{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T17:56:46Z","timestamp":1762624606594,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,17]],"date-time":"2021-04-17T00:00:00Z","timestamp":1618617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Sciences and Engineering Research Council of Canada","award":["2016-03803"],"award-info":[{"award-number":["2016-03803"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We develop several formal analogies between the logistic equation and the spatially homogeneous and isotropic relativistic cosmology described by the Einstein\u2013Friedmann equations. These analogies produce an effective Lagrangian and Hamiltonian and new symmetries for the logistic equation.<\/jats:p>","DOI":"10.3390\/sym13040704","type":"journal-article","created":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T21:59:49Z","timestamp":1618869589000},"page":"704","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Analogies between Logistic Equation and Relativistic Cosmology"],"prefix":"10.3390","volume":"13","author":[{"given":"Steve","family":"Dussault","sequence":"first","affiliation":[{"name":"Department of Physics &amp; Astronomy, Bishop\u2019s University, Sherbrooke, QC J1M 1Z7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2601-1870","authenticated-orcid":false,"given":"Valerio","family":"Faraoni","sequence":"additional","affiliation":[{"name":"Department of Physics &amp; Astronomy, Bishop\u2019s University, Sherbrooke, QC J1M 1Z7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrea","family":"Giusti","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hawking, S.W., and Ellis, G.F.R. (1973). The Large Scale Structure of Space-Time, Cambridge University Press.","DOI":"10.1017\/CBO9780511524646"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wald, R.M. (1984). General Relativity, Chicago University Press.","DOI":"10.7208\/chicago\/9780226870373.001.0001"},{"key":"ref_3","unstructured":"Carroll, S.M. (2004). Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley."},{"key":"ref_4","unstructured":"Kolb, E.W., and Turner, M.S. (1990). The Early Universe, Addison-Wesley."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Liddle, A.R., and Lyth, D.H. (2000). Cosmological Inflation and Large-Scale Structure, Cambridge University Press.","DOI":"10.1017\/CBO9781139175180"},{"key":"ref_6","unstructured":"Liddle, A.R. (2015). An Introduction to Modern Cosmology, Wiley."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Eisenhart, L.P. (1949). Riemannian Geometry, Princeton University Press.","DOI":"10.1515\/9781400884216"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1103\/RevModPhys.58.689","article-title":"Scale factors R(t) and critical values of the cosmological constant \u039b in Friedmann universes","volume":"58","author":"Felten","year":"1986","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1119\/1.19361","article-title":"Solving for the dynamics of the universe","volume":"67","author":"Faraoni","year":"1999","journal-title":"Am. J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1119\/1.4731258","article-title":"Qualitative study of perfect-fluid Friedmann-Lema\u00eetre-Robertson-Walker models with a cosmological constant","volume":"80","author":"Sonego","year":"2012","journal-title":"Am. J. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"063517","DOI":"10.1103\/PhysRevD.65.063517","article-title":"Symmetry and inflation","volume":"65","author":"Chimento","year":"2002","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.physletb.2011.08.018","article-title":"A symmetry of the spatially flat Friedmann equations with barotropic fluids","volume":"703","author":"Faraoni","year":"2011","journal-title":"Phys. Lett. B"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Faraoni, V. (2020). A Symmetry of the Einstein\u2013Friedmann Equations for Spatially Flat, Perfect Fluid, Universes. Symmetry, 12.","DOI":"10.3390\/sym12010147"},{"key":"ref_14","first-page":"571","article-title":"Equation of state and equation symmetries in cosmology","volume":"B14","author":"Szydlowski","year":"1983","journal-title":"Acta Phys. Polon."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s10714-006-0265-6","article-title":"Equation of state for the Universe from similarity symmetries","volume":"38","author":"Szydlowski","year":"2006","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1088\/0264-9381\/12\/7\/013","article-title":"New inflation from old","volume":"12","author":"Parsons","year":"1995","journal-title":"Class. Quantum Grav."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1007\/s10714-018-2402-4","article-title":"Reconstructions of the dark-energy equation of state and the inflationary potential","volume":"50","author":"Barrow","year":"2018","journal-title":"Gen. Rel. Gravit."