Accession Number : ADP006606
Title : Analytic Solution of the Period Four Quadratic Recursion Polynomial,
Corporate Author : ARMY LAB COMMAND WHITE SANDS MISSILE RANGE NM ATMOSPHERIC SCIENCES LAB
Personal Author(s) : Auvermann, Harry J.
Report Date : MAR 1992
Pagination or Media Count : 12
Abstract : This paper is concerned with stable points of iterates of the function F(z,d) = d - z2. The number of these stable points bifurcates successively as the real parameter d varies from -1/4 to 2. The number of stable points of a particular bifurcation is termed the period. Period one stable points are roots of the polynomial that result from substituting z for F(z,d) in the above. Two applications of F(z,d) produce a fourth order polynomial. Period two stable points are roots of this ,polynomial that are easily obtained. Four applications of F(z,d) produce a sixteenth order polynomial. Period four stable points are the roots of this polynomial. Four of these roots are known from analysis of the lower iterates. Solution of a twelfth order polynomial then determines the period four stable points. A general analytic solution method to recursion polynomials of this type has been given previously. This paper presents an alternate method for obtaining the analytical closed form expressions for the period four roots as a function of the parameter d.
Descriptors : *POLYNOMIALS, *ITERATIONS, FUNCTIONS, NUMBERS, PAPER, PARAMETERS, BIFURCATION(MATHEMATICS), STABILITY.
Subject Categories : Numerical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE