I am given a dynamical system
$$\dot x = f(x,y)= x - (1+\theta(x))x^3-y \\ \dot y =g(x,y)= y - 3x^2y + x$$
where $\theta(x)$ is a step function which is equal to $1$ when $x \geq 0$ and $0$ when $x<0$.
Now I am asked to prove whether or not the fixed point at the origin is asymptotically stable.
My thinking so far is as follows.
1) I know that when $x<|\sqrt{\frac{2}{3}}|$ that there can not exist a periodic orbit due to Bendixson's criterion
2) I have found that $\nabla \cdot \pmatrix{f(x,y) \\ g(x,y)} = 2$ when $x<0$ and $\nabla \cdot \pmatrix{f(x,y) \\ g(x,y)} = 2-3x^2$ when $x \geq 0$
3) I know that if there exists a strict Liapounov function around the fixed point then the fixed point is asymptotically stable
4) Not sure if this is relevant but Poincare bendixson states that if there exists a non empty closed and bounded omega limit set then there is either a fixed point or a periodic orbit. Now I know that there can't be a periodic orbit so there must be a fixed point
What I think I need to do
I think I need to find a strict Liapounov function which will then allow me to state the fixed point is asymptotically stable. How I am going to find this function is still up in the air, but maybe it's something to do with the fact that the orbital derviative
$$\frac{dV}{dt} = \int_{\phi(t,D)} d^nx(\nabla \cdot \pmatrix{f(x,y) \\ g(x,y)})$$ where $\phi(t,D)$ is the region obtained by evolving all the points in a set $D$.
Multiplying the first by $x$ and the second by $y$ we get
$$ x\dot x = x^2-(1-\theta(x))x^4-x y\\ y\dot y = y^2-3 x^2 y^2+x y $$
adding the equations
$$ \frac{1}{2}\frac{d}{dt}(x^2+y^2) = x^2+y^2 -((1+\theta(x))x^4+3 x^2 y^2) $$
and in polar coordinates
$$ \frac{1}{2}\frac{d}{dt}r^2 = r^2-r^4((1+\theta(x))\cos^4(\theta)+3\cos^2(\theta)\sin^2(\theta)) $$
and
$$ (1+\theta(x))\cos^4(\theta)+3\cos^2(\theta)\sin^2(\theta) \ge 0 $$
Now making $r^2 = u$ we have the differential equation
$$ \frac{1}{2}u'=u-\sigma(t)u^2 $$
with
$$ 0 \le \sigma(t) \lt 2 $$
This equation seems not to converge asymptotically to the origin.
I hope this helps.
Attached a stream plot near the origin with an orbit associated to $x(0) = 4, y(0) = 4$ in the original system.