Calculating limits of a system

Question

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

Answer 1

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).