System of differential equations, phase portraits and stability of fixed points

Question

Consider the system of differential equations:

$$x'=-x-y+4$$ $$y'=3-xy$$

a. Find the fixed points.

$x'=-x-y+4$

$x+y=4$

$x+3/x=4$

x=3,x=1

$y'=3-xy$

$y=3/x$

fixed points: (1,3), (3,1)

b. Determine the type of the linearized system at each fixed point.

calculating the Jacobian:

\begin{array}{cc} -1 & -1 \\ -y & -x \\ \end{array}

for the fixed point (1,3): \begin{array}{cc} -1 & -1 \\ -3 & -1 \\ \end{array}

calculating eigenvalues:

$λ_1=-1-\sqrt 3$ (could be positive or negative) $λ_2=\sqrt 3-1$ (negative)

So it is unstable (I think, because if we use the negative root of 3, then the first eigenvalue is positive, is this correct?

for the fixed point (3,1): \begin{array}{cc} -1 & -1 \\ -1 & -3 \\ \end{array}

calculating eigenvalues:

$λ_1=-2-\sqrt 2$ (negative) $λ_2=\sqrt 2-2$ (negative)

So it is stable

C. Determine the nullclines and the signs of $x'$ and $y'$ on the nullclines and in the various regions determined by them. (I'm not sure I am calculating the signs of $x'$ and $y'$ correctly)

y-nullcine: $y'=3/x$ x-nullcine: $y=4-x$

R1

$x'<00$ $y'>0$

R2

$x'<0$ $y'>0$

R3

$x'>0$ $y'<0$

R4

$x'<00$ $y'<0$

R5

$x'<0$ $y'<0$

R6

$x'0$ $y'<0$

d. Draw the phase plane portrait

Answer 1

Phase portrait:

$\qquad\qquad\qquad\qquad\qquad$

Fixed points:

• At $(3,1)$, Jacobian matrix $\begin{pmatrix}-1&-1\\-1&-3\end{pmatrix}$, trace $-4$ (negative), determinant $+2$ (positive), discriminant $(-4)^2-4\cdot(+2)=8$ (positive), hence two real negative eigenvalues: the point $(3,1)$ is a stable node
• At $(1,3)$, Jacobian matrix $\begin{pmatrix}-1&-1\\-3&+1\end{pmatrix}$, trace $0$, determinant $-4$ (negative), hence two real eigenvalues of opposite signs: the point $(1,3)$ is a saddle point

...As explained there:

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