A remark about asymptotic stability in Duffing equations: lateral stability in Comb-drive finger MEMS

• Published in: Journal of inequalities and applications Vol. 2023; no. 1; pp. 142 - 11
• Main Authors: Núñez, D., Murcia, L.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 03.11.2023; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Antiperiodic eigenvalues; Applications of Mathematics; Asymptotic properties; Asymptotic stability; Comb-drive; Damping; Duffing equation; Eigenvalues; Fingers; Lateral stability; Linear damping; Lower and upper solutions method; Lower bounds; Mathematics; Mathematics and Statistics; MEMS; Periodic solution; Asymptotic stability; MEMS; Periodic solution; Comb-drive; Lower and upper solutions method; Antiperiodic eigenvalues
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-023-03050-9

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 -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 ) 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.