TY - JOUR

T1 - A DUAL PRINCIPLE FOR SYMMETRIC PERIODIC SOLUTIONS BI-STABILITY IN NONINTERDIGITATED COMB-DRIVE MEMS

AU - Núñez, Daniel

AU - Murcia, Larry

N1 - Publisher Copyright:
© 2024 American Institute of Mathematical Sciences. All rights reserved.

PY - 2024/8

Y1 - 2024/8

N2 - In this work, we introduce a new principle that provides a tool for searching for odd periodic responses of a general nonlinear oscillator with symmetries. This result arises as the dual version of the variational principle first introduced in Ortega (2016), because in this case the nonlinearity xD(t, x) verifies the reversed inequality, i.e., D(t, 0) < D(t, x) for all t ∈ R and x ≠ 0. Thus, our result reveals that under certain conditions there exists a family of odd periodic responses with prescribed nodal properties for the general oscillator. Indeed, contrary to that obtained in Ortega (2016), the number of zeros of the solutions given by this dual principle is at least bounded below. To illustrate the application of our result, we consider a real example where the reversed inequality arises naturally: a noninterdigitated Comb-drive MEMS modeled by a nonlinear version of the Mathieu equation. Then, we prove the existence of a bi-stability operation regime for this example, since under certain conditions, the positive subharmonic of order 2 given by our principle and the trivial solution are linearly stable. Our results are based on classical ODE tools and the perturbation approach in Cen et al. (2020).

AB - In this work, we introduce a new principle that provides a tool for searching for odd periodic responses of a general nonlinear oscillator with symmetries. This result arises as the dual version of the variational principle first introduced in Ortega (2016), because in this case the nonlinearity xD(t, x) verifies the reversed inequality, i.e., D(t, 0) < D(t, x) for all t ∈ R and x ≠ 0. Thus, our result reveals that under certain conditions there exists a family of odd periodic responses with prescribed nodal properties for the general oscillator. Indeed, contrary to that obtained in Ortega (2016), the number of zeros of the solutions given by this dual principle is at least bounded below. To illustrate the application of our result, we consider a real example where the reversed inequality arises naturally: a noninterdigitated Comb-drive MEMS modeled by a nonlinear version of the Mathieu equation. Then, we prove the existence of a bi-stability operation regime for this example, since under certain conditions, the positive subharmonic of order 2 given by our principle and the trivial solution are linearly stable. Our results are based on classical ODE tools and the perturbation approach in Cen et al. (2020).

KW - Comb-drive

KW - MEMS

KW - Nodal properties

KW - Odd Periodic Solution

KW - Ortega’s Principle

UR - http://www.scopus.com/inward/record.url?scp=85195636060&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2024014

DO - 10.3934/dcdsb.2024014

M3 - Article

AN - SCOPUS:85195636060

SN - 1531-3492

VL - 29

SP - 3557

EP - 3571

JO - Discrete and Continuous Dynamical Systems Series B

JF - Discrete and Continuous Dynamical Systems Series B

IS - 8

ER -