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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

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