Subsection 2.2.2 Finding periodic orbits
Continuing with the example of the last sub-section, suppose we'd like to find all orbits of period 2 for \(f(x) = \frac{1}{5}x^2 - 2x + 6\text{.}\) I guess we should let \(F(x)=f^2(x)\) and then find all fixed points of \(F\) or, equivalently, find all roots \(F(x)-x\text{.}\) We can automate this procedure like so:
Each root is returned as a (value,multiplicity) pair. Note that the 1.38 and 3.618 agree with our prior computation; those form the orbit of period 2. The other points are fixed points. Of course, those should be the points of intersection with the line \(y=x\text{.}\) We can illustrate that with Sage too: