University of Georgia symbol

Analyzing Single Loops and Triple Loops in the equation:


Joshua Traxler

Here we explore the set of equations that may be represented as where n is a real number. Let's start with a simple case of n=1. At this point, the equation becomes: . Now this can be rewritten as . We observe that x can not be negative because the left would always be negative and the right side of the equation is a square, which will always be positive. Now let's try to better understand this simple case by checking some points. If x equals zero. Then we see that must equal zero. If x=1, then 1- is 1. Therefore, must either be zero or 2. Meaning y must be either zero or the positive/negative square root of 2. The seems to indicateNow let's take a look at the graph for a clearer picture:

Looking at the graph, we see that the points we have chose reprsent key places where the graph shifts in nature. We may ask ourselves why the loop apears between zero and one but not elsewhere. This is because x to the fifth power will be smaller than x squared during this interval. This allows another solutions where is a positive value. The solutions where is a negative number represents the curves that go off to infinity.

Now let's observe another simpler case, where n=0. This generates the equation: , which may be simplified as . We notice that this has pretty much all the same properties as except the right side of the equation is now negative when y is greater than x. Therefore, we would expect this graph to look almost identical in form except that the non-looping curves go off in the negative direction rather than the positive direction. We observe the graph to confirm:

So, what happens inbetween n=zero and one that causes these two similar graphs to morph into one another? Well, we notice that the right side is no longer a perfect square, meaning that negative values can potentially occur for x. This occurs either when negative or when is negative.  The negative value is likely to be small, but since is very small for low values of x, we expect there to some low value solution.  The smaller the value of n, the greater the difference between and allowing for negative values of x to have greater magnititude.  The sign of y is arbitrary, giving four possible solutions for each valid value for x. We view the graph for various values of n to get a fuller picture.





So, we see that there is a smooth transition with the properties described. Eventually the loops on the left unwind into the non-looping curves of n=0. The non-looping curves on the right eventually approach a vertical line n approaches zero. This can be seen clearly in the following video of the n transitioning from zero to one and back.


By the same principle, when n grows beyond one, and may differ in sign, allowing to have a negative value.  So, we expect to see a similar transformation for when n grows beyond one:


The behavior of this graph is similar in this interval of n, as predicted. 


This problem goes through several interesting transformations.  The most interesting patterns are typically generated when the right side of equation is able to be negative under certain constraints. When this is not possible it looks about like a regular strophoid.