The Butterfly Effect: What is it really?
Almost everyone has heard of some statement of the butterfly effect, the most common being “It has been said that something as small as a butterfly flapping its wings can cause a typhoon halfway across the world”. While the extremity described in the quote is very rare, the butterfly effect is a just one example of a branch of mathematics called Chaos Theory, which we will be discussing at a very basic level in this article.
Chaos Theory, at an extremely basic level, studies how small changes can affect large scale systems. It has applications in weather forecasting, as described in the quote above, as well as many fields of science and engineering.
Chaos Theory is fundamentally based on Initial Value Problems (IVP’s), which consist of two parts: a differential equation, and an initial condition. A differential equation is a relation between a function and its derivatives, and an initial condition is a value that the function must take at a certain point. An example of an IVP is:
The solution to the above initial value problem would be:
which can be easily obtained by integrating and plugging in the initial condition. The above IVP would be categorized as “first order”, since only the first derivatives of the function are involved, An IVP that involves a function’s second derivative would be classified as “second order”, and similarly for third derivatives “third order”.
The fundamental idea of chaos theory is that for a high order IVP, (third order and higher), tiny changes in initial conditions can lead to drastically different solutions.
To examine this phenomenon, I will consider 3IVP’s of increasing order for a quartic equation, and describe how the solutions change with different initial conditions.
Case 1: First Order IVP
We want a first Order Differential Equation, so our ideal choice would be something similar to the equation , Note that because our Equation is first order, we only need one initial condition.
The general solution to this equation is shown below:
Above are shown two solutions to the first order differential equation: One with initial condition f(0) = 0 (red) , and f(0) = 1(blue). We can see from the two functions above that a small change in these initial conditions doesn’t really change too much about the solutions.
Now, let’s consider a second order differential equation to describe a quartic function. We can easily write:
Note that because this is a 2nd order differential equation, we need two initial conditions to obtain a unique solution. In general, a differential equation of degree n requires n initial conditions to solve for a unique solution.
The general solution to the second order D.E above is given by:
Above are two solutions to the 2nd order D.E. The red graph shows the solution when subject to conditions {y(0) = 0; y’(0) = 0], and the blue equation shows the solution subject to conditions {y(0) = 1; y’(0) = 1}. We can see that although the difference here is still not large, it is much more noticeable, primarily because of the shift by 1 to the left.
Let’s further explore this by considering a 3rd order D.E. A 3rd order differential equation. The 3rd order differential equation corresponding to a quartic equation is:
The general solution to the above equation is shown below, where p, q, and r are arbitrary constants.
Below are two solutions of the form above:
The red graph corresponds to a solution subject to conditions {y’’(0) = 0, y’(0)=0, y(0) = 0}, and the blue graph corresponds to the solution subject to conditions {y’’(0) = 1, y’(0) = 1, y(0) = 1}. We can already see how these two graphs differ, especially close to the origin. There is again a vertical and horizontal.
Overall, we can see how small changes in the initial conditions have affected the solutions in the three graphs above, especially in the 3rd order equation. While the changes aren’t as drastic in the examples above, small changes in these initial conditions can have large effects in weather, science, and engineering.
Chaos Theory is a fascinating subject. What I have described above is only an introduction, and a very basic one at that.
Thanks for Reading!