Existence and Stability of 3-Cycles in Quadratic Maps
Dan Comănescu
TL;DR
This work studies real quadratic maps $g(x)=ax^2+bx+c$ and shows that the existence, number, and stability of 3-cycles are governed by a single perturbed discriminant $\delta=b^2-4ac-2b-7$. It formalizes the cycle points as roots of cubic polynomials and provides an explicit framework using the coordinate map $T$ and polynomials $Q_\beta$ and $P_\alpha$ to classify and locate 3-cycles. The paper also analyzes stability via hyperbolicity conditions and Schwarzian derivatives, distinguishing several regimes including nonhyperbolic cases. Finally, it demonstrates the approach on two classical real quadratic families, recovering known results for $g_c(x)=x^2+c$ and the logistic map $g_\lambda(x)=\lambda x(1-x)$ with transparent, elementary proofs.
Abstract
For a discrete dynamical system on $\R$ generated by a quadratic function, we show, using elementary computations, that the existence, number, and stability of 3-cycles are determined by a single parameter depending on the coefficients of the function.