The question reads: For which real numbers k is the zero state a stable equilibrium of the dynamic

Winigefx 2022-07-01 Answered
The question reads: For which real numbers k is the zero state a stable equilibrium of the dynamic system x t + 1 = A x t ?
A = [ 0.1 k 0.3 0.3 ]
So, my thought is I need to find the eigenvalues. In order to do this I calculated the characteristic polynomial as
x 2 0.4 x + 0.03 0.3 k = 0, with x representing eigenvalues.
Using the quadratic formula I found that the (real) eigenvalues are
x = 2 ± 1 + 30 k 10
and for the zero state to be in stable equilibrium 1 + 30 k < 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 1 30 k < 8?
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Answers (2)

frethi38
Answered 2022-07-02 Author has 16 answers
In general, the zero state of a dynamical system of the form x t + 1 = A x 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 + b i, 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 1 30 k < 8, though.)

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George Bray
Answered 2022-07-03 Author has 12 answers
Good answer. Just to be clear, though, one has asymptotic stability if a 2 + b 2 < 1. The system is generally considered stable if a 2 + b 2 = 1.

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