Question 7.6.2: Consider the dynamical system x(t + 1) =[p −q q p] x(t), whe......
Consider the dynamical system
\vec{x}(t + 1) =\begin{bmatrix}p&−q\\q&p\end{bmatrix}\vec{x}(t),
where p and q are real, and q is nonzero. Examine the stability of this system. Sketch phase portraits. Discuss your results in terms of Theorem 7.6.2.
As in Theorem 2.2.4, we can write
\begin{bmatrix}p&−q\\q&p\end{bmatrix}=r\begin{bmatrix}\cosθ&−\sinθ\\ \sinθ&\cosθ\end{bmatrix},
representing the transformation as a rotation through an angle θ combined with a
scaling by r = \sqrt{p^2 + q^2}. Then
\vec{x}(t)=\begin{bmatrix}p&−q\\q&p\end{bmatrix}^t\vec{x}_0=r^t\begin{bmatrix}\cos(θ t)&−\sin(θ t)\\ \sin(θ t)&\cos(θ t)\end{bmatrix}\vec{x}_0,
representing a rotation through an angle θt combined with a scaling by r^t.
Figure 2 illustrates that the zero state is stable if r = \sqrt{p^2 + q^2} < 1.
Alternatively, we can use Theorem 7.6.2 to examine the stability of the system.
From Example 6 of Section 7.5, we know that the eigenvalues of \begin{bmatrix}p&−q\\q&p\end{bmatrix} are λ_{1,2} = p ± iq, with |λ_1|=|λ_2| = \sqrt{p^2 + q^2}. By Theorem 7.6.2, the zero state is stable if \sqrt{p^2 + q^2 < 1}.
