{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T10:11:19Z","timestamp":1771495879144,"version":"3.50.1"},"reference-count":88,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T00:00:00Z","timestamp":1771459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we consider a simple bistable motif constituted by a self-enhancing Transcription Factor (TF) and its mRNA with non-instantaneous dynamics. In particular, we mainly numerically investigated the impact of bounded stochastic perturbations of Sine\u2013Wiener type affecting the degradation rate\/binding rate constant of the TF on the phase-like transitions of the system. We show that the intrinsic exponential delay in the TF positive feedback, due to the presence of a mRNA with slow dynamics, deeply affects the above-mentioned transitions for long but finite times. We also show that, in the case of more complex delays in the feedback and\/or in the translation process, the impact of the extrinsic stochasticity is further amplified. We also briefly investigate the power-law behavior (PLB) of the averaged energy spectrum of the TF by showing that, in some cases, the PLB is simply due to the filtering nature of the motif. A similar analysis can also be applied to biological models having a qualitatively similar structure, such as the well-known Capasso and Paveri\u2013Fontana model of cholera spreading.<\/jats:p>","DOI":"10.3390\/e28020238","type":"journal-article","created":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T08:51:09Z","timestamp":1771491069000},"page":"238","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Interplay Between Non-Instantaneous Dynamics of mRNA and Bounded Extrinsic Stochastic Perturbations for a Self-Enhancing Transcription Factor"],"prefix":"10.3390","volume":"28","author":[{"given":"Lorenzo","family":"Cabriel","sequence":"first","affiliation":[{"name":"Department of Mathematics, Informatics and Geosciences, University of Trieste, Via Economo 12\/3, 34132 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4240-3265","authenticated-orcid":false,"given":"Giulio","family":"Caravagna","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Informatics and Geosciences, University of Trieste, Via Economo 12\/3, 34132 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2021-401X","authenticated-orcid":false,"given":"Sebastiano","family":"de Franciscis","sequence":"additional","affiliation":[{"name":"Instituto de Astrof\u00edsica de Andaluc\u00eda, Consejo Superior de Investigaciones Cient\u00edficas (IAA-CSIC), Glorieta de la Astronom\u00eda s\/n, 18008 Granada, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0264-4761","authenticated-orcid":false,"given":"Fabio","family":"Anselmi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Informatics and Geosciences, University of Trieste, Via Economo 12\/3, 34132 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2190-272X","authenticated-orcid":false,"given":"Alberto","family":"D\u2019Onofrio","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Informatics and Geosciences, University of Trieste, Via Economo 12\/3, 34132 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,19]]},"reference":[{"key":"ref_1","unstructured":"Klipp, E., Liebermeister, W., Wierling, C., and Kowald, A. (2016). Systems Biology: A Textbook, John Wiley & Sons. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Alon, U. (2019). An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429283321"},{"key":"ref_3","unstructured":"Ingalls, B.P. (2013). Mathematical Modeling in Systems Biology: An Introduction, MIT Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1063\/1.1668896","article-title":"Symmetry breaking instabilities in dissipative systems. II","volume":"48","author":"Prigogine","year":"1968","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1126\/science.201.4358.777","article-title":"Time, structure, and fluctuations","volume":"201","author":"Prigogine","year":"1978","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1017\/S0033583500000615","article-title":"Biological order, structure and instabilities","volume":"4","author":"Prigogine","year":"1971","journal-title":"Q. Rev. Biophys."},{"key":"ref_7","unstructured":"Horsthemke, W., and Lefever, R. (1984). Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer."