Periodic oscillations for a graphene-based electrostatic micro actuator

We study the mechanical oscillations for a novel model of a graphene-based electrostatic parallel plates micro actuator introduced by Wei et al.(2017), considering damping eﬀects when a periodic voltage with alternating current is applied. Our analysis starts from recent results about this MEMS model with constant voltage, and provides new in-sights on the periodic mechanical responses for a variable input voltage. We derive suﬃcient conditions on the system physical components for which periodic oscillations with constant sign exist together with their stability prop-Daniel erties. Speciﬁcally, under some conditions, the existence of three periodic solutions is estab-lished, one of them is negative and the others are positive in sign. The positive one nearby the origin is asymptotically locally stable, whilst the other two are unstable. Additionally, we prove that no further constant sign periodic solutions can be found. The existence of periodic solutions is approached from direct and reverse order Lower and Upper Solutions Method, and the stability assertions are derived from the Liapounoﬀ-Zukovskii criteria for Hill’s equations and the linearization principle. Theoretical results are complemented by numerical simulations and numerical continuation results. Fur-thermore, these numerical simulations evidence the robustness of the graphene-based MEMS model over the traditional ones.

The graphene can be defined as an allotrope of carbon (see [9]), which can be understood at a first sight as a monolayer of carbon atoms that are tightly bounded, and organized into a 2D mesh like a planar honeycomb lattice [13]. This novel material has astonishing properties mainly due to its sigma bonding; therefore, graphene can be even 100 times stronger than steel (see [12]) and it can have exceptional friction, elastic and thermal behavior that makes it suitable for applications in electronic devices.
In [6] the authors present a novel graphenepolyaniline nanocomposite material, which used in supercapacitors could surpass traditional ones with applications in high powered portable devices. A review of some electrical and mechanical properties, and some synthesis processes of graphene and carbon nanotubes focusing on applications in sensors and actuators can be found in [23]. For more applications see [9,13,20,24,15,11].
On the other hand, as mentioned in [22] researchers on MEMS are usually leaded to perform several experimental tests by a trial-anderror approach, which carries longer design times and higher costs. Thus, in order to reduce them, we need a more theoretical approach based on mathematical modeling and simulating, allowing the design of higher quality devices (with novel materials) and optimize the performance of those available. For that purpose one of most popular modeling strategies consists in the treatment of the system like a lumped spring-mass one. Thus in this paper we focus on the mechanical response of electrostatic MEMS based on graphene, and specifically their oscillations due to a periodic input voltage through a nonlinear differential equation of this kind inspired by [21] .
From a mathematical point of view there are some recent results about the qualitative behavior of the oscillations for these kind of electrostatic actuated MEMS, when considering constant or periodic input voltage. Perhaps the first paper in this line using an analytical approach is due to Pelesko in [1]. Later in [7] the authors study existence and stability of periodic solutions for a canonical MEMS model called the Nathanson's model with constant or T -periodic input voltage, using classical topological techniques, and obtaining the existence of a saddle-node bifurcation for T -periodic solutions.
In [14], some sufficient conditions for existence and stability properties of periodic solutions for two periodically forced canonical MEMS models: the Nathanson's model and the Combdrive finger model (see [22] chapter 3.5.1 and 3.5.3 respectively) are provided using Averaging Theory and Lower and Upper Solutions Method.
In [16] the authors consider the Nathanson's model and the Comb-drive finger model with cubic stiffness and without damping, and obtain existence, multiplicity and linear stability results for positive periodic solutions. A new stability phenomenon is underlying in the Comb-drive model for large cubic stiffness.
A recent lumped-mass model for a graphenebased microelectromechanical system is established in [21], taking into account the third order elastic-stiffness constant of the material. Furthermore, authors in [21] determine analytically the existence of periodic solutions, determine the static pull-in voltage, and then present a bifurcation analysis for this novel graphene-based MEMS model when a constant voltage is applied. For more related results about this model with constant voltage see [18,17,2].
Next, we introduce the central model of this paper, which is the damped non autonomous version of the one proposed in [21]. The device consists of two parallel plates separated by a distance d (known as gap), where one plate is fixed and the other is movable. The movable plate with mass m is attached to one end of a graphene strip, and in general the device is actuated by an input voltage. The mechanical axial displacement of the graphene strip (graphene-based material strip) is modeled as a nonlinear spring of length L. The figure 1 illustrates the idealized device.
Let E to denote the Young's modulus of graphene, D the absolute value of the third order elastic stiffness modulus of graphene, ǫ 0 the dielectric constant in free space, ǫ r the relative permittivity of the gap space medium with respect to the free space, ǫ = ǫ 0 ǫ r the dielectric constant of the gap medium, A the movable plate area, A c the graphene strip crosssectional area, and x the axial displacement of the movable plate which is measured positive pointing down from its equilibrium position, i.e., from the position for which the distance between the plates is the gap d (see figure 1). Fig. 1: Idealized parallel plates capacitor for the graphene-based MEM device with a constant input voltage V DC (which can be replaced with a variable and periodic input voltage V(t)).
Then the dimensional equation that rules the movement of the movable plate when we consider damping effects, and a damping coefficient c, is given by (see [21] and [8]) are, respectively, the restoration and the electrostatic forces actuating over the device. When V DC is replaced by a variable voltage V(t) the electrostatic force becomes The restoration force F res follows from the study of the graphene strip mechanical response proposed in [8], which takes into account the thirdorder elastic stiffness modulus for graphene, and the resulting isotropic elastic response of graphene given in [12]. The electrostatic force F c is derived from the well known parallel plates electrostatic force (see [22]). Note that gravity effects have been neglected.
For modeling and numerical reasons it is advisable to define dimensionless variables by means of the following change of scales where ω 0 is the natural frequency of the system and d is the gap. Returning to the original symbols we obtain as in [21] the following dimensionless equation Here b denotes the dimensionless damping coefficient, and the parameters α and β relate the elastic properties of the material with the geometry of the device.
Then, the main purpose of this paper is to study analytically the existence and stability properties of periodic solutions of (1) when a periodic voltage with alternating current is applied. Thus, we obtain some sufficient conditions that lead to the existence of exactly three periodic oscillations with constant sign for the model presented above, being the central one asymptotically locally stable (Theorems 1 and 2). These results also show the rising of a fold bifurcation for periodic solutions like in [7]. Consequently, we provide a new insight into the dynamics of a graphenebased electrostatic micro actuator modeled by means of this non autonomous version of the novel graphene-based MEMS model from literature.
This paper is organized as follows. In section 2 we introduce some preliminary definitions and main results about existence, multiplicity and stability of periodic solutions for the graphenebased MEMS model. In section 3 we provide the proofs of our main results. Finally in section 4 we discuss some numerical results that highlight the advantages of using graphene, through comparison between a canonical linear elastic response MEMS model and the graphenebased MEMS model. For a better readability of this document we have included some auxiliary results in Appendixes A and B.

