Nonlinear system containing the tangent function.

Finally, the system given in Eqs. 6 and 7 was tested:

$$\dot x_1 = -\tan x_1 + x_2^2$$ (6)

$$\dot x_2 = -x_2+x_1$$ (7)

The ranges on $x_1$ and $x_2$ are $x_1 \in [-1,1]$ and $x_2\in[-1,1]$. The nonterminal set was $\{+,-,\times,\div,\cdot ^2,\sqrt{\cdot },\sin,\cos,\tan\}$.

This, of all the systems, had the oddest-looking results. I ran four tests, and three returned results that appeared, from the graphs, to satisfy the Lyapunov criteria. The results of the three good test runs were:

$$\sin\sin\tan\sin(x_1 + x_2^2 - \tan x_1) + (x_2^2 - \tan x_1)^2$$

$$0.9955 + \cos \big( (\cos x_1 + \sin \sin \cos (x_2^2))^2 \big)$$

$$1.557 + (x_2^2 - \tan x_1)^2 - \tan\cos\tan x_2$$

Fig. 10 shows the first result listed above.

The unusual form of these results suggest that genetic programming is useful for expanding the range of possibilities when searching for Lyapunov functions. A human trying to find a Lyapunov function analytically would tend not to use oddities like $\sin\sin\tan\sin(x)$; yet genetic programming can search these obscure recesses of hierarchal function space.

Figure 10: Result of a genetic programming Lyapunov search for a nonlinear system containing the tangent function.

$$V(x_1,x_2) = \sin\sin\tan\sin(x_1 + x_2^2 - \tan x_1) + (x_2^2 - \tan x_1)^2$$

$$\dot x_1 = -\tan x_1 + x_2^2$$

$$\dot x_2 = -x_2+x_1$$