{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T20:37:33Z","timestamp":1790368653661,"version":"4.1.0"},"reference-count":35,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J CIRCUIT SYST COMP"],"published-print":{"date-parts":[[2026,3,15]]},"abstract":"<jats:p>This paper derived and proved the simplex explicit commutativity theories and conditions for second-order linear time-varying systems (LTVSs) with both zero and nonzero initial conditions (ICs). We consider Heun\u2019s LTVS as a case study to verify the explicit commutative results, which were supported by simulation. Furthermore, we investigate the sensitivity of Heun\u2019s LTVS, the robustness of Heun\u2019s LTVS, the stability of Heun\u2019s LTVS, the effects due to disturbance on Heun\u2019s LTVS and the problem associated with commutativity of Heun\u2019s LTVS.\u00a0These findings will tackle many problems related to the commutativity theory, the stability of LTVS, design and behavior of control systems, which have made an essential contribution and play a vital role in science and engineering. By considering a sinusoid of amplitude 5, bias [Formula: see text] and frequency 7, with parameters [Formula: see text] and an arbitrary choosing initial time (IT) [Formula: see text] to be and also the initial states [Formula: see text], several quantitative results obtained by simulation show that the Heun\u2019s LTVSs [Formula: see text] and [Formula: see text] give the same output response, [Formula: see text] and [Formula: see text] are commutative under certain conditions and proved to be unstable numerically. Moreover, the quantitative results proved that the Heun\u2019s LTVSs [Formula: see text] and [Formula: see text] are very sensitive toward changes in ICs and parameters. Disturbance between the connections also affects the systems [Formula: see text] and [Formula: see text], these give different responses as a result of tampering with the conditions, hence commutativity is not satisfied. Several examples have been given to support our fact explicitly and numerically. However, the explicit commutativity and stability for Heun\u2019s LTVS have not been in the literature yet, and this paper presents it for the first time. The results are well verified by simulation and treated with Wolfram Mathematica 11.<\/jats:p>","DOI":"10.1142\/s0218126625504523","type":"journal-article","created":{"date-parts":[[2025,8,14]],"date-time":"2025-08-14T03:57:03Z","timestamp":1755143823000},"source":"Crossref","is-referenced-by-count":1,"title":["Explicit Commutativity and Stability Theories for Second-Order Heun\u2019s LTVSs"],"prefix":"10.1142","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1467-5426","authenticated-orcid":false,"given":"Salisu","family":"Ibrahim","sequence":"first","affiliation":[{"name":"Department of Mathematics Education, Tishk International University, Erbil, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Ankara, Turkey"},{"name":"Institute of Space Science, Magurele, Bucharest, Romania"},{"name":"Department of Mathematics, Lebanese American University, Beirut 11022801, Lebanon"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2025,9,27]]},"reference":[{"key":"S0218126625504523BIB001","volume-title":"Ordinary Differential Equations with Applications","author":"Chicone C.","year":"2006","edition":"2"},{"key":"S0218126625504523BIB002","unstructured":"G. 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