Consider the system
$x ^ { \prime } = 4 x + 2y ^ { 3 } \\ y ^ { \prime } = - 3 x$
Question: Find the stable and unstable manifolds around the fixed point $(0,0)$ and and sketch the phase portrait of the system.
Attempt: I've found the Jacobian matrix at the origin.
$J = \left[ \begin{array} { c c } 4 & 0 \\ - 3 & 0 \end{array} \right]$
$\lambda ^ { 2 } - 4 \lambda = \lambda ( \lambda - 4 )$
However, eigenvalues are $\lambda = 0$ and $ \lambda = 4$. So, the origin $(0,0)$ is not hyperbolic. Moreover, since there is no eigenvalue with negative real part, I can't apply the Local Center Manifold theorem here. I'm open to suggestions to find these manifolds.
Hint.
Considering instead after a rotation of $\frac{\pi}{2}$ necessary to handle conveniently the polynomial approaching the central manifold,
$$ \cases{ x' = 3y\\ y' = 4y-2x^3 } $$
we have for $n=5$ and with $y = h(x) = \sum_{k=1}^n a_k x^k$
$$ h_x(x)(3h(x))-(4h(x)-2x^3)=0 $$
and solving for $a_k$'s we have the solutions
$$ \cases{ h_1(x) =\frac{x^3}{2}+\frac{9 x^5}{16}+O_1(x^6)\\ h_2(x) = \frac{4 x}{3}-\frac{x^3}{6}-\frac{x^5}{80}+O_2(x^6) } $$
we follow with $h_1(x)$ with unstable flow given by $\dot x = \frac{27 x^5}{16}+\frac{3 x^3}{2}$