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$(-1,0),(1,0)$.

For the critical point at $(0, 0)$ just evaluate the dynamical system close to the origin. For example, take $x = 0.1$, and $y = 0$, you see that at that location ${\rm d y}/{\rm d}t < 0$, that means that a that location $y$ will be decreasing. In other words, the orbit going through $(0.1, 0)$ will rotate clock-wise. Same argument can be applied to the other critical points.

Here's a sketch to confirm it

![enter image description here](