Power-law behavior around bifurcation points of 1D maps: A supertracks approach

The convergence toward asymptotic states at bifurcation points (BPs) r=rb of 1D mappings of a free parameter r presents scaling laws whose characteristic exponents in principle should depend on the maps non-linear features. Aiming to better understand such comportment, we investigated the logistic-like and sine-like family of maps by studying transcritical, pitchfork, period-doubling, and tangent BPs. For this, we employed the supertracks framework, where continuous functions of r are generated, having the 1D map critical point as the initial condition. Analyzing these functions we obtained, from numerical and analytical procedures, four exponents to describe the asymptotic behavior when r=rb as well as another exponent typifying the case of r>rb. Moreover, we confirmed the universality classes of transcritical and pitchfork BPs proposed in the literature and unveiled novel universality results for period-doubling and tangent BPs. Our findings highlighted the usefulness of the supertracks method, for instance, helping to uncover universality in dynamical systems and allowing to establish parallels with critical phenomena.