{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T17:05:18Z","timestamp":1784653518013,"version":"3.55.0"},"reference-count":53,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T00:00:00Z","timestamp":1784592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bioengineering"],"abstract":"<jats:p>Sleep regulation is generated by interacting nonlinear feedback loops involving homeostatic pressure, circadian forcing, mutual inhibition, and neuromodulatory mechanisms. This work presents a nonlinear-dynamical interpretation of sleep with emphasis on bifurcation structure, Lyapunov stability, deterministic intermittency, and hyperchaotic fragmentation. After summarizing representative sleep models and the literature on sleep-stage dynamics, we combine bifurcation and Lyapunov exponent analyses of representative sleep-population models with a four-dimensional model in which cortical activation, sleep-promoting activity, homeostatic sleep drive, and the circadian pacemaker form a slow\u2013fast feedback system. The model maps coordinate-dependent feedback, separated homeostatic accumulation and clearance, external sleep-debt forcing, refractory dynamics, and on\u2013off intermittency onto physiological sleep\u2013wake processes. The baseline diagrams identify routes from regular oscillations to period-doubling, chaotic bands, periodic windows, and positive Lyapunov regions, while the new Lyapunov maps of the four-dimensional model identify parameter regions where the largest and second-largest Lyapunov exponents are positive, supporting the hyperchaotic classification. The formulation also shows how chronic sleep debt and disease-associated or stress-related insomnia can be represented as shifts in homeostatic drive, inhibitory gain, and circadian coupling. Bifurcation theory therefore provides a useful framework for interpreting irregular sleep transitions and for inspiring future sleep-technology and digital twin applications.<\/jats:p>","DOI":"10.3390\/bioengineering13070833","type":"journal-article","created":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T16:16:39Z","timestamp":1784650599000},"page":"833","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bifurcations and Hyperchaos in Mathematical Models of Sleep"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1521-2108","authenticated-orcid":false,"given":"Adriano","family":"Scibilia","sequence":"first","affiliation":[{"name":"Institute of Intelligent Industrial Systems and Technologies for Advanced Manufacturing, National Research Council, 20133 Milan, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2285-2979","authenticated-orcid":false,"given":"Luigi","family":"Fortuna","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Palermo, 90128 Palermo, Italy"},{"name":"Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City 727000, Vietnam"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,21]]},"reference":[{"key":"ref_1","unstructured":"Walker, M. (2017). Why We Sleep: Unlocking the Power of Sleep and Dreams, Scribner."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.tree.2022.10.004","article-title":"The evolution and diversification of sleep","volume":"38","author":"Rattenborg","year":"2023","journal-title":"Trends Ecol. Evol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1111\/j.1365-2869.2006.00559.x","article-title":"A phylogenetic analysis of the correlates of sleep in birds","volume":"15","author":"Roth","year":"2006","journal-title":"J. Sleep Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1111\/j.1365-2869.2004.00393.x","article-title":"Sleep deprivation in honey bees","volume":"13","author":"Sauer","year":"2004","journal-title":"J. Sleep Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1016\/j.anbehav.2005.01.008","article-title":"Sleeping under the risk of predation","volume":"70","author":"Lima","year":"2005","journal-title":"Anim. Behav."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Vyazovskiy, V.V., Tobler, I., and Gilestro, G.F. (2012). The temporal structure of behaviour and sleep homeostasis. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0050677"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.tins.2007.01.004","article-title":"Sensing and expressing homeostatic synaptic plasticity","volume":"30","author":"Rich","year":"2007","journal-title":"Trends Neurosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1016\/S1474-4422(22)00210-1","article-title":"Sleep function: An evolutionary perspective","volume":"21","author":"Siegel","year":"2022","journal-title":"Lancet Neurol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1126\/science.1082795","article-title":"Altered patterns of sleep and behavioral adaptability in NPAS2-deficient mice","volume":"301","author":"Dudley","year":"2003","journal-title":"Science"},{"key":"ref_10","first-page":"3884","article-title":"The multiple time scales of sleep dynamics as a challenge for modelling the sleeping brain","volume":"369","author":"Olbrich","year":"2011","journal-title":"Philos. Trans. A Math. Phys. Eng. