I am currently studying a paper on the determination of regions of attraction for autonomous systems. The paper I'm referring to is available at the following link: Estimation of the regions of attraction for nonlinear systems.
However, I'm facing difficulty understanding how the initial conditions are calculated and where the Hamilton-Jacobi equation comes into play. The examples provided for determining the region of attraction of a system are somewhat unclear to me. Does anyone have insights or suggestions to clarify these concepts?
For instance, considering the system described by the equations:
$$
\begin{align*}
\dot{x}_1 &= -\sin(x_1) - 0.5 \sin(x_1 - x_2) + 0.01 \\
\dot{x}_2 &= -0.5 \sin(x_2) - 0.5 \sin(x_2 - x_1) + 0.05
\end{align*}
$$
I'm particularly interested in understanding the proposed method for calculating the regions of attraction, as shown in the provided image:
