Could someone please help?
The question reads:
For which real numbers $k$ is the zero state a stable equilibrium of the dynamic system
$x_{t+1} = Ax_t$?
$A = \begin{bmatrix} 0.1 &k \\ 0.3 & 0.3 \end{bmatrix}$
So, my thought is I need to find the eigenvalues. In order to do this I calculated the characteristic polynomial as
$x^2 – 0.4x + 0.03 – 0.3k = 0$, with $x$ representing eigenvalues.
Using the quadratic formula I found that the (real) eigenvalues are
$$x = \frac{2 \pm \sqrt{1+30k}}{10}$$
and for the zero state to be in stable equilibrium $\sqrt{1+30k}<8$. Hence, $k<21/10$ (for stable equilibrium).
My question is how do I figure out the values for $k$ if the eigenvalues are complex?
Do I solve the inequality $\sqrt{-1-30k} < 8$?
Thanks!
Best Answer
In general, the zero state of a dynamical system of the form $x_{t+1} = Ax_t$ is a stable equilibrium when all of the eigenvalues of $A$ are inside the unit circle in the complex plane. So if you have an eigenvalue of the form $a+bi$, you get stability when $a^2 + b^2 < 1$. Since this is a homework question, I'll let you work out the rest of the problem from here. (This isn't going to be equivalent to solving $\sqrt{-1-30k} < 8$, though.)