# Limit of a dynamic system

• Jan 10th 2010, 12:17 AM
albus
Limit of a dynamic system
I have a dynamic system and initial condition is $u(0)=\left(\begin{array}{ccc}1&2&3\end{array}\righ t)$

System goes like this
$u(k+1)=A.u(k)$

$A = \left(\begin{array}{ccc}-1&2&0\\2&-4&0\\0&0&-3\end{array}\right)$

What happens if;

$\lim_{k\to\infty}u(k)=$
• Jan 10th 2010, 01:00 AM
CaptainBlack
Quote:

Originally Posted by albus
I have a dynamic system and initial condition is $u(0)=\left(\begin{array}{ccc}1&2&3\end{array}\righ t)$

System goes like this
$u(k+1)=A.u(k)$

$A = \left(\begin{array}{ccc}-1&2&0\\2&-4&0\\0&0&-3\end{array}\right)$

What happens if;

$\lim_{k\to\infty}u(k)=$

This limit does not exist (the sequence diverges).

As long as [1,2,3]' contains a component in the direction of an eigen vector corresponding to the largest absolute value eigen value and that is greater than 1 this will diverge with absolute value eventually growing by a factor equal to the largest absolute value of the eigen values. The iterates will eventually rotate so that they point closer and closer to the direction of the eigen vector in question (after adjusting for the sign anyway).

In this case the eigen value of largest absolute value is $\lambda=-5$

CB
• Jan 10th 2010, 01:19 AM
albus
Thank you, I had also found eigenvalues 0, -3 and -5 but thought that it will also goes to infinity.