We describe the remarkable appearance of the dynamics associated to the tangent bifurcation in low dimensional nonlinear maps in central problems in condensed matter and in complex systems. We first recall the basic features of the intermittency route to chaos via this kind of bifurcation and then turn into the description of two precise equivalences between apparently different physical problems. The first one concerns the electronic transport properties obtained via the scattering matrix of a solid defined on a double Cayley tree. This strict analogy reveals in detail the nature of the mobility edge normally studied near (not at) the metal-insulator transition in electronic systems. We provide an analytical expression for the conductance as a function of the system size. This manifests as power-law decay (with universal exponent) or few and far between large spike oscillations [1]. The second relates to the laws of Zipf and Benford, obeyed by scores of numerical data generated by many and diverse kinds of natural phenomena and human activity. The analogy effortlessly, and quantitatively, reproduces the bends or tails observed in real data for small and large rank. It explains the generic form of the degree distribution of scale-free networks and also suggests a possible thermodynamic structure underlying these empirical laws [2].
[1] M. Martínez-Mares, A. Robledo, Phys. Rev. E80, 045201(R) (2009).
[2] C. Altamirano, A. Robledo, LNICST Vol. 5 (Complex Sciences) pp. 2232-2237, Springer-Verlag (2009).