{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T03:06:36Z","timestamp":1771038396570,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2013,11,27]],"date-time":"2013-11-27T00:00:00Z","timestamp":1385510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann\u2013Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results.<\/jats:p>","DOI":"10.3390\/e15125178","type":"journal-article","created":{"date-parts":[[2013,11,27]],"date-time":"2013-11-27T14:43:20Z","timestamp":1385563400000},"page":"5178-5222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Generalized Statistical Mechanics at the Onset of Chaos"],"prefix":"10.3390","volume":"15","author":[{"given":"Alberto","family":"Robledo","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica y Centro de Ciencias de la Complejidad, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Circuito de la Investigaci\u00f3n Cient\u00edfica Ciudad Universitaria, Ciudad de Mexico 04510, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2013,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Beck, C., and Schlogl, F. (1993). Thermodynamics of Chaotic Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511524585"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1143\/PTPS.99.1","article-title":"Statistical mechanics of dynamical systems","volume":"99","author":"Mori","year":"1989","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Baldovin, F., and Robledo, A. (2002). Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions. Europhys. Lett., 60.","DOI":"10.1209\/epl\/i2002-00249-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"026209","DOI":"10.1103\/PhysRevE.72.026209","article-title":"Tsallis q index and Moris q phase transitions at the edge of chaos","volume":"72","author":"Mayoral","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"036213","DOI":"10.1103\/PhysRevE.77.036213","article-title":"q-deformed statistical-mechanical property in the dynamics of trajectories en route to the Feigenbaum attractor","volume":"77","author":"Robledo","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.physa.2006.03.018","article-title":"Fluctuating dynamics at the quasiperiodic onset of chaos, Tsallis q-statistics and Mori\u2019s q-phase thermodynamics","volume":"370","author":"Robledo","year":"2006","journal-title":"Physica A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.physa.2006.08.057","article-title":"Routes to chaos, universality and glass formation","volume":"372","author":"Baldovin","year":"2006","journal-title":"Physica A"},{"key":"ref_8","unstructured":"Schuster, H.G. (1988). Deterministic Chaos: An Introduction, VCH Publishers. [2nd ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hilborn, R.C. (2000). Chaos and Nonlinear Dynamics, Oxford University Press. [2nd ed.].","DOI":"10.1093\/acprof:oso\/9780198507239.001.0001"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_11","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/S0375-9601(00)00180-8","article-title":"Criticality and intermittency in the order parameter space","volume":"268","author":"Contoyiannis","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"035701","DOI":"10.1103\/PhysRevLett.89.035701","article-title":"Intermittent dynamics of critical fluctuations","volume":"89","author":"Contoyiannis","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.physa.2004.06.043","article-title":"Critical fluctuations, intermittent dynamics and Tsallis statistics","volume":"344","author":"Robledo","year":"2004","journal-title":"Physica A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3025","DOI":"10.1080\/00268970500185989","article-title":"Unorthodox properties of critical clusters","volume":"103","author":"Robledo","year":"2005","journal-title":"Mol. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"045201","DOI":"10.1103\/PhysRevE.80.045201","article-title":"Equivalence between the mobility edge of electronic transport on disorderless networks and the onset of chaos via intermittency in deterministic maps","volume":"80","author":"Robledo","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.physleta.2004.06.062","article-title":"Universal glassy dynamics at noise-perturbed onset of chaos: A route to ergodicity breakdown","volume":"328","author":"Robledo","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.physa.2004.04.065","article-title":"Aging at the edge of chaos: Glassy dynamics and nonextensive statistics","volume":"342","author":"Robledo","year":"2004","journal-title":"Physica A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"066213","DOI":"10.1103\/PhysRevE.72.066213","article-title":"Parallels between the dynamics at the noise-perturbed onset of chaos in logistic maps and the dynamics of glass formation","volume":"72","author":"Baldovin","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"012007","DOI":"10.1088\/1742-6596\/394\/1\/012007","article-title":"A dynamical model for hierarchy and modular organization: The trajectories en route to the attractor at the transition to chaos","volume":"394","author":"Robledo","year":"2012","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Fuentes, M.A., and Robledo, A. (2010). Renormalization group structure for sums of variables generated by incipiently chaotic maps. J. Stat. Mech., P01001.","DOI":"10.1088\/1742-5468\/2010\/01\/P01001"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"012002","DOI":"10.1088\/1742-6596\/201\/1\/012002","article-title":"Stationary distributions of sums of marginally chaotic variables as renormalization group fixed points","volume":"201","author":"Fuentes","year":"2012","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Fuentes, M.A., and Robledo, A. (2013). Sums of variables at the onset of chaos. Eur. Phys. J. B, submitted for publication.","DOI":"10.1140\/epjb\/e2014-40882-1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1140\/epjb\/e2011-10968-5","article-title":"Possible thermodynamic structure underlying the laws of Zipf and Benford","volume":"81","author":"Altamirano","year":"2011","journal-title":"Eur. Phys. J. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3625","DOI":"10.1016\/j.physleta.2012.10.050","article-title":"Feigenbaum graphs at the onset of chaos","volume":"376","author":"Luque","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_26","unstructured":"Ergodicity is referred here in relation to the total phase space on which the map is defined."