Harmonic Interpretation of Period-Doubling Bifurcations at the Edge of Chaos

In the classical theory of dynamical systems, period-doubling bifurcations are traditionally identified through strict periodicity conditions imposed on stable orbits. However, in the immediate vicinity of the chaos threshold such Boolean criteria become numerically unstable: exact periodicity is destroyed, while the universal structure of the cascade remains intact. In this work, an alternative spectral approach to bifurcation identification is proposed. Instead of requiring strict periodicity, we consider the moment of activation of a harmonic component with period in the system’s dynamics. A normalized harmonic score is introduced, based on a demeaned harmonic amplitude and median block aggregation of the time series. It is shown that the harmonic activation points , defined via an ε-criterion , are robust with respect to variations in ε and to the number of aggregation blocks. Numerical results for the logistic map demonstrate that harmonic activation provides a stable and physically interpretable signature of bifurcations even at the edge of chaos, where standard periodic criteria lose applicability.