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Finding the Local Equilibrium Point.
I am working on this problem. Could someone help me?
Problem:
Find the equilibrium points for the system $\begin{cases} x’=-2x-y \\\ y’=-2y-x \end{cases}$
Solution:
I need a more detailed guide on this topic. If someone could explain how to sketch this type of problem, I would be extremely grateful.
Thanks in advance
A:
The system is a coupled system of linear equations and is then equivalent to a single linear equation and its solution is given by $\alpha x + \beta y = 0$ (the reduced form) with $\alpha = \frac{1}{2}\pm \frac{\sqrt{3}}{6}i$ and $\beta = – \frac{1}{2}\mp \frac{\sqrt{3}}{6}i$. You can then find your eigenvalues and proceed with the solution. Since the system is autonomous, you can solve for $x$ and $y$ in terms of $x’$, $y’$, which you can then plug into the system.
$x’ = (\alpha – \frac{\alpha}{2})x – \alpha y = – \frac{1}{2}\alpha (x + \beta y) – \alpha y
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