Regarding the image below
I am struggling to see how differentiation of the Lyapunov function is carried out. I can see how
$$ \dot x_1 \frac {\partial V}{\partial x_1} + \dot x_2 \frac {\partial V}{\partial x_2} $$
is calculated in the first stage, but beyond that, the last $2$ lines seem like a big jump.
Can anyone provide clarity on this or a more detailed step by step method to using Lyapunov functions to test for stability?
Your system is $$m\ddot{x}+c|\dot{x}|\dot{x}+k_1 x+k_2 x^3=0$$ Define the two state variables as $$x_1=x\qquad x_2=\dot{x}$$ to get: $$\begin{align}\dot{x}_1&=x_2\\m\dot{x}_2&=-c|x_2|x_2-k_1 x_1-k_2 x_1^3\end{align}\tag{$*$}\label{*}$$ Also note that if $$V=\frac 12m x_2^2+\frac 12k_1 x_1^2+\frac14k_2x_1^4$$ then $$\begin{align}\frac{\partial V}{\partial x_1}&=k_1x_1+k_2x_1^3\\ \frac{\partial V}{\partial x_2}&=mx_2\end{align}$$ Now, to calculate $$\frac{\partial V}{\partial x_1}\dot{x}_1+\frac{\partial V}{\partial x_2}\dot{x}_2$$ you need to replace $\dot{x}_1$ and $\dot{x}_2$ from the equation $\eqref{*}$. The rest should be straightforward.