# Non-Linear Behavior of Iterated Functional Maps

The universal behavior of certain iterated nonlinear function maps (ie period doubling bifurcation route to chaos): $$x_{i+1}=f(x_i)$$ have been known since Feigenbaum: (see http://theworldismysterious.wordpress.com/2013/10/03/441/)

The usual method of solving the Hartree-Fock equations for interacting fermions is a nonlinear iterated functional mapping: $$f_{i+1}(x)=F[f_{i+1}(x),V_i[f_i(x)]]$$ where $F$ is a non-linear functional of the function $f(x)$. Here $f(x)$ represents the fermion orbitals and $F$ is an integro-differential operator that is non-linear in $f$ because it contains both $f$ and an effective potential function $V$ that also depends upon $f$. The usual method of solution is to guess an effective potential $V_0$ that is used to generate a set of orbitals $f_0(x)$. These orbitals are then used to generate a new effective potential $V_1$ and together they are used to generate a new set or orbitals via the functional iteration above. A solution is obtained when the iteration yields an equivalent set of orbitals on two successive iterations.

Since this mapping is nonlinear it is conceivable that convergence of the iteration may not occur and that something related to the period doubling bifurcations of nonlinear iterated function maps may result. In fact, I found such a situation during research I conducted in 1975.

1. My question is: has anyone else encountered such situations either in Hartree-Fock calculations or any other physics calculations that employ nonlinear iterated functional mappings?

2. A secondary question is: are there published mathematical investigations of nonlinear behavior in iterated functional maps?

• @centralcharge Thanks very much. The post looks very promising. – Lewis Miller Jan 1 '16 at 15:07
• Yes, Functional iteration is a well-developed field. The standard prototypes were worked out through functional conjugacy in the Schröder equation. – Cosmas Zachos Nov 14 '17 at 0:18