# Stable, but not asymptotically stable

• Feb 25th 2009, 02:35 AM
Naena
Stable, but not asymptotically stable
Hi everyone.

My task from difference equations is:

Lets suppose we have a nonlinear dynamic system x(t+1) = f(x(t)) , the status space X from R (real) and a fixed point xHAT.
Find a function f from C1 and a set X from R (real) , so that f ' (xHAT) = 1 and xHAT is stable, but NOT ASYMPTOTICALLY stable fixed point of our system.

Any suggestions?

Thx,
Naena
• Feb 25th 2009, 07:49 AM
HallsofIvy
Quote:

Originally Posted by Naena
Hi everyone.

My task from difference equations is:

Lets suppose we have a nonlinear dynamic system x(t+1) = f(x(t)) , the status space X from R (real) and a fixed point xHAT.
Find a function f from C1 and a set X from R (real) , so that f ' (xHAT) = 1 and xHAT is stable, but NOT ASYMPTOTICALLY stable fixed point of our system.

Any suggestions?

Thx,
Naena

Look at f(x)= x- 1.
• Feb 25th 2009, 10:06 AM
Naena
and a fixed point?
I doubt its correct, cause the task is not only to find a function with f ' = 1 in fixet point xHAT, but it also has to have a stable fixed point xHAT, which means xHAT=f(xHAT). :(