{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T11:55:05Z","timestamp":1787313305288,"version":"build-2736575974"},"reference-count":99,"publisher":"World Scientific Pub Co Pte Ltd","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,10]]},"abstract":"<jats:p>We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria.<\/jats:p>\n                  <jats:p>We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure.<\/jats:p>\n                  <jats:p>We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincar\u00e9 return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.<\/jats:p>","DOI":"10.1142\/s0218127414300286","type":"journal-article","created":{"date-parts":[[2014,10,31]],"date-time":"2014-10-31T01:52:45Z","timestamp":1414720365000},"page":"1430028","source":"Crossref","is-referenced-by-count":6,"title":["Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives"],"prefix":"10.1142","volume":"24","author":[{"given":"Vitor","family":"Araujo","sequence":"first","affiliation":[{"name":"Instituto de Matem\u00e1tica, Universidade Federal da Bahia, Av. Adhemar de Barros, S\/N, Ondina 40170-110, Salvador, BA, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefano","family":"Galatolo","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica Applicata, Via Buonarroti 1 Pisa, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maria Jos\u00e9","family":"Pacifico","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970 Rio de Janeiro, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2014,10,30]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1063\/1.166073"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/s002220000057"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385703000397"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2004.12.002"},{"key":"rf6","first-page":"1","volume":"90","author":"Anosov D. V.","year":"1967","journal-title":"Proc. Steklov Math. Inst."},{"key":"rf7","first-page":"107","volume":"22","author":"Anosov D. V.","year":"1967","journal-title":"Uspehi Mat. Nauk"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-006-9183-y"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/s00574-007-0049-y"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04595-9"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-012-1445-8"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-013-1231-0"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2007.19.67"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-006-0001-5"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-04-07671-3"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.2307\/2324899"},{"key":"rf20","doi-asserted-by":"crossref","first-page":"265","DOI":"10.17323\/1609-4514-2006-6-2-265-297","volume":"6","author":"Bautista S.","year":"2006","journal-title":"Mosc. Math. J."},{"key":"rf21","first-page":"1999","volume":"25","author":"Boichenko V. A.","year":"1989","journal-title":"Differentsial' nye Uravneniya"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(97)80131-0"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2003.158.355"},{"key":"rf24","series-title":"Mathematical Physics, III. Encyclopaedia of Mathematical Sciences","volume-title":"Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective","volume":"102","author":"Bonatti C.","year":"2005"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389848"},{"key":"rf26","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0081279","volume-title":"Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms","volume":"470","author":"Bowen R.","year":"1975"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(76)90052-5"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385706000976"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-008-0596-0"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-013-1835-6"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1007\/BF01215756"},{"key":"rf32","volume-title":"An Introduction to Chaotic Dynamical Systems","author":"Devaney R.","year":"1989"},{"key":"rf34","doi-asserted-by":"publisher","DOI":"10.2307\/121012"},{"key":"rf35","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385798117431"},{"key":"rf36","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700000572"},{"key":"rf38","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2007.166.269"},{"key":"rf39","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2007.v14.n5.a8"},{"key":"rf40","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-10-10275-5"},{"key":"rf41","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385709000856"},{"key":"rf42","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/24\/11\/005"},{"key":"rf43","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/6\/6\/014"},{"key":"rf45","doi-asserted-by":"publisher","DOI":"10.1007\/BF02684769"},{"key":"rf46","doi-asserted-by":"publisher","DOI":"10.4213\/rm17"},{"key":"rf48","doi-asserted-by":"publisher","DOI":"10.1016\/0893-9659(94)90058-2"},{"key":"rf49","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00327-0"},{"key":"rf50","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0092042"},{"key":"rf51","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385704000343"},{"key":"rf52","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdm060"},{"key":"rf53","first-page":"261","volume":"91","author":"Hopf