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Chaotic Dynamical Systems Project
$\text{Consider the following two maps:}$
$$ \text{map 1: } x_{n+1} = a - x_n^2 \\ \text{map 2: } x_{n+1} = asin(x_n),~ a \in [0, \pi] $$
Suppose $x^$ satisfies $f(x^) = x^$. Then $x^$ is a fixed point, for if $x_n = x^$ then $x_{n+1} = f(x_n)=f(x^)=x^$; hence the orbit remains at $x^$ for all future iterations.
$$ FP : f(x) = x \implies x_{n+1} = x_n \implies x = a - x^2 \iff x^2 + x - a = 0 \\[10pt] \therefore ~ x^* = \frac {-1\pm \sqrt{1+4a}} {2} \\[5pt] \text{(by quadratic theorem)} $$