Stability Systems - Duffing oscillator In the case a=1,b=-1 this is the system:
     $$ dx=y $$
$$ dy=-x + x^3$$
I have to draw the phase space with the trajectories of the orbits. And I don´t know who to demonstrate the direction in the orbits. I only know is a circle for the $(0,0)$ and hyperbola for$(-1,0),(1,0)$.

Multiply
$$
\ddot x+x-x^3=0
$$
with $2\dot x$ and integrate to get
$$
\dot x^2+x^2-\frac12x^4=R^2
$$
and parametrize this as a circle equation to get polar coordinates
$$(R\cos \phi(t),R\sin(\phi))=\left(\dot x,x\sqrt{1-\frac12x^2}\right).$$
Close to $x=0$ the equation $u=f(x)=x\sqrt{1-\frac12x^2}$ in the second component is monotone and can be inverted as $$x=g(u)=g(R\sin(\phi))$$ with time derivative
$$
\dot x = g'(R\sin(\phi))R\cos \phi(t)\,\dot\phi(t)\implies 0=\dot x=R\cos \phi(t)\text{ or }1=g'(R\sin(\phi))\dot\phi(t),
$$
so that $$\dot\phi(t)=g'(R\sin(\phi))^{-1}=1+O(R^2).$$ This means counter-clock wise rotation in a $(\dot x,x)$ diagram or, diagonally reflected, clockwise rotation in a $(x,\dot x)$ diagram.
