Question

In Exercises 9–14, classify the origin as an attractor, repeller, or saddle point of the dynamical system \({x_{k + 1}} = A{x_k}\). Find the directions of greatest attraction and/or repulsion.

13. \(A = \left( {\begin{aligned}{}{.8}&{}&{.3}\\{ - .4}&{}&{1.5}\end{aligned}} \right)\)

Short answer

The origin is a repellor as both the eigenvalues are greater than one.

The direction of greatest repulsion is through the origin, and the eigenvector is \({v_1} = \left( {\begin{aligned}{}3\\4\end{aligned}} \right)\).

Step-by-step solution

Discrete Dynamical System

For anyDynamical System with matrix\(A\), the eigenvalues of\(A\)will determine the system of:

\(\begin{aligned}{}{\rm{Greatest attraction }} \to {\rm{if }}\lambda < 1\\{\rm{Greatest repulsion }} \to {\rm{if }}\lambda > 1\end{aligned}\)

Evaluating Eigenvalues of a given matrix 

The given matrix for the dynamic system \({{\bf{x}}_{k + 1}} = A{{\bf{x}}_k}\) is:

\(A = \left( {\begin{aligned}{}{.8}&{}&{.3}\\{ - .4}&{}&{1.5}\end{aligned}} \right)\)

For eigenvalues, we have:

\(\begin{aligned}{}\det \left( {A - \lambda I} \right) &= 0\\\left| {\begin{aligned}{}{0.8 - \lambda }&{}&{0.3}\\{ - 0.4}&{}&{1.5 - \lambda }\end{aligned}} \right| &= 0\\{\lambda ^2} - 2.3\lambda + 1.32 &= 0\\\lambda &= 1.1,1.2\end{aligned}\)

Here, both the eigenvalues are greater than 1.

Hence, the origin is a repellor.

Eigenvector for eigenvalue 1.2

Now, for eigenvectors, we have:

At \(\lambda = 1.2\).

\(\begin{aligned}{}\left( {A - \left( {1.2} \right)I} \right){v_1} = 0\\\left( {\begin{aligned}{}{0.8 - 1.2}&{}&{0.3}\\{ - 0.4}&{}&{1.5 - 1.2}\end{aligned}} \right)\left( {\begin{aligned}{}x\\y\end{aligned}} \right) = 0\\\left( {\begin{aligned}{}{ - 0.4}&{}&{0.3}\\{ - 0.4}&{}&{0.3}\end{aligned}} \right)\left( {\begin{aligned}{}x\\y\end{aligned}} \right) = 0\end{aligned}\)

Also, the system of equation will be:

\(\begin{aligned}{l} - 0.4x + 0.3y = 0\\ - 0.4x + 0.3y = 0\end{aligned}\)

For this, the general solution, we have:

\(\begin{aligned}{}{v_1} &= \left( {\begin{aligned}{}x\\y\end{aligned}} \right)\\ &= \left( {\begin{aligned}{}{\frac{3}{4}y}\\y\end{aligned}} \right)\\ &= \left( {\begin{aligned}{}3\\4\end{aligned}} \right)\end{aligned}\)

Here, the eigenvector is: \({v_1} = \left( {\begin{aligned}{}3\\4\end{aligned}} \right)\).

Hence, the direction of greatest repulsion is through the origin and eigenvector: \({v_1} = \left( {\begin{aligned}{}3\\4\end{aligned}} \right)\).