{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:56:53Z","timestamp":1760057813722,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T00:00:00Z","timestamp":1740096000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article aims to study parallelizable random dynamical systems by examining them through the terms of dissipation and stochastic Lyapunov functions. It is demonstrated that any random variable that is not a random fixed point admits a tube, and every non-wandering point is within one. The Lyapunov function is employed to characterize the asymptotic stability of compact and closed random sets. The section of a random dynamical system is used to define the parallelizable random dynamical system, and it is proven that a random dynamical system is parallelizable if and only if it admits a section. Furthermore, the principle of Lyapunov used this characterization to study the parallelizability of random dynamical systems. The concept of symmetry is defined, and then its impact on the behavior of stochastic dynamic systems, particularly the Lorenz system, is discussed. In addition, by using an appropriate stochastic Lyapunov function, we have shown that the random Lorenz system is parallelizable.<\/jats:p>","DOI":"10.3390\/sym17030325","type":"journal-article","created":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T09:26:32Z","timestamp":1740129992000},"page":"325","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Lyapunov Stability, Parallelizablity, and Symmetry of Random Dynamical Systems"],"prefix":"10.3390","volume":"17","author":[{"given":"Ihsan Jabbar","family":"Kadhim","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq"}]},{"given":"Asmahan Abed","family":"Yasir","sequence":"additional","affiliation":[{"name":"Open Educational College, Al-Qadisiyah Center, Ministry of Education, Al Diwaniyah 58001, Iraq"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"543","DOI":"10.2307\/1970316","article-title":"Parallelizable Flows and Lyapunov\u2019s Second Method","volume":"73","author":"Dugundji","year":"1961","journal-title":"Ann. 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