},{"key":"ref_8","first-page":"367","article-title":"White and coloured external noise and transition phenomena in nonlinear systems","volume":"29","author":"Arnold","year":"1978","journal-title":"Z. Phys. B Condens. Matter"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"De Kepper, P., and Horsthemke, W. (1979). Experimental evidence of noise-induced transitions in an open chemical system. Synergetics, Springer.","DOI":"10.1007\/978-3-642-67262-0_10"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2056","DOI":"10.1016\/j.physa.2017.11.123","article-title":"Bounded noise induced first-order phase transitions in a baseline non-spatial model of gene transcription","volume":"492","author":"Caravagna","year":"2018","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_11","unstructured":"Prigogine, I. (1971). Dissipative structures in biological systems. De la Physique Th\u00e9orique \u00e0 la Biologie, Centre National de la Recherche Scientifique."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s11047-014-9424-y","article-title":"Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks","volume":"13","author":"Caravagna","year":"2014","journal-title":"Nat. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"26980","DOI":"10.1038\/srep26980","article-title":"Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif","volume":"6","author":"Caravagna","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"C531","DOI":"10.1152\/ajpcell.1998.274.2.C531","article-title":"Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems","volume":"274","author":"Smolen","year":"1998","journal-title":"Am. J. Physiol.-Cell Physiol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5529","DOI":"10.1093\/nar\/gkf682","article-title":"Genome-wide analysis of mRNA decay in resting and activated primary human T lymphocytes","volume":"30","author":"Raghavan","year":"2002","journal-title":"Nucleic Acids Res."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Lee, J.E., Lee, J.Y., Wilusz, J., Tian, B., and Wilusz, C.J. (2010). Systematic analysis of cis-elements in unstable mRNAs demonstrates that CUGBP1 is a key regulator of mRNA decay in muscle cells. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0011201"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1093\/dnares\/dsn030","article-title":"Database for mRNA half-life of 19 977 genes obtained by DNA microarray analysis of pluripotent and differentiating mouse embryonic stem cells","volume":"16","author":"Sharova","year":"2009","journal-title":"DNA Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"13339","DOI":"10.1038\/srep13339","article-title":"Heterogeneous lineage marker expression in naive embryonic stem cells is mostly due to spontaneous differentiation","volume":"5","author":"Nair","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"248102","DOI":"10.1103\/PhysRevLett.106.248102","article-title":"Determining the stability of genetic switches: Explicitly accounting for mRNA noise","volume":"106","author":"Assaf","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.jtbi.2012.10.004","article-title":"Type of noise defines global attractors in bistable molecular regulatory systems","volume":"317","author":"Jaruszewicz","year":"2013","journal-title":"J. Theor. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/0065-2571(65)90067-1","article-title":"Oscillatory behavior in enzymatic control processes","volume":"3","author":"Goodwin","year":"1965","journal-title":"Adv. Enzym. Regul."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/0022-5193(68)90189-6","article-title":"Mathematics of cellular control processes I. Negative feedback to one gene","volume":"20","author":"Griffith","year":"1968","journal-title":"J. Theor. Biol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/0022-5193(68)90190-2","article-title":"Mathematics of cellular control processes II. Positive feedback to one gene","volume":"20","author":"Griffith","year":"1968","journal-title":"J. Theor. Biol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1137\/0136019","article-title":"Mathematical analysis of a cellular control process with positive feedback","volume":"36","author":"Selgrade","year":"1979","journal-title":"SIAM J. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/0022-5193(78)90079-6","article-title":"Stability of cyclic gene models for systems involving repression","volume":"74","author":"Banks","year":"1978","journal-title":"J. Theor. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1098\/rspb.1995.0153","article-title":"A model for circadian oscillations in the Drosophila period protein (PER)","volume":"261","author":"Goldbeter","year":"1995","journal-title":"Proc. R. Soc. Lond. Ser. B Biol. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8614","DOI":"10.1073\/pnas.151588598","article-title":"Intrinsic noise in gene regulatory networks","volume":"98","author":"Thattai","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"031908","DOI":"10.1103\/PhysRevE.72.031908","article-title":"Core genetic module: The mixed feedback loop","volume":"72","author":"Hakim","year":"2005","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1109\/TAC.2007.911347","article-title":"Noise in gene regulatory networks","volume":"53","author":"Lestas","year":"2008","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"22096","DOI":"10.1073\/pnas.1008965107","article-title":"Bimodal gene expression in noncooperative regulatory systems","volume":"107","author":"Tabaka","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"058102","DOI":"10.1103\/PhysRevLett.106.058102","article-title":"Intrinsic noise in stochastic models of gene expression with molecular memory and bursting","volume":"106","author":"Jia","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"061913","DOI":"10.1103\/PhysRevE.85.061913","article-title":"Stochastic single-gene autoregulation","volume":"85","author":"Aquino","year":"2012","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"012715","DOI":"10.1103\/PhysRevE.89.012715","article-title":"Distribution of population-averaged observables in stochastic gene expression","volume":"89","author":"Bhattacharyya","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3116","DOI":"10.1016\/S0006-3495(01)75949-8","article-title":"Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations","volume":"81","author":"Kepler","year":"2001","journal-title":"Biophys. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7402","DOI":"10.1073\/pnas.0610468104","article-title":"Deterministic characterization of stochastic genetic circuits","volume":"104","author":"Scott","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1016\/j.jmb.2010.06.051","article-title":"Positive autoregulation shapes response timing and intensity in two-component signal transduction systems","volume":"401","author":"Mitrophanov","year":"2010","journal-title":"J. Mol. Biol."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Hermsen, R., Erickson, D.W., and Hwa, T. (2011). Speed, sensitivity, and bistability in auto-activating signaling circuits. PLoS Comput. Biol., 7.","DOI":"10.1371\/journal.pcbi.1002265"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Frigola, D., Casanellas, L., Sancho, J.M., and Iba\u00f1es, M. (2012). Asymmetric stochastic switching driven by intrinsic molecular noise. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0031407"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3776","DOI":"10.1016\/j.febslet.2008.10.005","article-title":"Robustness analysis of cellular memory in an autoactivating positive feedback system","volume":"582","author":"Cheng","year":"2008","journal-title":"FEBS Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1038\/nature02089","article-title":"A positive-feedback-based bistable \u2018memory module\u2019that governs a cell fate decision","volume":"426","author":"Xiong","year":"2003","journal-title":"Nature"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"6638","DOI":"10.1073\/pnas.0806137106","article-title":"Bistable switches control memory and plasticity in cellular differentiation","volume":"106","author":"Wang","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1088\/1478-3975\/4\/1\/004","article-title":"Positive feedback, stochasticity and genetic competence","volume":"4","author":"Karmakar","year":"2007","journal-title":"Phys. Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1038\/nrmicro1381","article-title":"Phenotypic variation in bacteria: The role of feedback regulation","volume":"4","author":"Smits","year":"2006","journal-title":"Nat. Rev. Microbiol."},{"key":"ref_44","unstructured":"Ptashne, M. (1992). A Genetic Switch: Phage Lambda and Higher Organisms, Blackwell."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1186\/1745-6150-5-50","article-title":"A kinetic model of TBP auto-regulation exhibits bistability","volume":"5","author":"Gokhale","year":"2010","journal-title":"Biol. Direct"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1038\/nature03524","article-title":"Enhancement of cellular memory by reducing stochastic transitions","volume":"435","author":"Acar","year":"2005","journal-title":"Nature"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1146\/annurev.biochem.69.1.183","article-title":"Two-component signal transduction","volume":"69","author":"Stock","year":"2000","journal-title":"Annu. Rev. Biochem."