Testing and Results

I applied my genetic programming implementation to three different nonlinear systems: a simple polynomial model, a damped pendulum, and a nonlinear system containing the tangent function. All three were two-dimensional, so that the returned function could be plotted.

The parameter settings used in the tests were as follows:

• The generation size varied. It was usually 500; but for some runs was 1000, others 200.

• The maximum number of generations was 20.

• The number of random points in a test set was 20. The maximum number of sets that could be tested was 10, making the maximum number of random points tested 200.

• The number of individuals selected for the tournament was usually 4, although it was 2 in a few runs.

• 90 percent of crossovers happen at a nonterminal node.

• The mutation rate was 2 percent.

• For each generation, 10 percent of the individuals are direct copies of one parent in the previous generation; the other 90 percent are formed by crossover.

The results, in general, were good, if uncertain. There were more false positives that I would have liked. In most cases, convergence was faster than I expected.

The functions produced were sometimes too complicated to analyse; therefore, many of the results are graphed. Figures 5-10 show graphs of some of the functions produced by the tests. Each figure shows four graphs. In each figure, the graph on the top-left plots the returned function $V(x)$, which is hopefully a Lyapunov function. Next to it, the graph on the top-right plots $\mathrm{min}(V(x),0)$; this plot makes it easy to see where $V(x)<0$. The graph on the bottom-left plots $\dot V(x)$. Next to it, the graph on the bottom-right plots $\mathrm{max}(\dot V(x),0)$, to make it easy to see where $\dot V(x)>0$. We want both graphs on the right to be flat.

Needless to say, because of the granularity of the plots, it is possible that there are regions that cross the plane but are missed by the plotting, especially near zero. In fact, I noted such missed details a few times when examining the returned function for simple systems. For most systems, absolute proof of these functions' Lyapunovness requires painstaking algebraic or numerical analysis.