Introduction
It all started with a short story titled “The Butterfly Effect.” I came across it one day and really enjoyed it, and decided to research more about what exactly the butterfly effect was. Like most people, I had only heard about the butterfly effect in a metaphorical and literary sense – something small can lead to a series of changes that eventually results in a big change. I started reading articles about it, and I soon realized that the metaphorical interpretation was in fact very similar to the scientific derivation. However, to explain this further, I will zoom out to the chaos theory, which is what the butterfly effect is part of.
To understand the chaos theory, one would first have to understand what dynamical and deterministic systems are. A dynamical system is a mathematical concept where state changes over time in geometrical space according to a fixed rule. In other words, the dependency on time a position of a point has is expressed by a function. For example, in the case of a deterministic system such as a pendulum, a function can be used to calculate the state of a pendulum (angle and angular velocity) at any given instant. However, this is also because a pendulum is a deterministic system, as opposed to a random system, meaning that each starting point can have only one outcome so it is predictable if you know the initial starting conditions.
The chaos theory studies the behaviour of deterministic dynamical systems that have these three characteristics: dense periodic orbits, topologically mixing phase space, and high sensitivity to initial conditions. However, while there is some more information about the first two characteristics in my discussion, my main focus is on high sensitivity to initial starting conditions.
High sensitivity to initial starting conditions is when a minute change in these conditions will yield a significantly different result. This sensitivity is what we know as the butterfly effect. It was named for the ability of a seemingly small event (such as the flap of a butterfly’s wings) to, by altering the initial starting conditions of, have a great effect on a large event (such as the path of a tornado). However, due to this sensitivity, we cannot predict the future of a chaotic system despite them being deterministic. In fact, it is because of their sensitivity to their deterministic nature that they are unpredictable. Their heavy dependency on the preciseness of initial starting conditions drives the need for exactness to a precision we cannot pinpoint, rendering chaotic systems virtually impossible to recreate. For a double pendulum, this means that a miniscule difference in the initial angle or velocity can result in a drastically different path.
For my project, I decided to determine whether or not a double pendulum is a chaotic system by observing its behaviours (via computer simulation) and comparing them with the characteristics of a chaotic system. In order to classify a system as chaotic, it must fit the three requirements above. However, for my experiment, I will only be focusing on high sensitivity to initial starting conditions for the sake of time. I’ve researched the topological transitivity and periodic orbit density of a double pendulum, and it meets both requirements. In my hypothesis, I will assume that these are true. For a double pendulum, a high sensitivity to initial starting conditions would mean that the difference in the paths taken by two pendulums with different initial starting angles/velocity must increase as time goes on. I chose a double pendulum because it seemed to be the most plausible choice due to its accessibility (as it is possible to do it virtually) and simplicity compared to other systems.