# How to determine the region in a state plane where the equilibrium state is asymptotically stable

I have a non-linear system shown:-

$$\dot{x}_1 = x_2$$

$$\dot{x}_2 = -(1-\vert x_1\vert)x_2 - x_1$$

How do I determine the region in the state plane where the equilibrium state at the origin is asymptotically stable?

• What are $x_1$ and $x_2$ functions of? Time only? Jun 21, 2017 at 17:37

As you might already know your system is nonlinear, which means that it is not trivial at all. See below for the plot of the system. The linearization around $(0,0)$ gives you the information that the eigenvalues have both negative real parts, which guarantees asymptotic stability in the sense of Lyapunov for the origin. From the plot, one can conjuncture a half-stable limit cycle. Which would imply that trivial equilibrium solution is stable inside the half-stable limit cycle. I doubt that you will be able to get the exact equation for the limit cylce.
I suggest you graph $x_1$ and $x_2$ for half unit intervals between -3 and 3 for both variables.
Then work out the derivative of $x_1$ and $x_2$, and mark these on the graph.