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This article presents the exact solution of fixed points functions for thecycle of period four of the quadratic recurrence equations. The solution isdemonstrated for the quadratic map and the logistic map. These recurrenceequations, presenting the real domain, as well as the Mandelbrot set,presenting the complex domain, are at the very heart of dynamical systems andchaos theory. Up to now, the closed explicit solutions of fixed pointsfunctions have only been known for three bifurcation ranges: for the cycles ofperiod 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 ofsimultaneous complexity and simplicity which represents the beauty of nature.

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