Universal Dimensionless Invariants in Period-Doubling Dynamics

This work presents a compact yet numerically robust investigation of the period-doubling cascade in one-dimensional nonlinear maps.The primary objective is to confirm the universality of scaling behavior without a priori fixing critical parameter values and without relying on a single numerical procedure. In Phase I, the logistic map is analyzed, where superstable parameters are defined by the condition that the critical point belongs exactly to a stable cycle of period .In Phase II, an analogous procedure is applied to the quadratic map providing an independent test of universality for a family with different normalization and phase-space geometry. For both families, sequences of superstable parameters, associated observables, Feigenbaum ratios , and accumulation points are computed.In addition, an independent harmonic sweep analysis based on parameter scanning and the detection of harmonic activation thresholds at high cascade levels is employed as a consistency check. All three procedures consistently converge to the universal Feigenbaum constant while the accumulation points and remain family-dependent.This triple numerical agreement excludes algorithmic artifacts and demonstrates that the universality of the period-doubling cascade represents a genuine structural property of nonlinear dynamical systems.