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"97257","DOI":"10.1109\/ACCESS.2024.3426942","article-title":"A nonlinear modeling framework for force estimation in human-robot interaction","volume":"12","author":"Scibilia","year":"2024","journal-title":"IEEE Access"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1016\/j.conb.2013.02.008","article-title":"Sleep deprivation and neurobehavioral dynamics","volume":"23","author":"Basner","year":"2013","journal-title":"Curr. Opin. Neurobiol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"e61","DOI":"10.1161\/STR.0000000000000453","article-title":"Impact of Sleep Disorders and Disturbed Sleep on Brain Health: A Scientific Statement from the American Heart Association","volume":"55","author":"Gottesman","year":"2024","journal-title":"Stroke"},{"key":"ref_14","unstructured":"Allen, M. (2026, July 10). Can\u2019t Sleep After COVID? How to Treat COVID Insomnia. Cognitive FX. Updated 28 January 2026. Available online: https:\/\/www.cognitivefxusa.com\/blog\/cant-sleep-after-covid-how-to-treat-covid-insomnia."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1038\/s41746-024-01125-5","article-title":"Five million nights: Temporal dynamics in human sleep phenotypes","volume":"7","author":"Viswanath","year":"2024","journal-title":"npj Digit. Med."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"111892","DOI":"10.1016\/j.jtbi.2024.111892","article-title":"Data-driven mathematical modeling of sleep consolidation in early childhood","volume":"593","author":"Athanasouli","year":"2024","journal-title":"J. Theor. Biol."},{"key":"ref_17","first-page":"025520019","article-title":"The Dynamical Models of Sleep: A Brief Review","volume":"2","author":"Scibilia","year":"2026","journal-title":"Nonlinear Sci. Control Eng."},{"key":"ref_18","first-page":"195","article-title":"A two process model of sleep regulation","volume":"1","year":"1982","journal-title":"Hum. Neurobiol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1177\/0748730406297512","article-title":"A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system","volume":"22","author":"Phillips","year":"2007","journal-title":"J. Biol. Rhythm."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"056127","DOI":"10.1103\/PhysRevE.73.056127","article-title":"Sleep dynamics: A self-organized critical system","volume":"73","author":"Comte","year":"2006","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.5665\/sleep.2880","article-title":"Sleep transitions in hypocretin-deficient narcolepsy","volume":"36","author":"Sorensen","year":"2013","journal-title":"Sleep"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Prerau, M.J., Brown, R.E., Bianchi, M.T., Ellenbogen, J.M., and Purdon, P.L. (2014). Tracking the sleep onset process: An empirical model of behavioral and physiological dynamics. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003866"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yetton, B.D., McDevitt, E.A., Cellini, N., Shelton, C., and Mednick, S.C. (2018). Quantifying sleep architecture dynamics and individual differences using Bayesian networks. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0194604"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1007\/s00285-009-0276-5","article-title":"A mathematical model of the sleep\/wake cycle","volume":"60","author":"Rempe","year":"2010","journal-title":"J. Math. Biol."},{"key":"ref_25","first-page":"3840","article-title":"Quantitative modelling of sleep dynamics","volume":"369","author":"Robinson","year":"2011","journal-title":"Philos. Trans. A Math. Phys. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1038\/s44323-025-00039-z","article-title":"The complexity and commonness of the two-process model of sleep regulation from a mathematical perspective","volume":"2","author":"Skeldon","year":"2025","journal-title":"npj Biol. Timing Sleep"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"e2109339118","DOI":"10.1073\/pnas.2109339118","article-title":"Competitive dynamics underlie cognitive improvements during sleep","volume":"118","author":"Chen","year":"2021","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ginsberg, A.G., Cruz, M.E.C., Weber, F., Booth, V., and Diniz Behn, C.G. (2024). A predictive propensity measure to enter REM sleep. Front. Neurosci., 18.","DOI":"10.3389\/fnins.2024.1431407"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"McCauley, M.E., McCauley, P., Kalachev, L.V., Banks, S., Dinges, D.F., and Van Dongen, H.P.A. (2024). The dynamics of neurobehavioral impairment and recovery after sleep loss. Front. Environ. Health, 3.","DOI":"10.3389\/fenvh.2024.1362755"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kinoshita, F.L., Yamada, R.G., Ode, K.L., and Ueda, H.R. (2025). A unified framework to model synaptic dynamics during the sleep\u2013wake cycle. PLoS Biol., 23.","DOI":"10.1371\/journal.pbio.3003198"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"97","DOI":"10.3390\/clockssleep6010008","article-title":"Can the Brain\u2019s Thermostatic Mechanism Generate Sleep-Wake and NREM-REM Sleep Cycles? A Nested Doll Model of Sleep-Regulating Processes","volume":"6","author":"Putilov","year":"2024","journal-title":"Clocks Sleep"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/0375-9601(85)90444-X","article-title":"Evidence of chaotic dynamics of brain activity during the sleep cycle","volume":"111","author":"Babloyantz","year":"1985","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"R\u00f6schke, J., and Ba\u015far, E. (1988). The EEG is Not a Simple Noise: Strange Attractors in Intracranial Structures. Chaos in Brain Function, Springer.","DOI":"10.1007\/978-3-642-71531-0_13"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/BF00226197","article-title":"Deterministic chaos and the first positive Lyapunov exponent: A nonlinear analysis of the human electroencephalogram during sleep","volume":"69","author":"Fell","year":"1993","journal-title":"Biol. Cybern."