Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram

TL;DR

This paper derives the exact explicit solutions for fixed points of period four in quadratic recurrence equations, advancing understanding of bifurcation diagrams in dynamical systems and chaos theory.

Contribution

It provides the first closed-form solutions for cycle four fixed points in quadratic maps, extending previous results for cycles of periods one to three.

Findings

Exact solutions for period four fixed points are derived.

Solutions are demonstrated for quadratic and logistic maps.

Enhances understanding of bifurcation structures in chaos theory.

Abstract

This article presents the exact solution of fixed points functions for the cycle of period four of the quadratic recurrence equations. The solution is demonstrated for the quadratic map and the logistic map. These recurrence equations, presenting the real domain, as well as the Mandelbrot set, presenting the complex domain, are at the very heart of dynamical systems and chaos theory. Up to now, the closed explicit solutions of fixed points functions have only been known for three bifurcation ranges: for the cycles of period one, two and three. With the discovery of the solution for cycle four, disclosed in this paper, further step has been made in our comprehension of simultaneous complexity and simplicity which represents the beauty of nature.