I am trying to understand attracting, Liapunov stable, asymptotically stable for given coupled system. I dont have any Liapunov function. Just two coupled systems such as
$\dot{x} = y$, $\dot{y} = -4x$
or sometimes normal systems
$\dot{x} = -x$, $\dot{y} = -5y$
How can I approach to this problem. Do I have to find the eigenvalues and then eigenvectors, write the solution etc or can it be determined just by looking at the eigenvalues ?
Or is it useful to use this diagram ?
I guess it can be determined from
$\lambda_{1,2} = \frac{1}{2} (\tau \pm \sqrt{\tau^2 - 4\Delta})$
$\tau = \lambda_1 + \lambda_2$ and $\Delta = \lambda_1 \lambda_2$
This is a new subject to me as a physics student so I am kind of confused.
A little of geometric qualitative insight also is worth.
For the first problem we have
$$ \cases{x\dot x = 4x y\\ 4 y\dot y = - 4x y} $$
after addition
$$ \frac 12\frac{d}{dt}(x^2+4y^2) = 0 $$
or
$$ x^2+4y^2 = C $$
so this characterizes a center. A conservative movement which turns forever around the origin describing an ellipse.
For the second we have analogously
$$ \frac 12\frac{d}{dt}(x^2+y^2) = -(x^2+5y^2) $$
and as $x^2+5y^2 > 0$ for $(x,y) \ne (0,0)$
we have that the movement is dissipative and goes to the origin describing stable orbits.