Density function associated with nonlinear bifurcating map

Abstract

In the class of nonlinear one-parameter real maps we study those with bifurcation that exhibits period doubling cascade. The fixed points of such a map form a finite discrete real set with dimension (2^n)m, where m is the (odd) number of "primary branches" of the map in the non-chaotic region and n is a non-negative integer. We associate with this map a nonlinear dynamical system whose Hamiltonian matrix is real, tridiagonal and symmetric. The density of states of the system is calculated and shown to have a number of separated bands equals to (2^n-1)m for n not equal 0, in which case the density has m bands. The location of the bands depends only on the map parameter and the odd/even ordering of the fixed points in the set. Polynomials orthogonal with respect to this density (weight) function are constructed. The logistic map is taken as an illustrative example.