E.","year":"1939","journal-title":"Ber. Verh. S\u00e4chs. Akad. Wiss. Leipzig"},{"key":"rf54","doi-asserted-by":"publisher","DOI":"10.1007\/BF00532744"},{"key":"rf55","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-8181-9"},{"key":"rf56","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-1025756-7"},{"key":"rf57","doi-asserted-by":"publisher","DOI":"10.2307\/1971328"},{"key":"rf58","first-page":"804","volume":"24","author":"Leonov G. A.","year":"1988","journal-title":"Diff. nye Uravneniya"},{"key":"rf59","first-page":"155","volume":"13","author":"Leonov G. A.","year":"2001","journal-title":"Algebra i Analiz"},{"key":"rf61","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2004.159.1275"},{"key":"rf62","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"rf63","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-005-1411-9"},{"key":"rf64","doi-asserted-by":"publisher","DOI":"10.2307\/2007021"},{"key":"rf65","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-70335-5"},{"key":"rf66","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-002-0676-5"},{"key":"rf67","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-005-1407-5"},{"key":"rf68","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04520-0"},{"key":"rf69","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-08-09751-7"},{"key":"rf70","doi-asserted-by":"publisher","DOI":"10.1016\/S0294-1449(00)00111-6"},{"key":"rf71","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385707000995"},{"key":"rf72","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1995-00558-6"},{"key":"rf73","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-98-00945-4"},{"key":"rf74","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4722-7_6"},{"key":"rf75","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(97)82717-6"},{"key":"rf76","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2004.160.375"},{"key":"rf77","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2006.15.225"},{"key":"rf78","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-010-9939-2"},{"key":"rf79","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5703-5"},{"key":"rf80","doi-asserted-by":"publisher","DOI":"10.1017\/S014338570000170X"},{"key":"rf81","doi-asserted-by":"publisher","DOI":"10.3934\/jmd.2008.2.397"},{"key":"rf82","series-title":"American Mathematical Society: Memoirs of the American Mathematical Society","volume-title":"Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables","author":"Philipp W.","year":"1975"},{"key":"rf83","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2011.07.001"},{"key":"rf84","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700002327"},{"key":"rf85","doi-asserted-by":"publisher","DOI":"10.1007\/BF01049724"},{"key":"rf86","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385799120911"},{"key":"rf87","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700004004"},{"key":"rf88","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385707000478"},{"key":"rf89","series-title":"Studies in Advanced Mathematics","volume-title":"Dynamical Systems: Stability, Symbolic Dynamics, and Chaos","author":"Robinson C.","year":"1999"},{"key":"rf90","volume-title":"An Introduction to Dynamical Systems: Continuous and Discrete","author":"Robinson C.","year":"2004"},{"key":"rf91","doi-asserted-by":"publisher","DOI":"10.2307\/2373810"},{"key":"rf92","first-page":"191","volume":"296","author":"Ruelle D.","year":"1983","journal-title":"C. R. Acad. Sci. Paris, S\u00e9r. I Math."},{"key":"rf93","doi-asserted-by":"publisher","DOI":"10.1007\/BF01217908"},{"key":"rf94","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511617546"},{"key":"rf95","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385799146820"},{"key":"rf96","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-006-0072-7"},{"key":"rf97","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1947-5"},{"key":"rf98","first-page":"335","volume":"1","author":"Sina\u012d J. G.","year":"1960","journal-title":"Sov. Math. Dokl."},{"key":"rf99","doi-asserted-by":"publisher","DOI":"10.1070\/RM1972v027n04ABEH001383"},{"key":"rf100","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11798-1"},{"key":"rf101","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700000493"},{"key":"rf102","doi-asserted-by":"publisher","DOI":"10.2307\/2372774"},{"key":"rf103","doi-asserted-by":"publisher","DOI":"10.1007\/s002080010018"},{"key":"rf104","doi-asserted-by":"publisher","DOI":"10.1070\/SM1998v189n02ABEH000300"},{"key":"rf105","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2009.10.002"},{"key":"rf107","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2003.10.001"},{"key":"rf108","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5775-2"},{"key":"rf109","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1981-0590412-0"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127414300286","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T15:29:37Z","timestamp":1565105377000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127414300286"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10]]},"references-count":99,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2014,10,30]]},"published-print":{"date-parts":[[2014,10]]}},"alternative-id":["10.1142\/S0218127414300286"],"URL":"https:\/\/doi.org\/10.1142\/s0218127414300286","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,10]]}}}