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




  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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.