{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T06:31:06Z","timestamp":1766212266856,"version":"3.48.0"},"reference-count":22,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T00:00:00Z","timestamp":1766016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The principal problem with the analysis of nonlinear dynamical systems is that it is repetitive and inefficient to simulate every initial condition and parameter configuration individually. This not only raises the cost of computation but also constrains scalability in the exploration of a large parameter space. To solve this, we restructured and extended the computational framework so that variation in the parameters and initial conditions can be automatically explored in a unified structure. This strategy is implemented in the brain-inspired nonlinear dynamical model that has three parameters and multiple coupling strengths. The framework enables detailed categorization of the system responses through statistical analysis and through eigenvalue-based assessment of the stability by considering multiple initial states of the system. These results reveal clear differences between periodic, divergent, and non-divergent behavior and show the extent to which the strength of the coupling kij can drive transitions to stable periodic behavior under all conditions examined. This method makes the analysis process easier, less redundant, and provides a scalable tool to study nonlinear dynamics. In addition to its computational benefits, the framework provides a general method that can be generalized to models with more parameters or more complicated network structures.<\/jats:p>","DOI":"10.3390\/a18120805","type":"journal-article","created":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T08:26:36Z","timestamp":1766132796000},"page":"805","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Scalable Framework with Modified Loop-Based Multi-Initial Simulation and Numerical Algorithm for Classifying Brain-Inspired Nonlinear Dynamics with Stability Analysis"],"prefix":"10.3390","volume":"18","author":[{"given":"Haseeba","family":"Sajjad","sequence":"first","affiliation":[{"name":"IT4Innovations, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6747-425X","authenticated-orcid":false,"given":"Adil","family":"Jhangeer","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic"},{"name":"Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan"}]},{"given":"Lubom\u00edr","family":"\u0158\u00edha","sequence":"additional","affiliation":[{"name":"IT4Innovations, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,18]]},"reference":[{"key":"ref_1","first-page":"42","article-title":"Nonlinear Dynamics in complex systems: A Mathematical Approach","volume":"1","author":"Raza","year":"2024","journal-title":"Front. Appl. Phys. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1088\/0951-7715\/1\/1\/005","article-title":"Stability of dynamical systems","volume":"1","author":"Zeeman","year":"1988","journal-title":"Nonlinearity"},{"unstructured":"Hale, J.K. (1968). Dynamical Systems and Stability, Brown University. (No. NASA-CR-95868).","key":"ref_3"},{"unstructured":"Stuart, A., and Humphries, A.R. (1998). Dynamical Systems and Numerical Analysis, Cambridge University Press.","key":"ref_4"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s10485-015-9409-8","article-title":"Dynamical systems in categories","volume":"25","author":"Behrisch","year":"2017","journal-title":"Appl. Categ. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1137\/090749761","article-title":"Dynamical state and parameter estimation","volume":"8","author":"Abarbanel","year":"2009","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"022918","DOI":"10.1103\/PhysRevE.89.022918","article-title":"Experimental observation of extreme multistability in an electronic system of two coupled R\u00f6ssler oscillators","volume":"89","author":"Patel","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4031","DOI":"10.1007\/s11071-024-09943-8","article-title":"Parameter identification of dynamical systems based on short-term prediction by the generalized cell mapping method with deep learning","volume":"113","author":"Yue","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"10335","DOI":"10.1007\/s11071-024-10781-x","article-title":"Comprehensive classification of multistability and Lyapunov exponent with multiple dynamics of nonlinear Schr\u00f6dinger equation","volume":"113","author":"Ali","year":"2024","journal-title":"Nonlinear Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"073151","DOI":"10.1063\/5.0159675","article-title":"Framework for global stability analysis of dynamical systems","volume":"33","author":"Datseris","year":"2023","journal-title":"Chaos"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2350007","DOI":"10.1142\/S021987622350007X","article-title":"A dynamical systems approach to machine learning","volume":"20","author":"Neisy","year":"2023","journal-title":"Int. J. Comput. Methods"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"112114","DOI":"10.1016\/j.chaos.2022.112114","article-title":"Multi-parameter analysis of transition from conservative to dissipative behaviors for a reversible dynamic system","volume":"159","author":"Li","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"033501","DOI":"10.1063\/1.5027718","article-title":"Multistability and tipping: From mathematics and physics to climate and brain","volume":"28","author":"Feudel","year":"2018","journal-title":"Chaos"},{"doi-asserted-by":"crossref","unstructured":"Onuki, A. (2002). Phase Transition Dynamics, Cambridge University Press.","key":"ref_14","DOI":"10.1017\/CBO9780511534874"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"043303","DOI":"10.1103\/PhysRevE.95.043303","article-title":"Classification framework for partially observed dynamical systems","volume":"95","author":"Shen","year":"2017","journal-title":"Phys. Rev. E"},{"unstructured":"Gorban, A.N. (2004). Singularities of Transition Processes in Dynamical Systems, Elsevier. Science Direct Working Paper (S1574-0358)-04.","key":"ref_16"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1023\/B:OPTE.0000042035.67293.92","article-title":"Robust parameter estimation in dynamic systems","volume":"5","author":"Kostina","year":"2004","journal-title":"Optim. Eng."},{"doi-asserted-by":"crossref","unstructured":"Schittkowski, K. (2000). Parameter estimation in dynamic systems. Progress in Optimization: Contributions from Australasia, Springer.","key":"ref_18","DOI":"10.1007\/978-1-4613-0301-5_13"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1140\/epjs\/s11734-021-00353-0","article-title":"Characterizing multistability regions in the parameter space of the Mackey\u2013Glass delayed system","volume":"231","author":"Tarigo","year":"2022","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"105165","DOI":"10.1016\/j.cnsns.2019.105165","article-title":"Multistability in a quasiperiodically forced piecewise smooth dynamical system","volume":"84","author":"Li","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"12554","DOI":"10.3934\/math.2025566","article-title":"Dynamic analysis and multistability of a discontinuous Jerk-like system","volume":"10","author":"Alharthi","year":"2025","journal-title":"AIMS Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20227","DOI":"10.1007\/s11071-025-11198-w","article-title":"From connectome to silicon: A biologically inspired complex network of CMOS chaotic oscillators for analog brain emulation","volume":"113","author":"Kuate","year":"2025","journal-title":"Nonlinear Dyn."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/805\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T05:17:27Z","timestamp":1766207847000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/805"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,18]]},"references-count":22,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["a18120805"],"URL":"https:\/\/doi.org\/10.3390\/a18120805","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2025,12,18]]}}}