Calculating limits of a system For the following system
$$x'=x(-x^2-y+4)=f(x,y)$$
$$y'=y(y^2+8x-1)=g(x,y)$$
I need to find the location of the critial points and determine each points type and stability.
I then need help finding calculating the limits
$$\lim\limits_{t \to \infty} x(t), \lim\limits_{t \to \infty} y(t)$$
if $x(0)=\frac{-5}{2}, y(0)=1$
I found the critical points and the type/stability but it seems like a large amount
$(0,0)$  eignevalues are $(4,-1)$ unstable saddle point
$(0,1)$ eigenvalues are $(2,3)$ unstable improper node
$(0,-1)$  eigenvalues are $(2,5)$ unstable improper node
$(2,0)$ eigenvalues are $(-8,15)$ Unstable saddle point
$(-2,0)$ eignevalues are $(-17,-8)$ Asymp stable improper node
$(-1, 3)$ eigenvalues are $(-2\sqrt 31 +8, 2\sqrt 31 +8)$ Unstable Saddle point
$(-3,-5)$ eigenvalues are $(-2\sqrt 259 +16, 2\sqrt 259 +16)$ unstable saddle point
$(2+i, 1-4i)$ eigenvalues are $(-29.22-18.774i, -6.78-5.226i)$ asymp stable spiral
$(2-i, 1+4i)$ eigenvalues are $(-29.22+18.774i, -6.78+5.226i)$ aspym stable spiral
In regards to the limits, I'm unsure of what $x(t)$ and $ y(t)$ values I am to use
 A: For the system
$$x'=x(-x^2-y+4) \\ y'=y(y^2+8x-1)$$
We find seven real critical points by setting $x' = y' = 0$ as
$$(x, y) = (-3,-5),(-2,0),(-1,3),(0,-1),(0,0),(0,1),(2,0)$$
We find the Jacobian as
$$J(x, y) = \dfrac{\partial f(x,y)}{\partial g(x,y)} = \begin{pmatrix} \dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y} \\\dfrac{\partial g}{\partial x} & \dfrac{\partial g}{\partial y}\end{pmatrix} = \begin{pmatrix}
 -3 x^2-y+4 & -x \\
 8 y & 8 x+3 y^2-1 \\
\end{pmatrix}$$
We find the eigenvalues of the Jacobian at each critical point and determine their stability (you can also include the type of point)
$$\begin{array} {|r|r|}\hline \mbox{Critical Point (x,y)} & \mbox{Eigenvalues}~~~(\lambda_1, ~\lambda_2) & \mbox{Stability} \\ \hline (-3,-5) & (2 \left(\sqrt{259}+8\right),2\left(8-\sqrt{259}))\right. & \mbox{Unstable} \\ \hline (-2,0) & (-17, -8) & \mbox{Stable} \\ \hline (-1,3) & (2 \left(\sqrt{31}+4\right),2 \left(4-\sqrt{31}\right)) & \mbox{Unstable} \\ \hline (0,-1) & (5, 2) & \mbox{Unstable} \\ \hline (0,0) & (4, -1) & \mbox{Unstable} \\ \hline (0,1) & (3, 2) & \mbox{Unstable} \\ \hline (2,0) & (15,-8) & \mbox{Unstable} \\ \hline  \end{array}$$
From this information, we can plot the phase portrait

For the last question,what do you notice using the IC as a starting point as $t$ approaches infinity (see the red line).
