Suppose we have a third order system, reduced to three first orders in the form
$\dot x_1 = x_2 \\ \dot x_2 = x_1 + x_3F(x_1) \\ \dot x_3 = x_3F(x_1)$
Suppose we know $F(0) = 0$
How do we find the stability around the equilibrium point?
My guess was to try and use Lyupunov but I am having trouble seeing how it works. I believe the only equilibrium point is $x = 0$, but I am lost where to go from here. If the system was second order, I could use phase portraits, is there something similar for third order?
Also, there seems to be a lack of resources on this material, any suggestions would be helpful!
Thank you
An equilibrium point requires that all time derivatives are zero. Setting all three differential equations to zero gives
\begin{align} \dot{x}_1 &= x_2 = 0 \\ \dot{x}_2 &= x_1 + x_3\,F(x_1) = 0 \\ \dot{x}_3 &= x_3\,F(x_1) = 0 \end{align}
The first equation gives $x_2=0$. By substituting the third equation into the second you get $x_1 + 0 = 0$ so $x_1=0$. Since $F(0)=0$ and $x_1=0$ at an equilibrium point we get $x_3\,F(x_1)=x_3\,0=0$, so $x_3$ can be anything.