I have the dynamical system $$\left\{ \begin{array}{c l} \dot{x} &= y\\ \dot{y} &= -x-y-y^3 \end{array}\right.$$
And I want to show that there exists no periodic solutions.
Is my reasoning and procedure below correct?
The only equilibrium point is $(0,0)$. Using the Lyapunov test function $V(x,y)=x^2/2+y^2/2$ we get that $\dot{V}=-(y^4+y^2) <0 \ \forall \ (x,y)\in\mathbb{R}^2\setminus\{(0,0)\}.$ This implies asymptotic stability for the origin. This also means that the positive invariant set for this equation is the whole $\mathbb{R}^2$, thus wherever on the plane we start we will eventually go to the origin as $t\rightarrow \infty$. Doesn't this now mean that no periodic orbits/solutions can exists?
I don't really understand why Pioncare-Bendixsons is needed here.
Yes you are right.
$$ x\dot x = x y\\ y\dot y = -x y - y^2-y^4 $$
adding both equations
$$ \frac 12\frac{d}{dt}(x^2+y^2) = -y^2(1+y^2) $$
considering that
$$ x = r\cos\theta\\ y = r\sin\theta $$
we have
$$ \frac 12\frac{d}{dt}(r^2) = -r^2\sin^2\theta(1+r^2\sin^2\theta) $$
This represents a kind of spiral sink ($r$ is always decreasing) without periodic solutions