# How to rotate a coordinate system to find the unstable manifold.

I am considering the dynamical system: $$u'=v-0.25(v-u)^2$$ $$v'=u(1+v)+0.25(u+v)^2$$

I have calculated the linear stable and unstable manifold as, $$E^s=sp(1,1)$$ and $$E^u=sp(-1,1)$$ for eigenvalues $$-1$$ and $$1$$ respectively. I did this by calculating the linear eigenvalues and vectors and then using the defintions for the linear stable and unstable manifolds:

$$E^s$$ is the eigenspace corresponding to the eigenvalues with negative real parts. Let $$v_1, . . . , v_k ⊂ R^n$$ be the eigenvectors corresponding to the eigenvalues of $$D(\bar{x})f$$ having negative real parts. $$E^s = sp{v_1, . . . , v_k}$$.

$$E^u$$ is equivalent for a positive eigenvalue.