$$x'=y$$ $$y'=-x+x^3$$
from above system, one gets hyperbolic equilibria $(1,0)$ and $(-1,0)$. and both equilibria have same eigenpairs $(\lambda,v)$, such as $(\sqrt{2},(1,\sqrt{2})^T)$ and $(-\sqrt{2},(1,-\sqrt{2})^T)$.
and here I tried to find its stable/unstable manifolds start with letting $y=f(x)=ax+bx^2+...$ and differentiate both side with respect to time $t$. then
$$y'=f'(x)x'$$
$$-x+x^3=(a+2bx+...)(ax+bx^2+...)$$
and I got $a^2=-1$ which is a contradiction. So it makes me to think that this system has no invariant manifolds at hyperbolic points, but I learned that every hyperbolic equilibrium has at least local invariant manifolds. It makes me very confused. Where was I wrong?