Suppose next that we wish to look at period-2 cycles. That is, we want to find two points and such that, and .
Let us start by writing, and see where trying to solve this leads.
Thus, the equation we wish to solve is actually .
This equation is a polynomial of degree 4, and so has 4 (possibly non-distinct) solutions. However, actually, we already know 2 of the solutions. They are and, computed above. It is simple to see why this is; if these points are left unchanged by 1 application of, then clearly they will be unchanged by 2 applications (or more).
Our 4th-order polynomial can therefore be factored in 2 ways :
Read more about this topic: Periodic Points Of Complex Quadratic Mappings
Famous quotes containing the word cycles:
“The stars which shone over Babylon and the stable in Bethlehem still shine as brightly over the Empire State Building and your front yard today. They perform their cycles with the same mathematical precision, and they will continue to affect each thing on earth, including man, as long as the earth exists.”
—Linda Goodman (b. 1929)