Our general aim is to advance understanding of complexity via nonlinear dynamics, specifically through the transitions to chaos in dissipative systems. We have worked mainly with dissipative low-dimensional nonlinear mappings, as they exhibit three routes to chaos: Intermittency, period doubling, and quasiperiodicity. One circumstance we emphasize is that of complex systems described via criticality (critical fluctuations) and by intermittency close to transitions to chaos in deterministic maps near tangent bifurcations [1,2]. We have made efforts to find and exploit analogies between the description of particular complex system problems or systems and the transitions to chaos in nonlinear dissipative systems. For example: Ranked data sets that originate from many different fields: astrophysical, geophysical, ecological, biological, technological, financial, urban, social, etc., suggesting some kind of universality. Over the years this situation has attracted much attention and the rationalization of Zipf empirical law has become a common endeavor in the study of complex systems [3,4]. In our work, we have linked on the one hand experimental data for size-rank functions with trajectories from nonlinear dynamical maps at the tangent bifurcation. On the other hand we have made connection with generalized entropy expressions. So we have provided a bridge between experimental data and theory for size-rank distributions based both on stochastic processes and deterministic nonlinear dynamical systems. Also, this work delivers a working explanation for the existing duality for Tsallis-type entropy expressions. Since there are many real sets of data that can be arranged as size-rank distributions, we provide numerous examples where generalized entropies apply, and this indicates the importance of this line of research work [5,6].

[1] Alberto Robledo, “Unorthodox Properties of Critical Clusters”, Mol. Phys., 103(21), 3025, 2005.

[2] Alberto Robledo, “Laws of Zipf and Benford, Intermittency, and Critical Fluctuations”, Chinese Science Bulletin, 56(34), 3643, 2011.

[3] Manfred Schroeder, “Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise”, Freeman, New York, 1991.

[4] Murray Gell-Mann, “The Quark and the Jaguar: Adventures in the Simple and the Complex”, Freeman, New York, 1994.

[5] G.Cigdem Yalcin, Carlos Velarde, Alberto Robledo, “Entropies for Severely Contracted Configuration Space” Heliyon-Elsevier, 1(3), 1-14, 2015.

[6] G.Cigdem Yalcin, Alberto Robledo, Murray Gell-Mann, “Incidence of q-statistics in Rank Distributions, PNAS, 111(39), 14082, 2014.