On a bi-stability regime and the existence of odd subharmonics in a Comb-drive MEMS model with cubic stiffness

A dual principle for symmetric periodic solutions Bi-stability in noninterdigitated Comb-drive MEMS

“…Furthermore, since (15) holds and ϕ m,p (t) is a nontrivial solution of the null Dirichlet problem associated with equation ( 17), we obtain as a direct consequence of Theorem 3.1 in [2] that there exist δ m,p > 0 and a unique smooth function ∆ m,p (δ) defined on [0, δ m,p [ such that ∆ m,p (0) = v N and Φ m,p (t, δ) := x(t, ∆ m,p (δ), δ) for δ ∈ [0, δ m,p [ is a nontrivial solution of the corresponding null Dirichlet problem associated with (16) that emanates from ϕ m,p (t). Here x(t, v, δ) denotes the general solution of (16),i.e., x(0, v, δ) = 0, ẋ(0, v, δ) = v. Moreover, because of the uniqueness of the branch given above by the Implicit Function Theorem, we have that for δ > 0 small enough Φ m,p (t, δ) ≡ x m,N (t), where x m,N (t) ≡ x(t, ωm,N , δ) is the solution of (16) given by Proposition 2.8 in the former section. This reveals a direct connection between the odd periodic oscillations provided by dual principle and those obtained by means of the corresponding perturbation of the nonlinear center at the origin.…”

“…We notice that the variational equation for (16) along the solution x m,N (t) is the Hill's equation ü + q(t, δ)u = 0, q(t, δ) := ∂F (t, x, δ) ∂x…”

“…Recently, works like [14,15,16] have considered the ideas in [17] to introduce some extensions and generalizations of the Ortega's principle in different contexts. In fact, the author in [14] obtained an extension of the variational principle to study the existence of even periodic solutions for the generalized Sitnikov problem.…”

“…Then, the authors in [15] provided an extension of the principle for equations with finite domain and also studied the existence of odd periodic responses for a Comb-drive finger MEMS model with a linear stiffness term and interdigitated fingers, and the case with a cubic nonlinear stiffness was considered in [11]. Later, a generalization of the Ortega's principle was obtained in [16] by means of the truncation technique and a priori bounds of period solutions. The general result in [16] was combined with the perturbation technique in [2] to demonstrate the existence of a bi-stability operation regime for the Comb-drive finger MEMS model with cubic stiffness.…”

“…Later, a generalization of the Ortega's principle was obtained in [16] by means of the truncation technique and a priori bounds of period solutions. The general result in [16] was combined with the perturbation technique in [2] to demonstrate the existence of a bi-stability operation regime for the Comb-drive finger MEMS model with cubic stiffness. Thus, the main purpose of this work is to introduce a dual variational principle for the existence of symmetric periodic solutions of (1) under the hypotheses H1-H2.…”