An elementary derivation of 3-cycles for a quadratic map
Arpad Benyi, Ioan Casu
TL;DR
This work provides an elementary, algebraic derivation of 3-cycles for the real quadratic map $f_c(x)=x^2+c$ by exploiting symmetric polynomials to obtain explicit existence and stability conditions without computer algebra. A key result is that nontrivial 3-cycles arise only when $c\le -\tfrac{7}{4}$, with a detailed bifurcation structure governed by the critical value $\tilde{c}\approx -1.768529$ and accompanied by stability analyses using the product of derivatives $8s_3$ and, in borderline cases, second-derivative/Schwarzian tests. The authors then leverage a conjugacy $h(x)=-rx+\frac{r}{2}$ with $c=-\frac{r(r-2)}{4}$ to translate these results to the logistic map $g_r(y)=ry(1-y)$, recovering classical logistic thresholds for the emergence and stability of 3-cycles: $r\in(-\infty,1-2\sqrt{2}]\cup[1+2\sqrt{2},\infty)$ for existence and $r_{min},r_{max}$ (approximately $-1.841499$ and $3.841499$) for stability. This provides a transparent, algebraically grounded path from quadratic to logistic dynamics and clarifies the onset of chaos in these canonical one-dimensional maps.
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.