$\dot{x}=-(x-1)\cdot(x-2)^2$
I want to find the stability with the 2 Lyapunov methods ( linearization and appropriate Lyapunov function). I solved similar exercises with the first method of linearization but I don't understand how to solve this because is not a system of two equations. Is there a variable change involved? And also I don't understand how to choose a suitable Lyapunov function. Thanks!
The system comprises two equilibrium points $x=1$ and $x=2$
To study the first equilibrium point $x=1$, consider the variable transformation $z=x-1$. Then the original dynamical system is rewritten as follows:
$$\dot{z}=-z(z-1)^2$$
Let $V(z)=\frac{1}{2}z^2$. Its time derivative: $$\dot{V}(z)=z\dot{z}=-z^2(z-1)^2<0\;\;\forall\;\; |z|<1$$
then the equilibrium point $z=0$ (or equivalently $x=1$) is stable.
For the second equilibrium $x=2$, we can not construct a Lyapunov function because it is not stable.