# Math Help - Liapunov stability exercise

1. ## Liapunov stability exercise

What can you say about the (Liapunov) stability of the zero solution of the following system?

$\left\{\begin{array}{l}x'_1=e^{-t}x_1^3-x_2\\\\\displaystyle x'_2=\frac{t^2}{1+t^2}x_1+\alpha x_2\end{array}\right\}$

???

First of all, what the heck is the "zero solution"?

Second, does anyone know how to answer this? I don't even know where to start. If it were a linear system then I'd convert to a first-order vector form, but it's not, so I can't.

Any help would be much appreciated!

2. When
$
x_1=0 \; \; x_2=0
$

then
$
x'_1=0 \; \; x'_2=0
$

so point (0,0) is stable or unstable point.
One of the methods to discover this is to linearise this system
and find eigenvalues. If
$
| \; \lambda_i \; | < 0
$

it is stable.