Manifestations of the intermittency route to chaos in the physics of condensed matter and of complex systems

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).