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Gao, R., Brokaw, S.E., Li, Z., Helfant, L.J., Wu, T., Malik, M., and Stock, A.M. (2022). Exploring the mono-\/bistability range of positively autoregulated signaling systems in the presence of competing transcription factor binding sites. PLoS Comput. Biol., 18.","DOI":"10.1371\/journal.pcbi.1010738"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1126\/science.1178962","article-title":"Noise can induce bimodality in positive transcriptional feedback loops without bistability","volume":"327","author":"To","year":"2010","journal-title":"Science"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2528","DOI":"10.1093\/emboj\/20.10.2528","article-title":"Positive feedback in eukaryotic gene networks: Cell differentiation by graded to binary response conversion","volume":"20","author":"Becskei","year":"2001","journal-title":"EMBO J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1039\/C6IB00242K","article-title":"Measurement of bistability in a multidimensional parameter space","volume":"9","author":"Jaquet","year":"2017","journal-title":"Integr. Biol."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"d\u2019Onofrio, A. (2013). Bounded Noises in Physics, Biology, and Engineering, Birkh\u00e4user\/Springer.","DOI":"10.1007\/978-1-4614-7385-5"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"MSB200831","DOI":"10.1038\/msb.2008.31","article-title":"Colored extrinsic fluctuations and stochastic gene expression","volume":"4","author":"Shahrezaei","year":"2008","journal-title":"Mol. Syst. Biol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"033301","DOI":"10.1063\/1.4977081","article-title":"Boundedness vs unboundedness of a noise linked to Tsallis q-statistics: The role of the overdamped approximation","volume":"58","author":"Domingo","year":"2017","journal-title":"J. Math. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/07362994.2019.1694416","article-title":"Properties of bounded stochastic processes employed in biophysics","volume":"38","author":"Domingo","year":"2020","journal-title":"Stoch. Anal. Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.physd.2004.01.017","article-title":"Effect of non-Gaussian noise sources in a noise-induced transition","volume":"193","author":"Wio","year":"2004","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0378-4371(01)00435-6","article-title":"Effective Markovian approximation for non-Gaussian noises: A path integral approach","volume":"303","author":"Fuentes","year":"2002","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/S0378-4371(01)00062-0","article-title":"Enhancement of stochastic resonance: The role of non Gaussian noises","volume":"295","author":"Fuentes","year":"2001","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3532","DOI":"10.1016\/j.physleta.2009.07.069","article-title":"Stability regions for Mathieu equation with imperfect periodicity","volume":"373","author":"Bobryk","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.physa.2005.03.055","article-title":"Transitions induced by bounded noise","volume":"358","author":"Bobryk","year":"2005","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1629","DOI":"10.1016\/j.jsv.2013.11.019","article-title":"Stochastic stability of a fractional viscoelastic column under bounded noise excitation","volume":"333","author":"Deng","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.bpc.2009.07.001","article-title":"Non-Gaussian noise optimized spiking activity of Hodgkin\u2013Huxley neurons on random complex networks","volume":"144","author":"Gong","year":"2009","journal-title":"Biophys. Chem."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.physa.2013.09.021","article-title":"Impact of bounded noise and shortcuts on the spatiotemporal dynamics of neuronal networks","volume":"393","author":"Yang","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Ridolfi, L., D\u2019Odorico, P., and Laio, F. (2011). Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press.","DOI":"10.1017\/CBO9780511984730"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.ecocom.2007.06.011","article-title":"Stochastic analysis of predator\u2013prey type ecosystems","volume":"4","author":"Cai","year":"2007","journal-title":"Ecol. Complex."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"2402","DOI":"10.1016\/j.optcom.2012.01.004","article-title":"Noise induced oscillations in time delayed semiconductor laser system","volume":"285","author":"Banerjee","year":"2012","journal-title":"Opt. Commun."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"5632650","DOI":"10.1155\/2018\/5632650","article-title":"Subthreshold Periodic Signal Detection by Bounded Noise-Induced Resonance in the FitzHugh\u2013Nagumo Neuron","volume":"2018","author":"Yao","year":"2018","journal-title":"Complexity"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1688","DOI":"10.1007\/s10955-018-2145-3","article-title":"Improving the stochastic resonance in a bistable system with the bounded noise excitation","volume":"173","author":"Zhao","year":"2018","journal-title":"J. Stat. