Genetic algorithms to search the Lyapunov space

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Offline marcm200

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« on: July 04, 2019, 12:14:55 PM »
I tried the genetic algorithm approach by Ashlock (see fractalforum-link) with Lyapunov exponents. I started with the b*sin²(x+r) formula and sequence AAAAAABBBBBB.

The "chromosome" for one individual consists of the parameter b, the position (x0,y0) and a size. I used the tournament-7-rule: In each round, pick 7 individuals randomly, assess their fitness and breed the two best: now the two children replace the two least fit individuals of that seven (after some cross-over mutations with swapping b between children or x,y). Afterwards a mutational round for all 7 starts with slight changes to the parameters in decreasing maximal amount as the population ages.

I ran the program 15 times, took the fittest and displayed it below. Also included is the palette (upper row) and the fitness function (upper left black/white image), which works as a 20x20 grid imposed on the rectangle of each individual and calculates the square distance to a pre-defined "target" 20x20 grid of Lyapunov exponents between -1 and +1.

It proved very difficult to find any fitness function that produces some kind of convergent behaviour in the sense that the best indivuduals of separate runs share some similarity. The current function likes a big ordered corner region (silver) going into more or less smooth chaos (red).

Current observation is, when using Lyapunov exponent as target value, that using a highly volatile fitness function like a forest with high trees (the first I started with), it always "converges" to more or less a featureless order region instead of what I expected to see: a chaos/order spot pattern.

Currently it looks like the grid fitness function does not work well with Lyapunov exponent images. Maybe a more statistical approach on the image itself (like spyke did, see above link) would be better suited.


Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/genetic-algorithms-to-search-the-lyapunov-space/2910/


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