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"045104","DOI":"10.1103\/PhysRevE.66.045104","article-title":"Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics","volume":"66","author":"Baldovin","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1016\/j.physa.2006.06.003","article-title":"Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points","volume":"370","author":"Robledo","year":"2006","journal-title":"Physica A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/S0378-4371(02)01177-9","article-title":"The renormalization group and optimization of non-extensive entropy: Criticality in non-linear one-dimensional maps","volume":"314","author":"Robledo","year":"2002","journal-title":"Physica A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.physd.2004.01.016","article-title":"Criticality in nonlinear one-dimensional maps: RG universal map and nonextensive entropy","volume":"193","author":"Robledo","year":"2004","journal-title":"Physica D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1103\/PhysRevLett.48.1645","article-title":"Exact solutions to the feigenbaum renormalization-group equations for intermittency","volume":"48","author":"Hu","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"104101","DOI":"10.1103\/PhysRevLett.90.104101","article-title":"Aging in subdiffusion generated by a deterministic dynamical system","volume":"90","author":"Barkai","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4591","DOI":"10.1073\/pnas.85.13.4591","article-title":"Sporadicity: Between periodic and chaotic dynamical behaviors","volume":"85","author":"Gaspard","year":"1988","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"045202","DOI":"10.1103\/PhysRevE.69.045202","article-title":"Nonextensive Pesin identity: Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map","volume":"69","author":"Baldovin","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Anania, G., and Politi, A. (1988). Dynamical behaviour at the onset of chaos. Europhys. Lett., 7.","DOI":"10.1209\/0295-5075\/7\/2\/005"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1143\/PTP.82.897","article-title":"Dynamic description of the critical 2\u221e attractor and 2m-band chaos","volume":"82","author":"Hata","year":"1989","journal-title":"Progr. Theor. Phys."},{"key":"ref_37","first-page":"591","article-title":"Labyrinthine pathways towards supercycle attractors in unimodal maps","volume":"7","author":"Moyano","year":"2009","journal-title":"Cent. Eur. J. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"057201","DOI":"10.1103\/PhysRevE.79.057201","article-title":"Proposed central limit behavior in deterministic dynamical systems","volume":"79","author":"Grassberger","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Diaz-Ruelas, A., and Robledo, A. (2013). Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos. Europhys. Lett., submitted for publication.","DOI":"10.1209\/0295-5075\/105\/40004"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4289","DOI":"10.1103\/PhysRevLett.81.4289","article-title":"Fractals at T = Tc due to instantonlike configurations","volume":"81","author":"Antoniou","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3125","DOI":"10.1103\/PhysRevE.62.3125","article-title":"Fractal geometry of critical systems","volume":"62","author":"Antoniou","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0370-1573(82)90089-8","article-title":"Fluctuations and simple chaotic dynamics","volume":"92","author":"Crutchfield","year":"1982","journal-title":"Phys. Rep."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"For a review see:\n\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              de Benedetti\n\t\t\t\t\t\t\t              P.G.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              Stillinger\n\t\t\t\t\t\t\t              F.H.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t          Supercooled liquids and the glass transition\n\t\t\t\t\t\t\t          Nature\n\t\t\t\t\t\t\t          2001\n\t\t\t\t\t\t\t          410\n\t\t\t\t\t\t\t          259\n\t\t\t\t\t\t\t          267","DOI":"10.1038\/35065704"},{"key":"ref_44","unstructured":"De Benedetti, P.G. (1996). Metastable Liquids. Concepts and Principles, Princeton University Press."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1063\/1.165869","article-title":"Overview of coupled map lattices","volume":"2","author":"Kaneko","year":"1992","journal-title":"Chaos"},{"key":"ref_46","unstructured":"See, for example:\n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              Chaikin\n\t\t\t\t\t\t\t\t              P.M.\n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              Lubensky\n\t\t\t\t\t\t\t\t              T.C.\n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t          Principles of Condensed Matter Physics\n\t\t\t\t\t\t\t\t          Cambridge University Press\n\t\t\t\t\t\t\t\t          Cambridge, UK\n\t\t\t\t\t\t\t\t          1995"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3643","DOI":"10.1007\/s11434-011-4827-y","article-title":"Laws of Zipf and Benford, intermittency, and critical fluctuations","volume":"56","author":"Robledo","year":"2011","journal-title":"Chin. Sci. Bull."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4972","DOI":"10.1073\/pnas.0709247105","article-title":"From time series to complex networks: The visibility graph","volume":"105","author":"Lacasa","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"046103","DOI":"10.1103\/PhysRevE.80.046103","article-title":"Horizontal visibility graphs: Exact results for random time series","volume":"80","author":"Luque","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Luque, B., Lacasa, L., Ballesteros, F., and Robledo, A. (2011). Feigenbaum graphs: A complex network perspective of chaos. PLoS One, 6.","DOI":"10.1371\/journal.pone.0022411"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"013109","DOI":"10.1063\/1.3676686","article-title":"Analytical properties of horizontal visibility graphs in the Feigenbaum scenario","volume":"22","author":"Luque","year":"2012","journal-title":"Chaos"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s00332-012-9153-2","article-title":"Quasiperiodic graphs: Structural design, scaling and entropic properties","volume":"23","author":"Luque","year":"2013","journal-title":"J. Nonlinear Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"052801","DOI":"10.1103\/PhysRevE.87.052801","article-title":"Horizontal visibility graphs generated by type-I intermittency","volume":"87","author":"Luque","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_54","unstructured":"N\u00fa\u00f1ez, A.M., Luque, B., Cordero, M., G\u00f3mez, M., and Robledo, A. (2013). Quasiperiodic graphs at the onset of chaos. Phys. Rev. E, submitted for publication."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"6390","DOI":"10.1073\/pnas.1103539108","article-title":"Generalized entropies and the transformation group of superstatistics","volume":"108","author":"Hanel","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"19151","DOI":"10.1073\/pnas.1216885109","article-title":"Generalized entropies and logarithms and their duality relations","volume":"109","author":"Hanel","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/12\/5178\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:55Z","timestamp":1760219455000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/12\/5178"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,27]]},"references-count":56,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2013,12]]}},"alternative-id":["e15125178"],"URL":"https:\/\/doi.org\/10.3390\/e15125178","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,27]]}}}