{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:46:02Z","timestamp":1774626362099,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,23]],"date-time":"2020-07-23T00:00:00Z","timestamp":1595462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"publisher","award":["OPUS 14   No. 2017\/27\/B\/ST8\/01330"],"award-info":[{"award-number":["OPUS 14   No. 2017\/27\/B\/ST8\/01330"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider a linear dynamical system under the action of potential and circulatory forces. The matrix of potential forces is positive definite, and the main question is when the circulatory forces induce instability to the system. Different approaches to studying the problem are discussed and illustrated by examples. The case of multiple eigenvalues also is considered, and sufficient conditions of instability are obtained. Some issues of the dynamics of a nonlinear system with an unstable linear approximation are discussed. The behavior of trajectories in the case of unstable equilibrium is investigated, and an example of the chaotic behavior versus the case of bounded solutions is presented and discussed.<\/jats:p>","DOI":"10.3390\/sym12081210","type":"journal-article","created":{"date-parts":[[2020,7,23]],"date-time":"2020-07-23T11:26:01Z","timestamp":1595503561000},"page":"1210","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"first","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 90-924 Lodz, Poland"}]},{"given":"Nataliya","family":"Losyeva","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Physics and Mathematics, Sumy State Pedagogical University named after A.S. Makarenko, 40002 Sumy, Ukraine"}]},{"given":"Volodymyr","family":"Puzyrov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Physics and Mathematics, Sumy State Pedagogical University named after A.S. Makarenko, 40002 Sumy, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lamb, J.S.W., and Brands, H. (1994). Symmetries and reversing symmetries in kicked systems. Dynamics, Bifurcation and Symmetry, Springer.","DOI":"10.1007\/978-94-011-0956-7_16"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1016\/S0021-8928(96)00116-5","article-title":"Invariant sets and symmetric periodic orbits of reversible mechanical systems","volume":"60","author":"Tkhai","year":"1996","journal-title":"J. Appl. Math. 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