Zero State, Stable Equilibrium, Dynamic System

[essie52]

Could someone please help?

The question reads:

For which real numbers "k" is the zero state a stable equilibrium of the dynamic system (vector(x))(t+1)=A(vector(x))(t)?

A= [0.1 k $0.3 0.3] --> a 2 x 2 matrix with$ separating the two rows.

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 (2 +/- (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 2 +/- (sqrt(-1-30k)) < 8?

Thanks!

 

[essie52]

Got it! Thanks anyway!