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"063524","DOI":"10.1103\/PhysRevD.102.063524","article-title":"The infinite dimensional symmetry groups of the Friedmann equation","volume":"102","author":"Pailas","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Christodoulakis, T., Karagiorgos, A., and Zampeli, A. (2018). Symmetries in classical and quantum treatment of Einstein\u2019s cosmological equations and mini-superspace actions. Symmetry, 10.","DOI":"10.3390\/sym10030070"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Tsamparlis, M., and Paliathanasis, A. (2018). Symmetries of differential equations in cosmology. Symmetry, 10.","DOI":"10.3390\/sym10070233"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1850093","DOI":"10.1142\/S0217732318500931","article-title":"Duality transformation and conformal equivalent scalar-tensor theories","volume":"33","author":"Gionti","year":"2018","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"043528","DOI":"10.1103\/PhysRevD.93.043528","article-title":"Classical and quantum solutions in Brans-Dicke cosmology with a perfect fluid","volume":"93","author":"Paliathanasis","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1650183","DOI":"10.1142\/S0217732316501832","article-title":"Noether symmetries and duality transformations in cosmology","volume":"31","author":"Paliathanasis","year":"2016","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1088\/1475-7516\/2016\/05\/066","article-title":"Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities","volume":"5","author":"Zampeli","year":"2016","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"095202","DOI":"10.1088\/1751-8113\/47\/9\/095202","article-title":"Lie point and variational symmetries in minisuperspace Einstein gravity","volume":"47","author":"Christodoulakis","year":"2014","journal-title":"J. Phys. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.geomphys.2013.12.001","article-title":"FLRW metric cosmology with a perfect fluid by generating integrals of motion","volume":"77","author":"Dimakis","year":"2012","journal-title":"J. Geom. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103512","DOI":"10.1103\/PhysRevD.83.103512","article-title":"Using the Noether symmetry approach to probe the nature of dark energy","volume":"83","author":"Basilakos","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"123514","DOI":"10.1103\/PhysRevD.84.123514","article-title":"Constraints and analytical solutions of f(R) theories of gravity using Noether symmetries","volume":"84","author":"Paliathanasis","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_29","first-page":"62","article-title":"Observational constraints on conformal time symmetry, missing matter and double dark energy","volume":"7","author":"Hee","year":"2018","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_30","first-page":"31","article-title":"Cosmology as a CFT1","volume":"12","author":"Livine","year":"2019","journal-title":"J. High Energy Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"215001","DOI":"10.1088\/1361-6382\/abb577","article-title":"The cosmological constant from conformal transformations: M\u00f6bius invariance and Schwarzian action","volume":"37","author":"Livine","year":"2020","journal-title":"Class. Quantum Grav."},{"key":"ref_32","first-page":"67","article-title":"Conformal structure of FLRW cosmology: Spinorial representation and the so(3,2) algebra of observables","volume":"3","author":"Livine","year":"2020","journal-title":"J. High Energy Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1002","DOI":"10.1140\/epjc\/s10052-020-08590-8","article-title":"A new symmetry of the spatially flat Einstein-Friedmann equations","volume":"80","author":"Dussault","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_34","first-page":"210","article-title":"Relativistic cosmology and the regularization of orbits","volume":"113","author":"Barrow","year":"1993","journal-title":"Observatory"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1142\/S0217732300000980","article-title":"Darboux class of cosmological fluids with time-dependent adiabatic indices","volume":"15","author":"Rosu","year":"2000","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"047602","DOI":"10.1103\/PhysRevE.65.047602","article-title":"Newton\u2019s laws of motion in form of Riccati equation","volume":"65","author":"Nowakowski","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1007\/s10773-006-9123-2","article-title":"Supersymmetry of FRW barotropic cosmologies","volume":"45","author":"Rosu","year":"2006","journal-title":"Int. J. Theor. Phys."