Cyclic and Chaotic Behavior in Genetic Algorithms

This paper demonstrates dynamical system models of genetic algorithms that exhibit cycling and chaotic behavior. The genetic algorithm is a binary-representation genetic algorithm with truncation selection and a density-dependent mutation. The density dependent mutation has a separate mutation rate for each bit position which is a function of the level of convergence at that bit position. Density-dependent mutation is a very plausible method to maintain diversity in the genetic algorithm. Further, the introduction of chaos can potentially be used as a source of diversity in a genetic algorithm. The cycling/chaotic behavior is most easily seen in a 1-bit genetic algorithm, but it also occurs in genetic algorithms over longer strings, and with and without crossover. Dynamical system models of genetic algorithms model the expected behavior or the algorithm, or the behavior in the limit as the population size goes to infinity. These models are useful because they can show behavior of a genetic algorithm that can be masked by the stochastic effects of running a genetic algorithm with a finite population. The most extensive development of dynamical systems models has been done by Michael Vose and coworkers. (See [Vose and Liepins, 1991], [Vose and Wright, 1994] and [Vose, 1999] for examples.) They have developed an elegant theory of simple genetic algorithms based on random heuristic search. Heuristic search theory is based on the idea of a heuristic map G, which is a map from a population space to itself. The map G includes all of the dynamics of the simple genetic algorithm. The map defines a discrete-time dynamical system which we call the infinite population model. The simple genetic algorithm heuristic G is called focused if G is continuously differentiable and if the sequence p,G(p),G(p), . . . converges for every p. In other words, G is focused if every trajectory of the dynamical system converges to a fixed point. With one exception, infinite population models of genetic algorithms always seem to converge to a fixed point. The exception is the result of Wright and Bidwell [Wright and Bidwell, 1997], who show stable cycling behavior corresponding to very “weird” mutation and crossover distributions that would never be used in practice. Cycling behavior has also been shown in biological population genetics models [Hastings, 1981]. The random heuristic search model also leads in a natural way to a Markov chain model where the states are (finite) populations. Vose has a number of results that connect the infinite population model to the finite population model in the limit as the population goes to infinity. These theorems assume that the heuristic G is focused. For example, he shows that ([Vose, 1999], theorem 13.1), the probability of being in a given neighborhood of the set of fixed points can be made arbitrarily high by choosing the population size to be sufficiently large. Thus, there is a lot of interest in knowing whether the heuristic that defines the infinite population model of a genetic algorithm is focused. This paper gives numerical examples where the infinite population model of a genetic algorithm exhibits stable cycling/chaotic behavior, which implies that the heuristic is not focused. We expect that the examples of this paper very well could arise in practice if the mutation and selection described in this paper was used. However, they are not examples of the simple genetic algorithm in that a density-dependent mutation scheme is used. Chaotic behavior could also be useful for restoring diversity in a run of a genetic algorithm that is not making progress. When the GA seems to have converged, the parameters could be adjusted to introduce chaotic behavior, which can move the algorithm from a local optimum. We are aware of one other paper that discusses chaos (more accurately fractals) and genetic algorithms. Juliany and Vose [Juliany and Vose, 1994] generated fractals by determining the basins of attractions of fixed points of G