A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS
Abstract In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on L p $L^{p}$ -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-11-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-023-03050-9 |
| _version_ | 1797636646223478784 |
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| author | D. Núñez L. Murcia |
| author_facet | D. Núñez L. Murcia |
| author_sort | D. Núñez |
| collection | DOAJ |
| description | Abstract In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on L p $L^{p}$ -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill’s equations of potential a ( t ) $a(t)$ employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS. |
| first_indexed | 2024-03-11T12:38:01Z |
| format | Article |
| id | doaj.art-d2e70733e52c4120ae3c2d0bb4695d61 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-03-11T12:38:01Z |
| publishDate | 2023-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-d2e70733e52c4120ae3c2d0bb4695d612023-11-05T12:32:07ZengSpringerOpenJournal of Inequalities and Applications1029-242X2023-11-012023111110.1186/s13660-023-03050-9A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMSD. Núñez0L. Murcia1Departamento de Ciencias Naturales y Matemáticas, Facultad de Ingeniería y Ciencias, Pontificia Universidad JaverianaDepartamento de Ciencias Naturales y Matemáticas, Facultad de Ingeniería y Ciencias, Pontificia Universidad JaverianaAbstract In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on L p $L^{p}$ -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill’s equations of potential a ( t ) $a(t)$ employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS.https://doi.org/10.1186/s13660-023-03050-9MEMSComb-drivePeriodic solutionAsymptotic stabilityLower and upper solutions methodAntiperiodic eigenvalues |
| spellingShingle | D. Núñez L. Murcia A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS Journal of Inequalities and Applications MEMS Comb-drive Periodic solution Asymptotic stability Lower and upper solutions method Antiperiodic eigenvalues |
| title | A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS |
| title_full | A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS |
| title_fullStr | A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS |
| title_full_unstemmed | A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS |
| title_short | A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS |
| title_sort | remark about asymptotic stability in duffing equations lateral stability in comb drive finger mems |
| topic | MEMS Comb-drive Periodic solution Asymptotic stability Lower and upper solutions method Antiperiodic eigenvalues |
| url | https://doi.org/10.1186/s13660-023-03050-9 |
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