Using the linearization theorem for the system of differential equations

Question

For the system of differential equations $$\dot{x} = xy - 1,$$ $$\dot{y} = x^2 - y^2$$ sketch the phase portrait of the system near the fixed points. Use the linearization theorem.

Attempt: To analyze the system near its fixed points using the linearization theorem, we first find the fixed points by setting $\dot{x} = 0$ and $\dot{y} = 0$. Solving the equations $xy - 1 = 0$ and $x^2 - y^2 = 0$ simultaneously, we obtain the fixed points $(1, 1)$ and $(-1, -1)$. Next, we linearize the system by calculating the Jacobian matrix:

$$ J = \begin{pmatrix} \frac{\partial \dot{x}}{\partial x} & \frac{\partial \dot{x}}{\partial y} \\ \frac{\partial \dot{y}}{\partial x} & \frac{\partial \dot{y}}{\partial y} \end{pmatrix} = \begin{pmatrix} y & x \\ 2x & -2y \end{pmatrix}. $$

Now that I got Jacobian, how can I sketch the phase portrait? Any ideas?

Answer 1

We will use LibreTexts: Stability and classification of isolated critical points as a guide.

You evaluate the eigenvalues of Jacobian at each critical point while making sure to watch for marginal cases like centers which require further analysis.

The eigenvalues for $(1,1)$ are

$$\left(\frac{1}{2} \left(-\sqrt{17}-1\right),\frac{1}{2} \left(\sqrt{17}-1\right)\right)$$

These are real and opposite sign which is an unstable saddle per the LibreText table.

As per a comment by @OscarLanzi, the eigenvalues are inverted about the origin, so the two sets of eigenvalues should also be negatives of each other, Thus for $(-1,-1)$, they are

$$\left(\frac{1}{2} \left(\sqrt{17}+1\right),\frac{1}{2} \left(1-\sqrt{17}\right)\right)$$

These are real and opposite sign which is an unstable saddle per the LibreText table.

So, our linearization shows we have two unstable saddles and do not have to worry about centers. That is generally enough to draw a rough order phase portrait from the LibreText link.

Since we know the critical points and their stability, now we can proceed to using A quick guide to sketching phase planes to draw a more detailed phase portrait and arrive at something like (try and fill in the manual details)

You can also use the second tab on this tool and others online to draw it. Click in the image to add streamlines.