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"109896","DOI":"10.1016\/j.automatica.2021.109896","article-title":"Deterministic error bounds for kernel-based learning techniques under bounded noise","volume":"134","author":"Maddalena","year":"2021","journal-title":"Automatica"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"3695","DOI":"10.1007\/s11071-020-05741-0","article-title":"An advanced perception model combining brain noise and adaptation","volume":"100","author":"Chholak","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"115214","DOI":"10.1088\/1402-4896\/ad7e5c","article-title":"Bounded diffusing diffusivities: Brownian yet non-Gaussian diffusion","volume":"99","author":"Luo","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Jin, H., Guan, Y., and Yao, L. (2017). Minimum entropy active fault tolerant control of the non-Gaussian stochastic distribution system subjected to mean constraint. Entropy, 19.","DOI":"10.3390\/e19050218"},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Flora, B.F.F., Ciancio, A., and d\u2019Onofrio, A. (2021). On systems of active particles perturbed by symmetric bounded noises: A multiscale kinetic approach. Symmetry, 13.","DOI":"10.3390\/sym13091604"},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Hui, C.H., Lo, C.F., and Liu, C.H. (2023). Equity Price Dynamics under Shocks: In Distress or Short Squeeze. Risks, 12.","DOI":"10.20944\/preprints202311.0517.v1"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1126\/science.1136678","article-title":"Relative impact of nucleotide and copy number variation on gene expression phenotypes","volume":"315","author":"Stranger","year":"2007","journal-title":"Science"},{"key":"ref_76","unstructured":"Alberts, B., Bray, D., Hopkin, K., Johnson, A.D., Lewis, J., Raff, M., Roberts, K., and Walter, P. (2015). Essential Cell Biology, Garland Science."},{"key":"ref_77","unstructured":"Coppel, W.A. (1965). Stability and Asymptotic Behavior of Differential Equations, Heath."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1038\/nature02298","article-title":"Multistability in the lactose utilization network of Escherichia coli","volume":"427","author":"Ozbudak","year":"2004","journal-title":"Nature"},{"key":"ref_79","first-page":"121","article-title":"A mathematical model for the 1973 cholera epidemic in the European Mediterranean region","volume":"27","author":"Capasso","year":"1979","journal-title":"Rev. D\u2019Epidemiol. Sante Publique"},{"key":"ref_80","unstructured":"Dimentberg, M.F. (1988). Statistical Dynamics of Nonlinear and Time-Varying Systems, Taunton Press."},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"d\u2019Onofrio, A. (2013). Dynamics of systems with randomly disordered periodic excitations. Bounded Noises in Physics, Biology, and Engineering, Springer.","DOI":"10.1007\/978-1-4614-7385-5"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Wedig, W.V. (1990). Analysis and simulation of nonlinear stochastic systems. Nonlinear Dynamics in Engineering Systems, Springer.","DOI":"10.1007\/978-3-642-83578-0_42"},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"d\u2019Onofrio, A. (2013). On bounded stochastic processes. Bounded Noises in Physics, Biology, and Engineering, Springer.","DOI":"10.1007\/978-1-4614-7385-5"},{"key":"ref_84","unstructured":"Cabriel, L. (2025, December 28). SDE-SS. Available online: https:\/\/github.com\/lcabriel\/SDE-SS."},{"key":"ref_85","unstructured":"Chua, L.O., Desoer, C.A., and Kuh, E.S.j. (1987). Linear and Nonlinear Circuits, McGraw-Hill."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/jae.659","article-title":"Computation and analysis of multiple structural change models","volume":"18","author":"Bai","year":"2003","journal-title":"J. Appl. Econom."},{"key":"ref_87","unstructured":"Kim, T., Lee, H., and Choi, H. (2024). Improved identification of breakpoints in piecewise regression and its applications. arXiv."},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Capasso, V. (1993). Mathematical Structures of Epidemic Systems, Springer.","DOI":"10.1007\/978-3-540-70514-7"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/2\/238\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T08:53:24Z","timestamp":1771491204000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/2\/238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,19]]},"references-count":88,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["e28020238"],"URL":"https:\/\/doi.org\/10.3390\/e28020238","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,19]]}}}