{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T14:36:31Z","timestamp":1762353391765,"version":"build-2065373602"},"reference-count":50,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"SECyT of Universidad Nacional de C\u00f3rdoba","award":["RTI2018-094409-B-I00"],"award-info":[{"award-number":["RTI2018-094409-B-I00"]}]},{"name":"Universidad Polit\u00e9cnica de Madrid","award":["RTI2018-094409-B-I00"],"award-info":[{"award-number":["RTI2018-094409-B-I00"]}]},{"name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades of Spain","award":["RTI2018-094409-B-I00"],"award-info":[{"award-number":["RTI2018-094409-B-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The traditional theory of chaotic intermittency developed for return maps hypothesizes a uniform density of reinjected points from the chaotic zone to the laminar one. In the past few years, we have described how the reinjection probability density function (RPD) can be generalized as a power law function. Here, we introduce a broad and general analytical approach to determine the RPD function and other statistical variables, such as the characteristic relation traditionally utilized to characterize the chaotic intermittency type. The proposed theoretical methodology is simple to implement and includes previous studies as particular cases. It is compared with numerical data, the M function methodology, and the Perron\u2013Frobenius technique, showing high accuracy between them.<\/jats:p>","DOI":"10.3390\/sym15081591","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T09:57:02Z","timestamp":1692179822000},"page":"1591","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Theoretical Evaluation of the Reinjection Probability Density Function in Chaotic Intermittency"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7250-0392","authenticated-orcid":false,"given":"Sergio","family":"Elaskar","sequence":"first","affiliation":[{"name":"Departamento de Aeron\u00e1utica, Instituto de Estudios Avanzados en Ingenier\u00eda y Tecnolog\u00eda (IDIT), FCEFyN, Universidad Nacional de C\u00f3rdoba and CONICET, C\u00f3rdoba 5000, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3384-9521","authenticated-orcid":false,"given":"Ezequiel","family":"del R\u00edo","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Aplicada, ETSI de Aeron\u00e1utica y Espacio, Universidad Polit\u00e9cnica de Madrid, 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Elaskar, S. (2023). Symmetry in Nonlinear Dynamics and Chaos. Symmetry, 15.","DOI":"10.3390\/sym15010102"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Schuster, H., and Just, W. (2005). Deterministic Chaos, Wiley VCH.","DOI":"10.1002\/3527604804"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Nayfeh, A., and Balachandran, B. (1995). Applied Nonlinear Dynamics, Wiley.","DOI":"10.1002\/9783527617548"},{"key":"ref_4","unstructured":"Marek, M., and Schreiber, I. (1995). Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press."},{"key":"ref_5","unstructured":"Strogatz, S. (1994). Nonlinear Dynamics and Chaos, Perseus Book Publishing."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Elaskar, S., and del Rio, E. (2017). New Advances on Chaotic Intermittency and Applications, Springer.","DOI":"10.1007\/978-3-319-47837-1"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Elaskar, S., and del Rio, E. (2023). Review of Chaotic Intermittency. Symmetry, 15.","DOI":"10.3390\/sym15061195"},{"key":"ref_8","unstructured":"Rasband, S. (1990). Chaotic Dynamics of Nonlinear Dynamics, John Wiley & Sons."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Guckenheimer, J., and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field, Springer.","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1098\/rspa.1949.0136","article-title":"The nature of turbulent motion at large wave-number","volume":"199","author":"Batchelor","year":"1949","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1017\/S0022112071002581","article-title":"Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid","volume":"50","author":"Kuo","year":"1971","journal-title":"J. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0375-9601(79)90255-X","article-title":"Intermittency and Lorenz model","volume":"75","author":"Manneville","year":"1979","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Irimiciuc, S., Saviuc, A., Tudose-Sandu-Ville, F., Toma, S., Nedeff, F., Marcela Rusu, C., and Agop, M. (2020). Non-Linear Behaviors of Transient Periodic Plasma Dynamics in a Multifractal Paradigm. Symmetry, 12.","DOI":"10.3390\/sym12081356"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1561","DOI":"10.1142\/S0218127408021178","article-title":"The intermittency route to chaos of an electronic digital oscillator","volume":"18","author":"Stavrinides","year":"2008","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Schmiegel, J., and Pons, F. (2021). Stochastic Intermittency Fields in a von K\u00e1rm\u00e1n Experiment. Symmetry, 13.","DOI":"10.3390\/sym13091752"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1080\/13647830.2011.620174","article-title":"Chaotic dynamics in premixed Hydrogen\/air channel flow combustion","volume":"16","author":"Pizza","year":"2012","journal-title":"Combust. Theor. Model"},{"key":"ref_17","unstructured":"Chian, A. (2007). Complex System Approach to Economic Dynamics. Lecture Notes in Economics and Mathematical Systems, Springer."},{"key":"ref_18","unstructured":"Bhansali, R., Holland, M., and Kokoszka, P. (2007). Long Memory in Economics, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.physa.2004.01.012","article-title":"Type-I intermittency in nonstationary systems: Models and human heart-rate variability","volume":"336","author":"Zebrowski","year":"2004","journal-title":"Physical A"},{"key":"ref_20","first-page":"151","article-title":"Scaling and intermittency of brains events as a manifestation of consciousness","volume":"1510","author":"Paradisi","year":"2012","journal-title":"AIP Conf. Proc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.chaos.2018.03.011","article-title":"Scaling and intermittency of brains events as a manifestation of consciousness","volume":"110","author":"Bashkirtseva","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"21037","DOI":"10.1038\/srep21037","article-title":"Periodic, quasi-periodic and chaotic dynamics in simple gene elements with time delays","volume":"6","author":"Suzuki","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1016\/j.cub.2010.04.053","article-title":"The function of bilateral odor arrival time differences in olfactory orientation of sharks","volume":"20","author":"Gardiner","year":"2010","journal-title":"Curr. Biol."