An elementary derivation of 3-cycles for a quadratic map

Abstract: We present an elementary derivation of the period-three cycles for the real quadratic map $x\mapsto x^2+c$, a fundamental model in one-dimensional discrete dynamics. Using symmetric polynomials, we obtain a complete algebraic characterization of 3-cycles and determine explicit conditions for their existence and stability, without reliance on computer algebra. Through conjugacy with the logistic map, we recover the classical threshold values of the logistic parameter corresponding to the emergence and loss of stability of the 3-cycle. Our methodology outlines a transparent and algebraically grounded route to understanding the onset of chaos in quadratic and logistic dynamics.