Periodic solutions of the Nathanson's and the Comb-drive models. (September 2018) - Llibre, Jaume; Nuñez, Daniel E.; Rivera, Andrés
Abstract: We provide sufficient conditions for the existence and stability properties of periodic solutions of the second-order non-autonomous differential equation of the Nathanson's model x ̈ + x + a x ̇ − b ( v 0 + δ v ( ω t ) ) 2 ( 1 − x ) 2 = 0, and of the Comb-drive model x ̈ + x + a x ̇ − 4 b ( v 0 + δ v ( ω t ) ) 2 x ( 1 − x 2 ) 2 = 0, where x ∈ R, a, b, v 0, δ are positive parameters and v ( ω t ) is a 2 π ∕ ω -periodic function. The results are obtained using the averaging theory and the lower and upper solution method. Highlights: We prove the existence of an unstable periodic orbit for the nathanson model via averaging theory. We give sufficient conditions for the stability of the equilibrium of the comb-drive model. The comb-drive model is studied via the upper and lower solution method.