analyse the double pendulum system, one can time how long it takes for either pendulum to flip over vertically (attain an angle with respect to the downward vertical with magnitude greater than 180 degrees), when varying the initial starting angles of the system. The angle to the vertical made by the first pendulum can be plotted on the x-axis and the angle of the second pendulum is on the y-axis.

This means that each pixel within the flip plot image represents the initial starting angle of each pendulum. The colour of each pixel is determined by the time taken for a flip to occur. Brighter pixels represent longer flip times.

This amazing plot contains a multitude of fascinating features, in addition to resembling the Eye of Sauron. The eye itself appears because in that region it is energetically impossible for a flip to occur. A double pendulum with angles in the central region does not possess enough potential energy for a flip to occur.

The real charm of this plot, however, is revealed in the sweeping arms protruding from the central region. This is chaos visualised! Configurations with very similar initial conditions (pixels next to each other), have massively varying flip times (large contrast in colour).

Another interesting feature are the so called ‘islands of stability’ scattered throughout the plot. These are regions that do not flip but are surrounded by chaos.

In the case of the double pendulum, the flip plot actually represents the phase space of the double pendulum system. To model the trajectory of the double pendulum, one only needs to know the two angles, theta 1 and theta 2. It is truly remarkable how much information is contained within this plot. In the following parts, we will explore the possibility of using a flip plot to actually measure chaos!