Main Results
In order to establish the main results of this paper we need first to consider some preliminaries. Notice that equation (1) can be rewritten asẍ . Additionally, here we consider a smooth positive and T -periodic input voltage function V(t). Thus we can define Along this paper we will consider two important auxiliary functions defined as follows.
For each α > 0, let us define φ α : IR → IR by This function will be referred to as the auxiliary function for equation (1). Notice that φ α ∈ C 1 (IR) and it has roots at x = 0, x = ± 1 /α and x = 1 (see figure 2). Additionally, the auxiliary function has exactly one negative local minimum in ] − ∞, 0[ and a positive local maximum in ]0, S α [ where We will tackle some properties and the importance of the auxiliary function φ α .
Let us consider now C α ∈ ]0, S α [ the unique critical point of φ α , such that Indeed, a direct computation shows that On the other hand, for M ∈ 0, ( π T ) 2 we define G : M, ( π T ) 2 → IR as Is is easy to verify that G is a continuous positive function vanishing only on the boundary. Therefore, we can define Function G arises from the methodology and will be useful to apply Theorem 4 in Appendix A.
Definition 1 Consider the auxiliary function φ α and assume that βV 2 max < K α . Thus, we denote the root in ]0, C α [ of function h with h 0 = βV 2 min by x U 1 and the root in ]0, Now we are able to establish our main results.
Theorem 1 Consider equation (1) and assume that where η α and Γ α are given by with b * α , x U 1 and x L1 as in Definition 1. Then there exist at least three T -periodic solutions ψ(t), ϕ 1 (t) and ϕ 2 (t) of (1), such that for all t ∈ [0, T ] Moreover, the solution ψ(t) is asymptotically locally stable, and the solution ϕ 2 (t) is unstable.
Then there exists exactly a negative T -periodic solutionφ(t) of (1) which is unstable.
Some additional restriction over the period T allows us to introduce a multiplicity result for positive T -periodic solutions and inquire about the stability properties of ϕ 2 (t).
Theorem 2 Consider the equation (1), assume hypothesis 1 of Theorem 1 and that Then the problem (1) has exactly two positive T -periodic solutions ψ(t), ϕ 1 (t) and exactly one negative T -periodic solution ϕ 2 (t), which are the solutions given by Theorem 1. Moreover, the solution ψ(t) is asymptotically locally stable and ϕ 1 (t), ϕ 2 (t) are unstable.