},{"key":"ref_38","first-page":"843","article-title":"FRW barotropic zero modes: Dynamical systems observability","volume":"1","author":"Rosu","year":"2007","journal-title":"Appl. Math. Sci."},{"key":"ref_39","first-page":"13","article-title":"Shifted Riccati procedure: Application to conformal barotropic FRW cosmologies","volume":"7","author":"Rosu","year":"2011","journal-title":"SIGMA"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1340017","DOI":"10.1142\/S0217732313400178","article-title":"Inhomogeneous barotropic FRW cosmologies in conformal time","volume":"28","author":"Rosu","year":"2013","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1550100","DOI":"10.1142\/S021773231550100x","article-title":"Barotropic FRW cosmologies with Chiellini damping in comoving time","volume":"30","author":"Rosu","year":"2015","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1450063","DOI":"10.1142\/S0218271814500631","article-title":"A Riccati equation based approach to isotropic scalar field cosmologies","volume":"23","author":"Harko","year":"2014","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Schuch, D. (2018). Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics, Springer.","DOI":"10.1007\/978-3-319-65594-9"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"062703","DOI":"10.1063\/1.1920287","article-title":"Lagrangian Formalism for nonlinear second-order Riccati Systems: One-dimensional Integrability and two-dimensional Superintegrability","volume":"46","author":"Santander","year":"2005","journal-title":"J. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1007\/s11232-006-0086-5","article-title":"Exact solution in a string cosmological model","volume":"148","author":"Koshelev","year":"2006","journal-title":"Theor. Math. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3936","DOI":"10.1103\/PhysRevD.42.3936","article-title":"Nonlinear evolution of long-wavelength metric fluctuations in inflationary models","volume":"42","author":"Salopek","year":"1990","journal-title":"Phys. Rev. D"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"082503","DOI":"10.1063\/1.3460856","article-title":"Friedman versus Abel equations: A connection unraveled","volume":"51","author":"Yurov","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"023523","DOI":"10.1103\/PhysRevD.66.023523","article-title":"The Ermakov-Pinney equation in scalar field cosmologies","volume":"66","author":"Hawkins","year":"2002","journal-title":"Phys. Rev. D"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1950022","DOI":"10.1142\/S0219887819500221","article-title":"A new look at the Schr\u00f6dinger equation in exact scalar field cosmology","volume":"16","author":"Fomin","year":"2018","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_50","unstructured":"Barbosa-Cendejas, N., and Reyes, M.A. (2009). The Schrodinger picture of standard cosmology. arXiv."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1088\/1475-7516\/2014\/12\/035","article-title":"Friedmann\u2019s Equations in All Dimensions and Chebyshev\u2019s Theorem","volume":"1412","author":"Chen","year":"2014","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1088\/1475-7516\/2015\/05\/020","article-title":"Explicit integration of Friedmann\u2019s equation with nonlinear equations of state","volume":"2015","author":"Chen","year":"2015","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1088\/1475-7516\/2015\/10\/056","article-title":"Friedmann-Lemaitre cosmologies via roulettes and other analytic methods","volume":"10","author":"Chen","year":"2015","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_54","first-page":"87","article-title":"L\u2019int\u00e9gration des diff\u00e9rentielles irrationnelles","volume":"18","author":"Chebyshev","year":"1853","journal-title":"J. Math. Pures Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1080\/07468342.1994.11973625","article-title":"An invitation to integration in finite terms","volume":"25","author":"Marchisotto","year":"1994","journal-title":"College Math. J."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"054002","DOI":"10.1088\/1361-6404\/ab9c93","article-title":"Natural phenomena described by the same equation","volume":"41","author":"Faraoni","year":"2020","journal-title":"Eur. J. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/704\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:49:07Z","timestamp":1760161747000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/704"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,17]]},"references-count":56,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["sym13040704"],"URL":"https:\/\/doi.org\/10.3390\/sym13040704","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,4,17]]}}}