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## Homework Statement

Give an example of a non-linear discrete-time system of the form

x

_{1}(k + 1) = f

_{1}(x

_{1}(k), x

_{2}(k))

x

_{2}(k + 1) = f

_{2}(x

_{1}(k), x

_{2}(k))

With precisely four singular points, two of which are unstable, and two other singular points which are asymptotically stable.

## Homework Equations

[tex]

J = \begin{bmatrix}

\frac{\partial f_1}{\partial x_1} &

\frac{\partial f_1}{\partial x_2} & \\[1ex]

\frac{\partial f_2}{\partial x_1} &

\frac{\partial f_2}{\partial x_2} &

\end{bmatrix}

[/tex]

## The Attempt at a Solution

I know that the singular is asymptotically stable if the eigenvalues of the Jacobian are under 1, and unstable if an eigenvalue is greater than 1. I've found a solution in MATLAB through brute force, but this is a practice exam question; how would I be able to do this by hand just by looking at it?