Remark 1 For Theorem 2 we have that if
then condition (5) of Theorem 2 is easily satisfied. Furthermore, if then the number is real and positive. Thus, condition (5) is equivalent to the restriction over the period i.e., the condition is basically like a high frequency result. T -periodic solutions ψ(t), ϕ 1 (t) and ϕ 2 (t) for equation (1).

Proofs
The proofs of the main results are presented in this section. First we discuss some properties of the auxiliary function φ α and its relevance along this section.
The auxiliary function.
An elementary computation shows that the positive critical point C α defined in section 1 has the following analytic expression On the other hand, for a T -periodic and variable input voltage the function φ α is worthy for the existence study of T -periodic solutions.
In fact, the x-coordinates of intersections of graph of the auxiliary function with horizontal lines, determinate the possible constant and non strict lower and upper solutions for the periodic problem (18) (see Appendix A). In other words, because Definition 2 in Appendix A, the non strict constant lower and upper solutions (denoted by x L and x U respectively ) are the roots in ]−∞, 1[ of equations The existence of such roots follows as particular cases from the existence of roots for the function h given in section 2. Figure 4 illustrates the auxiliary function and the location of non strict constant lower and upper solutions of (18) (see Appendix A) assuming that βV 2 max < K α . Additionally, strict lower and upper solutionsL andŪ can be viewed as solutions of the following inequalities Proof (Theorem 1) Notice that for equation (2) the function f can be written as and g is T -periodic in t. We will consider throughout the proof the non strict constant lower and upper solutions of the periodic problem (18) (see Appendix A) for which (6), (7) hold respectively. We divide the proof in three steps (existence, uniqueness and stability) but before we make a necessary remark; Then, hypothesis 1.c) is equivalent to Step 1. Existence of T -periodic solutions.
1. Assume hypothesis 1.a) and consider the auxiliary function φ α . Then there exist three roots for each equation (6) (L) and (7) Thus, we consider the following sets: Therefore, from hypothesis 1.b) combined with Remark 2 we obtain that On the other hand, ∂ v f (t, x, v) = b, and then assuming hypothesis 1.c) we get from Remark 2 that 0 < b < b * α . Thus, a direct application of Theorem 4 in Appendix x) satisfies the hypotheses of Theorem 3 in Appendix A onĒ 2 . Therefore, a direct application of Theorem 3 implies existence of at least one positive T -periodic 2. Assuming hypothesis 2), we can consider the following two cases: βV 2 max > K α ≥ βV 2 min and βV 2 min > K α . It is easy to verify that for both cases there exists exactly one root for each equation (6) and (7), such that Step 2. Uniqueness of negative T -periodic solutions.
Now we shall prove that if 1) holds then ϕ 2 (t) is the unique negative T -periodic solution of problem (1).
From here φ α (x) ≤ βV 2 max and therefore x L3 ≤ x (see figure 4). Similarly, we have that 0 ≥ figure 4). Hence we obtain that any negative T -periodic solution of problem (1) satisfies

Suppose now that there exists other negative
Since g is of class C 1 on D, and g(t, ·), ϕ 2 (t), ϕ 3 (t) are T -periodic functions, it is easy to verify thatâ(t) ∈ C(IR/TZ). Furthermore, for α . So, the Mean Value Theorem implies that for all t ∈ IR a(t) < 0. (10) can not admit nontrivial T -periodic solutions as consequence of Lemma 3 in Appendix B, therefore ϕ 2 (t) is the unique negative T -periodic solution of problem (1).

