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

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.