Density function associated with nonlinear bifurcating map

TL;DR

This paper investigates the density function related to nonlinear maps with period doubling bifurcations, analyzing the spectral properties of an associated Hamiltonian system and constructing orthogonal polynomials, with the logistic map as an example.

Contribution

It introduces a novel connection between nonlinear bifurcating maps and spectral density functions of associated Hamiltonian systems, including explicit band structure analysis.

Findings

Density of states exhibits separated bands depending on bifurcation level

Band locations depend on map parameters and fixed point ordering

Orthogonal polynomials are constructed with respect to the density function

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.