A zodiac of studies on complex systems

Abstract

We offer a brief description of a set of interrelated research lines on the physics of complex systems developed under a unifying methodology grown out from nonlinear dynamics of low dimensionality. The research lines were, and are, developed over a two-decade period (tacitly or not) under a simplifying assumption (and a posteriori corroboration) of a drastic reduction of degrees of freedom. The studies are conveniently grouped into twelve units, and these in turn into four groups, as in a zodiac. The studies in the first group, named Sensitivities, Glasses and Localizations, have in common a clear-cut original opening in the sense that, to our knowledge, the main tenet or result is not found elsewhere. Those in the second group, named Sums, Rankings and Fluctuations, have as a starting point previous stimulating studies or ideas that we followed up but then we converted into separate approaches. The subjects in the third group, named Networks, Measures and Games, involve preset work programs to be followed but ended up within unanticipated, perhaps deeper, grounds. The topics in the final fourth group, named Partitions, Diagonals and Windows, occurred, or are taking place, as specific technical goals that have become after belated realizations to be possible contributions towards the answer of fundamental quests. We discuss connections underlying different aspects of these investigations. Nonlinear dynamics

How, Why and When Tsallis Statistical Mechanics Provides Precise Descriptions of Natural Phenomena

“…We quote briefly evidence of q -statistical features obtained from nonlinear dynamical models employed for specific problems in condensed matter physics and in complex systems. A more detailed description can be found in the review article “A zodiac of studies on complex systems” [ 18 ].…”

“…Furthermore, over the last two decades a number of central problems for complex systems were modelled around the mentioned RG fixed-point maps that represent the three routes to chaos, the analysis of which led to both fresh insights and correct predictions with explicit presence of the Tsallis entropy, and their associated distributions or Lyapunov exponents [ 6 , 7 , 17 , 18 ]. Amongst condensed matter physics problems there are: dominant fluctuations in critical phenomena, dynamics of glass formation, and wave-scattering localization transitions.…”

“…See Section 4 . Finally, we give a brief description of several research line topics (already quoted above) in the field of complex systems with reference to their relationships with Tsallis statistics [ 18 ]. See Section 5 .…”

Logistic map trajectory distributions: Renormalization-group, entropy, and criticality at the transition to chaos

“…The evolution via sequential gap formation of uniformly distributed ensemble of trajectories toward supercycle and Misiurewicz point attractors display a "recapitulation" property, 13,15,16 i.e., progression toward 2 n -periodic or 2 n -band chaotic attractors repeats successively that toward those attractors with 2 k , k = 0, 1, 2, . .…”

Logistic map trajectory distributions: Renormalization-group, entropy and criticality at the transition to chaos

“…The evolution via sequential gap formation of uniformly distributed ensemble of trajectories towards supercycle and Misiurewicz point attractors display a 'recapitulation' property [13,15,16], i.e. progression towards 2 n -periodic or 2 n -band chaotic attractors repeats successively that towards those attractors with 2 k , k = 0, 1, 2, ..., n − 1.…”