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Atema, J., Br\u00f6nmark, C., and Hansson, L. (2012). Chemical Ecology in Aquatic Systems, Oxford University Press.","DOI":"10.1093\/acprof:osobl\/9780199583096.001.0001"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"041505-1","DOI":"10.1115\/1.4031405","article-title":"Intermittency route to combustion instability in a laboratory spray combustor","volume":"138","author":"Pawar","year":"2016","journal-title":"J. Eng. Gas Turbine Power"},{"key":"ref_26","first-page":"723","article-title":"Chaotic pulses for discrete reaction diffusion systems","volume":"4","author":"Nishiura","year":"2005","journal-title":"SIAM J. App. Dyn. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"184502","DOI":"10.1103\/PhysRevLett.110.184502","article-title":"Flow intermittency, dispersion and correlated continuous time random walks in porous media","volume":"110","author":"Dentz","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"042115","DOI":"10.1063\/1.3385796","article-title":"Analysis of the intermittency behavior in a low-temperature discharge plasma by recurrence plot quantification","volume":"17","author":"Stan","year":"2010","journal-title":"Phys. Plasmas"},{"key":"ref_29","first-page":"487","article-title":"Multichannel type-I intermittency in two models of Rayleigh-Benard convection","volume":"51","author":"Malasoma","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","first-page":"519","article-title":"Theory of intermittency","volume":"25","author":"Hirsch","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0167-2789(91)90050-J","article-title":"An experimental observation of a new type of intermittency","volume":"48","author":"Price","year":"1991","journal-title":"Physical D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1103\/PhysRevLett.70.279","article-title":"On-off intermittency: A mechanism for bursting","volume":"70","author":"Platt","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/PhysRevLett.79.47","article-title":"Attractor\u2013repeller collision and eyelet intermittency at the transition to phase synchronization","volume":"79","author":"Pikovsky","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1103\/PhysRevLett.81.321","article-title":"Phase jumps near a phase synchronization transition in systems of two coupled chaotic oscillators","volume":"81","author":"Lee","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1625","DOI":"10.1103\/PhysRevLett.68.1625","article-title":"New type of intermittency in discontinuous maps","volume":"68","author":"Bauer","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0375-9601(92)90133-7","article-title":"Type V intermittency","volume":"171","author":"He","year":"1992","journal-title":"Phys. Lett. A"},{"key":"ref_37","unstructured":"Ott, E. (2008). Chaos in Dynamical Systems, Cambridge University Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1142\/S0218127410026381","article-title":"New characteristic relation in type-II intermittency","volume":"20","author":"Elaskar","year":"2010","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1650228","DOI":"10.1142\/S021812741650228X","article-title":"On the theory of intermittency in 1D maps","volume":"26","author":"Elaskar","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Elaskar, S., del Rio, E., and Lorenzon, D. (2020, January 1\u20134). Chaotic intermittency in maps with infinite derivative. Proceedings of the 2020 IEEE Congreso Bienal de Argentina (ARGENCON), Resistencia, Argentina. Published as IEEE Xplore 9505502.","DOI":"10.1109\/ARGENCON49523.2020.9505502"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1007\/s11071-016-2951-6","article-title":"Evaluation of the statistical properties for type-II intermittency using the Perron-Frobenius operator","volume":"86","author":"Elaskar","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1051\/jphys:0198000410110123500","article-title":"Intermittency, self-similarity and 1\/f spectrum in dissipative dynamical systems","volume":"41","author":"Manneville","year":"1980","journal-title":"J. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2759","DOI":"10.1016\/j.physa.2011.03.016","article-title":"Reinjection probability density in type-III intermittency","volume":"390","author":"Elaskar","year":"2011","journal-title":"Physical A"},{"key":"ref_44","unstructured":"Bartle, R. (1976). The Elements of Real Analysis, John Wiley & Sons."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Elaskar, S., del Rio, E., and Gutierrez Marcantoni, L. (2020, January 1\u20134). Analytical evaluation of the reinjection probability density function in chaotic intermittency. Proceedings of the 2020 IEEE Biennial Congress of Argentina, ARGENCON 2020, Resistencia, Argentina. Published as IEEE Xplore 9505540.","DOI":"10.1109\/ARGENCON49523.2020.9505540"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Murillo Tsijli, M. (2015). Sobre las fracciones continuas: Aplicaciones y curiosidades. Matem\u00e1tica Educ. Internet, 15.","DOI":"10.18845\/rdmei.v15i2.2171"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Hensley, D. (2006). Continued Fractions, Word Scientific.","DOI":"10.1142\/5931"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1080\/00029890.1992.11995835","article-title":"Continued Fractions and Chaos","volume":"99","author":"Corless","year":"1992","journal-title":"Am. Math. Mon."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1093\/imamci\/18.2.281","article-title":"Analysis of performance of symmetric second-order line search algorithms through continued fractions","volume":"18","author":"Pronzato","year":"2001","journal-title":"IMA J. Math. Control Inf."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Elaskar, S., del Rio, E., and Schulz, W. (2022). Analysis of the Type V Intermittency Using the Perron-Frobenius Operator. Symmetry, 14.","DOI":"10.3390\/sym14122519"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1591\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:34:30Z","timestamp":1760128470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1591"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,16]]},"references-count":50,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081591"],"URL":"https:\/\/doi.org\/10.3390\/sym15081591","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,8,16]]}}}