{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:17:10Z","timestamp":1760059030906,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T00:00:00Z","timestamp":1747612800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this article, fourth-order systems of ordinary differential equations are studied. These systems are of a special form, which is used in modeling gene regulatory networks. The nonlinear part depends on the regulatory matrix W, which describes the interrelation between network elements. The behavior of solutions heavily depends on this matrix and other parameters. We research the evolution of trajectories. Two approaches are employed for this. The first approach combines a fourth-order system of two two-dimensional systems and then introduces specific perturbations. This results in a system with periodic attractors that may exhibit sensitive dependence on initial conditions. The second approach involves extending a previously identified system with chaotic solution behavior to a fourth-order system. By skillfully scanning multiple parameters, this method can produce four-dimensional chaotic systems.<\/jats:p>","DOI":"10.3390\/computation13050123","type":"journal-article","created":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T05:37:13Z","timestamp":1747633033000},"page":"123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modeling Networks of Four Elements"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-6438-0602","authenticated-orcid":false,"given":"Olga","family":"Kozlovska","sequence":"first","affiliation":[{"name":"Institute of Applied Mathematics, Riga Technical University, 1048 Riga, Latvia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5074-804X","authenticated-orcid":false,"given":"Felix","family":"Sadyrbaev","sequence":"additional","affiliation":[{"name":"Faculty of Natural Sciences and Health, Daugavpils University, 5401 Daugavpils, Latvia"},{"name":"Institute of Mathematics and Computer Science, University of Latvia, 1459 Riga, Latvia"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alakwaa, F.M. (2015). Modeling of Gene Regulatory Networks: A Literature Review. J. Comput. Syst. Biol., 1.","DOI":"10.15744\/2455-7625.1.102"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.2174\/1389203721666200213103350","article-title":"Overview of Gene Regulatory Network Inference Based on Differential Equation Models","volume":"21","author":"Yang","year":"2020","journal-title":"Curr. Protein Pept. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bhattacharya, P., Raman, K., and Tangirala, A.K. (2022). Discovering adaptation-capable biological network structures using control-theoretic approaches. PLOS Comput. Biol., 18.","DOI":"10.1371\/journal.pcbi.1009769"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Emmert-Streib, F., Dehmer, M., and Haibe-Kains, B. (2014). Gene regulatory networks and their applications: Understanding biological and medical problems in terms of networks. Front. Cell Dev. Biol., 2.","DOI":"10.3389\/fcell.2014.00038"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Menolascina, F. (2021). Using Models to (Re-)Design Synthetic Circuits. Synthetic Gene Circuits, Humana.","DOI":"10.1007\/978-1-0716-1032-9"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"223","DOI":"10.4236\/jbise.2013.62A027","article-title":"Modeling of gene regulatory networks: A review","volume":"6","author":"Vijesh","year":"2013","journal-title":"J. Biomed. Sci. Eng."},{"key":"ref_7","unstructured":"Smith, H. (2025, March 12). A Classical Gene Regulatory Network. Available online: https:\/\/sites.science.oregonstate.edu\/~deleenhp\/teaching\/spring06\/MAP4484\/hal-cyclicgene.pdf."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Horimoto, K., Regensburger, G., Rosenkranz, M., and Yoshida, H. (2008). An Algorithm for Qualitative Simulation of Gene Regulatory Networks with Steep Sigmoidal Response Functions. Algebraic Biology, Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-540-85101-1"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Maizels, R.J. (2024). A dynamical perspective: Moving towards mechanism in single-cell transcriptomics. Philos. Trans. R. Soc. B Biol. Sci., 379.","DOI":"10.1098\/rstb.2023.0049"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s41965-020-00046-y","article-title":"A survey of gene regulatory networks modelling methods: From differential equations, to Boolean and qualitative bioinspired models","volume":"2","author":"Barbuti","year":"2020","journal-title":"J. Membr. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12551-010-0041-4","article-title":"Network modelling of gene regulation","volume":"3","author":"Ho","year":"2010","journal-title":"Biophys. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13059-024-03264-0","article-title":"Biologically informed NeuralODEs for genome-wide regulatory dynamics","volume":"25","author":"Hossain","year":"2024","journal-title":"Genome Biol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"033102","DOI":"10.1063\/5.0172767","article-title":"Chaos in gene regulatory networks: Effects of time delays and interaction structure","volume":"34","author":"Atay","year":"2024","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_14","first-page":"1","article-title":"A Kinetic Finite Volume Discretization of the Multidimensional PIDE Model for Gene Regulatory Networks","volume":"86","year":"2024","journal-title":"Bull. Math. