Density function associated with nonlinear bifurcating map

Details Density function associated with nonlinear bifurcating map Density function associated with nonlinear bifurcating map

2005-07-29

arxiv.org/abs/math-ph/0507071v1

J. Phys. A: Math. Gen. 39 (2006) 6851-6859

Physics

Mathematical Physics

8 pages of text, 9 figures (one in color)

Scientific paper

10.1088/0305-4470/39/22/003

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.

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