Word of caution: this is research I did in high school - treat it very informally. To know more read James Gleick's Chaos, Steven Strogatz's Nonlinear Dynamics, or Deep Simplicity by John Gribbon. Anyways, grab a coffee or tea as this'll be a winding journey.
So, it started with my reading The Three Body Problem by Cixin Lui in which there was an extraterrestrial civilization that lived in a solar system with 3 suns that moved with chaotic motion. Every day there was a risk that one of the suns would come too close and fry them all. There was no way to predict whether or not a sun would even rise the next day. All of this seemed to contradict what I’ve learnt in physics classes so I decided to investigate chaos. To do so, I analyzed the motion of a double pendulum, a figurehead of chaos theory which consists of two regular pendula attached to each other with a movable joint.
So, if you raised the pendulum to different angles and released it, which angles would lead to it acting chaotically and which angles would lead to it acting like a normal single pendulum? To answer this we first need a quick summary of what chaos is.
It's a theory that describes dynamic (moving) systems which can act chaotically. Chaotic motion has no universally agreed upon mathematical definition, instead describing any dynamic system with some general characteristics:
In practice, though, we always have measurement errors and limits on precision which will dramatically affect the evolution of the system regardless of their magnitude.
Many systems in real life demonstrate chaotic motion. Some examples include how clouds form, the flow of any fluid, the stock market, why they make golf balls bumpy, the weather, and even our own solar system.
A key feature of any dynamic system is the attractor. An attractor is a state or set of states a system tends to over time. It can be represented as a point or line or region in phase space, a graph with multiple parameters of the system’s current state (e.g. velocity, position, angle, etc). A normal pendulum on Earth with air friction has a fixed-point attractor (a one-dimensional attractor) at 0 degrees to the vertical (straight down), and zero velocity. The pendulum will swing for a while but eventually be attracted to the motionless, vertical state.
On the other hand, the phase space of a chaotic system is not attracted to any certain point. This is because it follows a strange attractor, an attractor that is in the shape of a fractal. A fractal is a shape with infinite side length but within a confined space. This is possible because a fractal’s sides are infinitely jagged and with more jaggedness comes more side length. No matter how far you zoom in to the image below, you will always see more complexities on the side of the black shape. Each seemingly straight edge is just further turns and curls.
Strange attractors are fractal because the trajectories within a complete phase space (with all dimensions of motion) of a chaotic system cannot overlap. Overlapping in omni-dimensional phase space would correspond to the system having the exact same state, which would lead to continuing on the same path after that overlap in consideration of Newtonian mechanics and determinism, which chaos theory does not violate.
This means a frictionless double pendulum will never have the exact same state twice after releasing it. The same reason explains why the phase space trajectories for periodic systems like single pendula are identical and always overlapping.
After recording the first 5 to 10 seconds of the pendulum’s motion, after which it comes to rest anyway, I worked the footage with a software called ImageJ. ImageJ tagged each joint as indicated by the yellow tape individually in each frame and gave numerical values of their positions. Each frame ImageJ could analyze was separated by 0.02 seconds. The black background and yellow tape was to increase their contrast, making it easier for the software to follow the joints as they moved around.
I used the data analysis software Origin 8 to compile and process all of the data. I tested each starting angle starting from 90° and increasing with increments of 15° and ending at 180°. Although aligning the position of the joint with the hole of the angle board was done by eye, the inherent error was actually necessary because the small differences in initial angle are what triggered the exponential divergence that is the telltale sign of chaos, as will be explained later. The pendulum was released and recorded 3 times per angle, and one complete set of data was extracted for each frame of each trial. This filled over 132,000 cells of data. A short Origin 8 script calculates angles from the Cartesian positional data, shown below.
The easiest way to present all of the angles is to plot the angle of the lower arm against the angle of the upper arm in a phase space. Parameters such as acceleration and velocity are omitted so the phase space can stay 2-dimensional. This type of 2-angle phase space is common for showing the trajectory of a double pendulum. Beside the phase spaces are sample screenshots showing the actual tracks of each trial as pieced together by ImageJ that are useful for qualitative understanding. The tracks are two different colors, one for each joint.
In all of the phase spaces you can see the tracks converge onto the center (0, 0). This corresponds to the rest state of the pendulum, hanging straight down, and the point (0, 0) is therefore a fixed-point attractor. The fact it comes to a stop and is attracted to this point is an immutable feature of all dissipative systems, but you can still see the unique motion of the pendulum before it loses too much energy due to friction.
You can see in the phase spaces of 90°, 105°, 120°, and 150° the 3 different tracks look very similar, and indeed their motion was very similar as is qualitatively visible in the snapshots of their trials’ motions. Conversely, the phase spaces of angles 135°, 165°, and 180° all start at nearly exactly the same position on the graph, theoretically (165, 0) or (180, 0), and continue in a similar fashion towards the middle at which point they diverge and there is no similarity in their motions.