The map X(n+1) = [X(n)]^3 + a*Xn has a fixed point at the origin. For what
values of a is this fixed point stable?
So stable meaning is will always gradually be along the line of 0?
So would a have to be -[X(n)]^2
?
A point is a stable fixed point if when you start near the point all subsequent points are also near the point.
Near zero your difference equation becomes:
$\displaystyle x_{n+1}$=$\displaystyle ax_n$
This is clearly unstable when $\displaystyle |a|>1$. It is also unstable when $\displaystyle a=1$ and stable when $\displaystyle a=-1$ (but to show this you need to look at the sign of the non-linear term), and it is stable if $\displaystyle |a|<1$
CB