I am currenty studying the synchronization of Lorenz systems, in which one system transmits one of its coordinates to the other and this drives the other system to converge towards it exponentially, and I came across a very useful paper on this subject.
My question is regarding the proof of the exponential synchronization of the two Lorenz systems. In the paper, the author uses Lyapunov's stability theorem to prove the convergence of the coupled system error towards zero. I understand the theorem, and can numerically prove the convergence of the error between the two systems towards zero using python, but I can't find anything online to suggest how to even derive the Lyapunov function. In the paper it is given with no explanation. I searched the internet and it seems that in general, candidate functions are examined to see if they fit the system, but yet again there is no explanation or reasoning behind how you even find these candidate functions.
You are correct that in general, candidate Lyapunov functions are examined to see if they fit the system, and there is no universal method for finding these candidate functions. However, there are some general principles and strategies that can be used to guide the search for a Lyapunov function.
One approach is to use physical insight to guide the choice of a candidate function. The Lyapunov function should capture some physical property of the system, such as energy or potential, and its derivative should reflect the dissipation or loss of that property. For example, in the case of a mechanical system, the Lyapunov function might be chosen as the system's total energy, and its derivative would represent the rate of dissipation of energy due to friction or other forces.
Another approach is to use known properties of the system to guide the search for a Lyapunov function. For example, if the system is known to have stable equilibria or limit cycles, then the Lyapunov function might be chosen to be a measure of the distance to the equilibrium or the phase difference from the limit cycle. In this case, the derivative of the Lyapunov function would be chosen to be negative definite in a neighborhood of the equilibrium or limit cycle.
In your paper, I think your problem is the page 8, theorem 1 right? I found that based on the Lyapunov function and its derivative given in the paper, it is clear that the function satisfies the necessary conditions for a Lyapunov function, namely, $V(0,0,0,t) = 0$, $V(e1,e2,e3,t) > 0$ for $e1, e2, e3 ≠ 0$, and $\dot V(e1,e2,e3,t) < 0$ for $e1, e2, e3 ≠ 0$.
Since the derivative of the Lyapunov function is negative definite, the error between the two Lorenz systems will converge to zero asymptotically. This implies that the two systems are synchronized, and the synchronization error will approach zero exponentially.