The GA heuristic generically has hyperbolic fixed points

A continuously differentiable function 𝒢 models the behavior of a simple genetic algorithm, incorporating the genetic operators selec-tion, mutation and crossover. In this model, populations are repre-sented by points in ℜⁿ. (Typically, in genetic search the populations correspond to points in Λ = {x ∈ ℜⁿ: ∀i.x_i ≥ 0, ∑x_i = 1}). Given population p the expected next population of the GA is Gp). Sev-eral facts are known about the relationship between the GA and 𝒢. In particular, the GA spends much of its time near fixed points of 𝒢 in the large population case. Consequently, it is desirable to understand the behavior of 𝒢 near its fixed points. A fixed point x of 𝒢 is called hyperbolic if the differential d𝒢_x of 𝒢 at x has no eigenvalues of modulus one. If x is a hyperbolic fixed point of 𝒢, then d𝒢_x sheds light on the behavior of the sequence p, 𝒢(p), 𝒢²(p), in a neighborhood of x. Therefore, the differential of 𝒢 can be used to explain the behavior of 𝒢 near its hyperbolic fixed points. This report shows that in the positive mutation case, there is a dense open set in (ℜ+)ⁿ such that if the fitness parameter of 𝒢 belongs to this set, then all fixed points of 𝒢 in Λ are hyperbolic.