# Center, Stable and Unstable subspaces.

I'm trying to do an exercise in Perko's Differential equations and dynamical systems chapter 1 section 9 Stability theory that states the following.

Let $$A$$ be a nonsingular $$n\times n$$ matrix and let $$x(t)$$ be a solution of the IVP $$\dot{x}=Ax$$ with $$x(0)=x_0$$. Show that

1. if $$x_0\in E^s-{0}$$ then $$\lim_{t\to\infty}x(t)=0$$ and $$\lim_{t\to-\infty}|x(t)|=\infty$$
2. if $$x_0\in E^u-{0}$$ then $$\lim_{t\to\infty}|x(t)|=\infty$$ and $$\lim_{t\to-\infty}x(t)=0$$
3. if $$x_0\in E^c-{0}$$ and A is semisimple then there are positive constants m and M such that $$m\leq|x(t)|\leq M$$
4. if $$x_0\in E^c$$ and A is not semisimple, then there is an $$x_0\in \mathbb{R}^n$$ such that $$\lim_{t\to\pm\infty}|x(t)|=\infty$$
5. if $$E^s\neq{0}$$,$$E^u\neq{0}$$ and $$x_0\in E^s \oplus E^u-\{E^s \cup E^u\}$$, then $$\lim_{t\to\pm\infty}|x(t)|=\infty$$
6. if $$E^u\neq{0}$$,$$E^c\neq{0}$$ and $$x_0\in E^u \oplus E^c-\{E^u \cup E^c\}$$, then $$\lim_{t\to\infty}|x(t)|=\infty$$ and $$\lim_{t\to-\infty}x(t)$$ does not exist
7. if $$E^s\neq{0}$$,$$E^c\neq{0}$$ and $$x_0\in E^s \oplus E^c-\{E^s \cup E^c\}$$, then $$\lim_{t\to-\infty}|x(t)|=\infty$$ and $$\lim_{t\to\infty}x(t)$$ does not exist

Where if $$\lambda_j=a_j+ib_j$$ are eigenvalues of A and $$w_j=u_j+iv_j$$ are generalized eigenvectors of A corresponding to $$\lambda_j$$ then we define

$$E^s=Span\{u_j,v_j/a_j<0\}$$,

$$E^u=Span\{u_j,v_j/a_j>0\}$$,

$$E^c=Span\{u_j,v_j/a_j=0\}$$

1. and 2. are consequences of theorems 1 and 2 from the section, also I know that the coordinates of the solution $$x(t)$$ are linear combinations of functions of the form $$t^ke^{at}cos(bt)$$ or $$t^ke^{at}sin(bt)$$ where $$\lambda=a+bi$$ is an eigenvalue of $$A$$ and $$0\leq k\leq n-1$$, therefore if A is semisimple this coordinates are linear combinations of the following functions:

$$cos(bt)$$ or $$sin(bt)$$ so I also have 3.

I do not know how to do 4 any hints?.

In 5. since $$x_0\in E^s \oplus E^u-\{E^s \cup E^u\}$$ then we can write $$x_0=\alpha +\beta$$ with $$\alpha \in E^s$$ and $$\beta \in E^u$$ then $$\lim_{t\to\pm\infty}|x(t)|=\lim_{t\to\pm\infty}|e^{At}x_{0}|=\lim_{t\to\pm\infty}|e^{At}(\alpha +\beta)|=|\lim_{t\to\pm\infty}e^{At}\alpha +\lim_{t\to\pm\infty}e^{At}\beta|=\infty$$ because 1. and 2.

Also this doesn't seem to work with $$E^c$$ because I do not know what to do with those limits can you guide me in the right direction? Thanks