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.matcom.2023.08.007","article-title":"Input-to-state stability of stochastic Markovian jump genetic regulatory networks","volume":"222","author":"Cao","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_16","first-page":"1","article-title":"Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons","volume":"13","author":"Wilson","year":"1972","journal-title":"Biol. Cybern."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Noonburg, V.W. (2019). Differential Equations: From Calculus to Dynamical Systems, AMS\/MAA TEXTBOOKS. [2nd ed.].","DOI":"10.1090\/text\/025"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1096\/fj.00-0361com","article-title":"Neural network model of gene expression","volume":"15","author":"Vohradsky","year":"2001","journal-title":"FASEB J."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Furusawa, C., and Kaneko, K. (2008). A generic mechanism for adaptive growth rate regulation. PLOS Comput. Biol., 4.","DOI":"10.1371\/journal.pcbi.0040003"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"66","DOI":"10.21595\/vp.2022.22829","article-title":"Genetic engineering\u2014Construction of a network of four dimensions with a chaotic attractor","volume":"44","author":"Samuilik","year":"2022","journal-title":"Vibroengineering PROCEDIA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"273","DOI":"10.51537\/chaos.1513080","article-title":"A New 6D Two-wing Hyperchaotic System: Dynamical Analysis, Circuit Design, and Sinchronization","volume":"6","author":"Kopp","year":"2024","journal-title":"Chaos Theory Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11323","DOI":"10.1038\/ncomms11323","article-title":"A geometrical approach to control and controllability of nonlinear dynamical networks","volume":"7","author":"Wang","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/ncomms2939","article-title":"Realistic control of network dynamics","volume":"4","author":"Cornelius","year":"2013","journal-title":"Nat. Commun."},{"key":"ref_24","first-page":"393","article-title":"Control in Inhibitory Genetic Regulatory Network Models","volume":"1","author":"Ogorelova","year":"2020","journal-title":"Contemp. Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ogorelova, D., Sadyrbaev, F., and Samuilik, I. (2023). On Targeted Control over Trajectories of Dynamical Systems Arising in Models of Complex Networks. Mathematics, 11.","DOI":"10.3390\/math11092206"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2271","DOI":"10.1142\/S0218127402005820","article-title":"Chaos in a three-dimensional general model of neural network","volume":"12","author":"Das","year":"2002","journal-title":"Int. J. Bifurc. Chaos Appl. Sci. Eng."},{"key":"ref_27","first-page":"11056","article-title":"Is one dimensional Poincar\u00e9 map sufficient to describe the chaotic dynamics of a three dimensional system?","volume":"219","author":"Mukherjee","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kozlovska, O., Sadyrbaev, F., and Samuilik, I. (2023). A New 3D Chaotic Attractor in Gene Regulatory Network. Mathematics, 12.","DOI":"10.3390\/math12010100"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"13","DOI":"10.51537\/chaos.1380419","article-title":"In Search of Chaos in Genetic Systems","volume":"6","author":"Kozlovska","year":"2024","journal-title":"Chaos Theory Appl."},{"key":"ref_30","first-page":"78","article-title":"Numerical Calculation of Lyapunov exponents","volume":"6","author":"Sandri","year":"1996","journal-title":"Math. J."},{"key":"ref_31","unstructured":"Sayama, H. (2015). Introduction to the Modeling and Analysis of Complex Systems, Open SUNY."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Perko, L. (2001). Differential Equations and Dynamical Systems, Springer. [3rd ed.].","DOI":"10.1007\/978-1-4613-0003-8"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1720","DOI":"10.1109\/JLT.2010.2048412","article-title":"Adaptive virtual network topology control based on attractor selection","volume":"28","author":"Koizumi","year":"2010","journal-title":"J. Light. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"43","DOI":"10.37394\/232018.2022.10.6","article-title":"Models of genetic networks with given properties","volume":"10","author":"Kozlovska","year":"2022","journal-title":"WSEAS Trans. Comput. Res."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/5\/123\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:34:59Z","timestamp":1760031299000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/5\/123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,19]]},"references-count":34,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["computation13050123"],"URL":"https:\/\/doi.org\/10.3390\/computation13050123","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2025,5,19]]}}}