Simple Polynomial Model

The first system tested was the nonlinear system given by Eqs. 2 and 3 (which had appeared in a homework problem):

$$\dot z_1 = z_1 z_2$$ (2)

$$\dot z_2 = -z_1^2-z_2$$ (3)

The ranges of $z_1$ and $z_2$ are $z_1\in[-1,1]$ and $z_2\in[-1,1]$. For two of the runs, I used the nonterminal set $\{+,-,\times,\div\}$; for the rest, $\{+,-,\times,\div,\cdot ^2,\sqrt{\cdot }\}$.

One test run returned the function $V(z_1,z_2)=z_1^2+z_2^2$, which is indeed a Lyapunov function in the domain. It is clearly positive definite, and its derivative is

$$\dot V = 2z_1^2z_2 + 2(-z_1^2-z_2)z_2$$

$$= -2z_2^2 \leq 0 .$$

However, it only proves stability, not asymptotic stability, because $\dot V=0$ on the $z_1$-axis. Another test run returned the function $V(z_1,z_2)=(z_1^2+z_2^2)^2$, which is also a Lyapunov function, but again does not prove asymptotic stability. Fig. 5 shows the plots of this Lyapunov function.

There were a couple runs that produced functions that looked Lyapunov from the graphs, and seemed have $\dot V$ strictly less than zero except at the origin, which would prove asymptotic stability. They were somewhat more complex to analyse. Fig. 6 shows the plots for one of these functions. (However, I suspect this function has a small region where it fails to satisfy the Lyapunov criteria near the origin.)

But not all of the tests returned a Lyapunov-looking function. One test produced a function that (seemed as if it) met the three Lyapunov criteria; however, it wasn't continuously differentiable (see Fig. 7). On some other runs, the system didn't converge, but reached the maximum number of generations while only satisfying the Lyapunov criteria on 80% or so of the domain. The two runs I made without the square function in the nonterminal set did this. This is the nature of genetic algorithms, because of their randomness: getting a good result often requires several runs.

Figure 5: Result of a genetic programming Lyapunov search for a simple polynomial system.

$$V(z_1,z_2) = (z_1^2 + z_2^2)^2$$

$$\dot z_1 = z_1 z_2$$

$$\dot z_2 = -z_1^2-z_2$$

Figure 6: Result of a genetic programming Lyapunov search for a simple polynomial system.

$$V(z_1,z_2) = (-z_1^2 - z_2)^2 + z_1^2z_2^2$$

$$\dot z_1 = z_1 z_2$$

$$\dot z_2 = -z_1^2-z_2$$

Figure 7: Result of a genetic programming Lyapunov search for a simple polynomial system.

$$V(z_1,z_2) = \sqrt{z_1^2} + \sqrt{(-z_1^2 - z_2)^2} + z_1^2z_2^2$$

$$\dot z_1 = z_1 z_2$$

$$\dot z_2 = -z_1^2-z_2$$