Chebyshev Series Representation of Feigenbaum's Period-Doubling Function

TL;DR

This paper presents a Chebyshev series method to accurately approximate Feigenbaum's period-doubling function, providing detailed expansion coefficients for various solutions over a specific interval.

Contribution

It introduces a Chebyshev series approach to solve the Feigenbaum-Cvitanovic equation and tabulates precise coefficients for multiple solutions, enhancing computational understanding.

Findings

Accurate Chebyshev coefficients for solutions with even exponents from 2 to 14.

Improved numerical methods for solving functional equations related to chaos theory.

Detailed tabulation of expansion coefficients for various solutions.

Abstract

The Feigenbaum-Cvitanovic equation -lambda * g(x)= g(g(lambda * x)) is solved over the interval 0 <= x <= 1 with a Chebyshev series representation of g(x). Accurate expansion coefficients are tabulated for solutions g(x) = 1+O(x^z) with even exponents from z=2 up to z=14.