# stable fixed point

• Oct 24th 2009, 03:13 PM
chrisc
stable fixed point
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
?
• Oct 24th 2009, 11:25 PM
CaptainBlack
Quote:

Originally Posted by chrisc
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