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/0013-4694(96)95636-9","article-title":"Discrimination of sleep stages: A comparison between spectral and nonlinear EEG measures","volume":"98","author":"Fell","year":"1996","journal-title":"Electroencephalogr. Clin. Neurophysiol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"10122","DOI":"10.1523\/JNEUROSCI.23-31-10122.2003","article-title":"Interplay of electroencephalogram phase and auditory-evoked neural activity","volume":"23","author":"Kruglikov","year":"2003","journal-title":"J. Neurosci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1137\/110832823","article-title":"A fast\u2013slow analysis of the dynamics of REM sleep","volume":"11","author":"Behn","year":"2012","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1137\/16M1071328","article-title":"One-dimensional map for the circadian modulation of sleep in a sleep-wake regulatory network model for human sleep","volume":"16","author":"Booth","year":"2017","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s11571-009-9092-2","article-title":"Effects of the anesthetic agent propofol on neural populations","volume":"4","author":"Hutt","year":"2009","journal-title":"Cogn. Neurodyn."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Paul, K., Cauller, L.J., and Llano, D.A. (2016). Presence of a chaotic region at the sleep\u2013wake transition in a simplified thalamocortical circuit model. Front. Comput. Neurosci., 10.","DOI":"10.3389\/fncom.2016.00091"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.cels.2017.11.011","article-title":"Chaotic Dynamics Mediate Brain State Transitions, Driven by Changes in Extracellular Ion Concentrations","volume":"5","author":"Rasmussen","year":"2017","journal-title":"Cell Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"110837","DOI":"10.1016\/j.jtbi.2021.110837","article-title":"Chaos in memory function of sleep: A nonlinear dynamical analysis in thalamocortical study","volume":"528","author":"Foroutannia","year":"2021","journal-title":"J. Theor. Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"022412","DOI":"10.1103\/PhysRevE.94.022412","article-title":"Wake-sleep transition as a noisy bifurcation","volume":"94","author":"Yang","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1177\/0748730409346655","article-title":"A Mathematical Model of Homeostatic Regulation of Sleep-Wake Cycles by Hypocretin\/Orexin","volume":"24","author":"Postnova","year":"2009","journal-title":"J. Biol. Rhythm."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Duarte, C.D., Pacheco, M., Iaconis, F.R., Rosso, O.A., Gasaneo, G., and Delrieux, C.A. (2025). Statistical complexity analysis of sleep stages. Entropy, 27.","DOI":"10.3390\/e27010076"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Yao, C., and Yang, D. (2025). Mechanistic modeling of sleep\u2013wake transitions via circadian-modulated threshold dynamics. bioRxiv.","DOI":"10.1101\/2025.07.10.664059"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"686","DOI":"10.3390\/clockssleep5040046","article-title":"The wave model of sleep dynamics and an invariant relationship between non-REM and REM sleep","volume":"5","author":"Kharchenko","year":"2023","journal-title":"Clocks Sleep"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1046\/j.1365-2869.7.s1.6.x","article-title":"Temporal evolution of coherence and power in the human sleep electroencephalogram","volume":"7","author":"Achermann","year":"1998","journal-title":"J. Sleep Res."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"6862","DOI":"10.1523\/JNEUROSCI.1318-04.2004","article-title":"The Sleep Slow Oscillation as a Traveling Wave","volume":"24","author":"Massimini","year":"2004","journal-title":"J. Neurosci."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Granada, A.E., and Herzel, H. (2009). How to Achieve Fast Entrainment? The Timescale to Synchronization. PLoS ONE, 4.","DOI":"10.1371\/journal.pone.0007057"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"044215","DOI":"10.1103\/PhysRevE.111.044215","article-title":"Dynamical mechanism for the interplay of circadian, homeostatic, and ultradian rhythm in normal human sleep","volume":"111","author":"Ji","year":"2025","journal-title":"Phys. Rev. E"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"115732","DOI":"10.1016\/j.chaos.2024.115732","article-title":"A thermodynamic model of the REM\u2013NREM sleep cycle","volume":"190","author":"Sun","year":"2025","journal-title":"Chaos Solitons Fractals"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Qu, W., and Wang, M.-J.-S. (2026). Structured and factorized multi-modal representation learning for physiological affective state and music preference inference. Symmetry, 18.","DOI":"10.3390\/sym18030488"}],"container-title":["Bioengineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2306-5354\/13\/7\/833\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T16:33:23Z","timestamp":1784651603000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2306-5354\/13\/7\/833"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,21]]},"references-count":53,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2026,7]]}},"alternative-id":["bioengineering13070833"],"URL":"https:\/\/doi.org\/10.3390\/bioengineering13070833","relation":{},"ISSN":["2306-5354"],"issn-type":[{"value":"2306-5354","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,21]]}}}