It is often good to determine whether or not an equilibrium point of a differential equation model is globally stable. To do this, a Lyapunov function is often employed. May I know if there are methods of determining these Lyapunov functions or one will have to be doing trial and error until one gets a good function.
To my knowledges, no general methods exists. But if the differential equations represents the model of a physical system (e.g. machanical system, electrical system, etc.), the energy of the represented system can be used as Lyapunov function.
Since Wronskian of two solutions is a constant for the second order linear differential equations, one can construct approximate Lyapunov function for nonlinear equations by using Wronskians. See for example formula (2.14) and Remark 2.2 in my paper
Stability of planar nonautonomous dynamic systems. Advances in Difference Equations 2013, 2013:144. 38 pages.
In case of linear system, a quadratic function could be used as a Lyapunov function. But when it comes to nonlinear system, finding a Lyapunov function could be not so straight forward.
This might help in the case of nonlinear system
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-243j-dynamics-of-nonlinear-systems-fall-2003/lecture-notes/lec7_6243_2003.pdf
The function of total energy is a good candidate for the Lyapunov function
but in many cases, finding a suitable Lyapunov function still a tricky point.
Ivan Furlan is right.
In the context of complex organisms, Karl Friston states: "On a physicists view, any (weakly mixing random dynamical) process can be completely characterized by a function of its current state. Formally speaking, the current state corresponds to a location in some abstract state space and the function is known as a Lyapunov function (i.e., a function of the states that always increases, on average".
(doi: 10.3389/fpsyg.2018.00579)
The best way is formulate them to describe the abstract energy of the system. This way, they have strong physical meaning and it is easier to describe +/- definiteness, with correlation to dissipation and stability.
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