{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T08:10:09Z","timestamp":1767773409544,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T00:00:00Z","timestamp":1669680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"SECyT of Universidad Nacional de C\u00f3rdoba, 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>A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD functions is analyzed. Several tests were performed, in which the proposed technique was compared with the classical theory of intermittency, the M function methodology, and numerical data. The analysis exposed that the new technique can accurately capture the numerical data. Therefore, the scheme presented herein is a useful tool to theoretically evaluate the statistical variables for type V intermittency.<\/jats:p>","DOI":"10.3390\/sym14122519","type":"journal-article","created":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T02:09:58Z","timestamp":1669687798000},"page":"2519","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Analysis of the Type V Intermittency Using the Perron-Frobenius Operator"],"prefix":"10.3390","volume":"14","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 Rio","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0015-6211","authenticated-orcid":false,"given":"Walkiria","family":"Schulz","sequence":"additional","affiliation":[{"name":"Departamento de Aeron\u00e1utica, FCEFyN, Universidad Nacional de C\u00f3rdoba, C\u00f3rdoba 5000, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Schuster, H., and Just, W. (2005). Deterministic Chaos, Wiley VCH.","DOI":"10.1002\/3527604804"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Nayfeh, A., and Balachandran, B. (1995). Applied Nonlinear Dynamics, Wiley.","DOI":"10.1002\/9783527617548"},{"key":"ref_3","unstructured":"Marek, M., and Schreiber, I. (1995). Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Elaskar, S., and del Rio, E. (2017). New Advances on Chaotic Intermittency and Its Applications, Springer.","DOI":"10.1007\/978-3-319-47837-1"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1103\/PhysRevLett.68.553","article-title":"Return to type I intermittency","volume":"68","author":"Kaplan","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","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":"Physica D"},{"key":"ref_7","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_8","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_9","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_10","doi-asserted-by":"crossref","first-page":"114101","DOI":"10.1103\/PhysRevLett.97.114101","article-title":"Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization","volume":"97","author":"Hramov","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1446","DOI":"10.1103\/PhysRevLett.51.1446","article-title":"Experimental evidence of intermittencies associated with a subharmonic bifurcation","volume":"16","author":"Dubois","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","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. Bifurcation Chaos"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"082108","DOI":"10.1063\/1.2768513","article-title":"Damping models in the truncated derivative nonlinear Schr\u00f6dinger equation","volume":"14","author":"Sanmartin","year":"2007","journal-title":"Phys. Plasmas"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"023025","DOI":"10.1088\/1367-2630\/11\/2\/023025","article-title":"Controlling crisis-induced intermittency using its relation with a boundary crisis","volume":"11","author":"Zambrano","year":"2009","journal-title":"New J. Phys."},{"key":"ref_15","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_16","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_17","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_18","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_19","unstructured":"Chian, A. (2007). Complex System Approach to Economic Dynamics, Springer. Lecture Notes in Economics and Mathematical Systems."},{"key":"ref_20","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":"Physica A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1063\/1.4776519","article-title":"Scaling and intermittency of brains events as a manifestation of consciousness","volume":"1510","author":"Paradisi","year":"2013","journal-title":"AIP Conf. Proc."},{"key":"ref_22","first-page":"519","article-title":"Theory of intermittency","volume":"25","author":"Hirsch","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","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. Bifurcation Chaos"},{"key":"ref_24","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":"Physica A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Elaskar, S., del Rio, E., and Elaskar, S. (2022). Intermittency Reinjection in the Logistic Map. Symmetry, 14.","DOI":"10.3390\/sym14030481"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"121001","DOI":"10.1115\/1.4041577","article-title":"Discontinuous reinjection probability density function in type V intermittency","volume":"13","author":"Elaskar","year":"2018","journal-title":"J. Comp. Nonlinear Dynam."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s11071-018-4083-7","article-title":"Non-uniform reinjection probability density function in type V intermittency","volume":"92","author":"Elaskar","year":"2018","journal-title":"Nonlinear Dynam."},{"key":"ref_28","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_29","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_30","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/0375-9601(93)91063-B","article-title":"The distribution of laminar lengths in type V intermittency","volume":"182","author":"Fan","year":"1993","journal-title":"Phys. Lett. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1088\/0253-6102\/35\/3\/275","article-title":"Characteristics of period-doubling bifurcation cascades in quasi-discontinuous systems","volume":"35","author":"Wu","year":"2001","journal-title":"Commun. Theor. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"070503","DOI":"10.1088\/0256-307X\/27\/7\/070503","article-title":"Intermittent chaotic neural firing characterized by non-smooth like features","volume":"27","author":"Wang","year":"2010","journal-title":"Chin. Phys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1450082","DOI":"10.1142\/S0218127414500825","article-title":"Difference Between Intermittent chaotic bursting and spiking of neural firing patterns","volume":"24","author":"Gu","year":"2014","journal-title":"Int. J. Bifurcation Chaos"},{"key":"ref_34","unstructured":"Sternberg, S. (2010). Dynamical Systems, Dover Publications."},{"key":"ref_35","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 Dynam."},{"key":"ref_36","unstructured":"Lasota, A., and Mackey, M. (1998). Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Springer Science & Business Media."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Beck, C., and Schogl, F. (1993). Thermodynamics of Chaotic Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511524585"},{"key":"ref_38","unstructured":"Burden, R., Faires, J., and Burden, A. (2014). Numerical Analysis, Cengage Learning."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2519\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:28:57Z","timestamp":1760146137000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2519"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,29]]},"references-count":38,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122519"],"URL":"https:\/\/doi.org\/10.3390\/sym14122519","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,11,29]]}}}