Finally, equation in
The uniqueness for negative T -periodic solutions assuming hypothesis 2) is straightforward following the precedent ideas.
1. Assume hypothesis 1) and considerĒ 1 . Thus, variational equation along solution ψ(t) is given bÿ where a 1 (t) = ∂ x g(t, ψ(t)). Additionally, a 1 (t) is a T -periodic, continuous and not identically zero function. Therefore, using the first part of Lemma 2 in Appendix B we obtain the equivalent Hill's equation where z 1 (t) = e 1 2 bt y 1 (t) and Notice that Q(t) is T -periodic, continuous and satisfies for all t ∈ [0, T ] that Since x L1 y x U 1 satisfy, respectively, φ α (x L1 ) = βV 2 max and φ α (x U 1 ) = βV 2 min , it follows that Thus, the function Q(t) satisfies and assuming hypothesis 1.b) and 1.c), we obtain as consequence of Remark 2 and (8) that for M defined in (9). Then an application of Proposition 2 in Appendix B shows that equation (12) is elliptic. Moreover, second part of Lemma 2 in Appendix B implies that characteristic multipliers of (11) are inside of the unit disk. The conclusion follows from linearization principle.
2. Assume hypothesis 1.a) of Theorem 1, thus we know that onĒ 3 there exists exactly one negative T -periodic solution ϕ 2 (t) of problem (1), which has its range included in [x L3 , x U 3 ]. On the other hand, from the continuity and the decreasing character of φ α on −∞, − 1 α one has the existence of ǫ > 0 such that respectively, strict constant lower and upper solutions of (18) (see Appendix A). The conclusion follows applying Proposition 1 in Appendix A.
A similar procedure, assuming hypothesis 2) implies that the T -periodic solutionφ(t) of the problem (1) is unstable. This completes the proof.

Remark 3
In practice, if the inequality in condition 1.b) is strict, then we can choose ℓ in Theorem 4 as in order to obtain an easily computable Γ α since (9) holds. Furthermore, b * α can be replaced by G(ℓ).

Remark 4
The proof of Theorem 1 also provides us a posteriori bounds over the found Tperiodic solutions ψ(t), ϕ 1 (t) and ϕ 2 (t). Thus assuming hypothesis 1) we obtain that Proof (Theorem 2) Assume hypothesis 1) of Theorem 1. Then a direct application of Theorem 1 implies the existence of at least three T -periodic solutions ψ(t), ϕ 1 (t) and ϕ 2 (t) of problem (1), that are located as in Remark 4. Moreover, we know that solution ψ(t) is locally asymptotically stable and that solution ϕ 2 (t) is unstable and the unique negative T -periodic solution. Thus, it remains to prove that ψ(t) and ϕ 1 (t) are the only two positive T -periodic solutions and that solution ϕ 1 (t) is unstable. We divide the proof in two steps.
Let D = IR × ]0, 1[, then g is continuous on D and for all (t, x) ∈ D we have that Therefore, assuming hypothesis (5) of Theorem 2 we obtain that Additionally, for all (t, x) ∈ D it verifies the application of second part of Lemma 1 in Appendix A implies that ψ(t) and ϕ 1 (t) are the only T -periodic solutions of (1) in ]0, 1[.
Assume the hypothesis of Theorem 2, then we know that there exists exactly one positive Tperiodic solution ϕ 1 (t) of (1), which has its range included in [x L2 , x U 2 ]. On the other hand, we have that a similar argument on ]C α , S α [ as in step 3 of the proof of Theorem 1 leads to the existence of a small ǫ > 0 such that L = x L2 − ǫ > C α andŪ = x U 2 + ǫ are, respectively, strict constant lower and upper solutions of (1) (see definition above). Thus, conclusion follows from Proposition 1 in Appendix A since we know that ϕ 1 (t) is the unique Tperiodic solution such that for all t ∈ [0, T ] This completes the proof.
Periodic oscillations for a graphene-based electrostatic micro actuator 11 4 Numerical Results: application to a graphene-based micro actuator In this section we present some numerical results for a graphene-based micro actuator modeled by the damped non autonomous version of the graphene-based MEMS model introduced in section 1. Thus, we aim to verify theoretical results obtained in previous sections and to show some gains of using graphene by means of a comparison between the nonlinear elastic response model of this paper and the linear elastic response parallel plates MEMS model (Nathanson's model). First we justify the parameter values used in the simulations and then we compare the two models.

Parameters values for the graphene MEMS model
In this section we summarize the parameter values for the numerical results starting with the mechanical response of the graphene. The experimental values of the second order and third order elastic stiffness modulus for 2D monolayer graphene can be found in [12]. Thus, assuming a graphene thickness of 0.335 nm (see [12]) we obtain the corresponding values of E = 1.0(1) TPa and D = 2.0(4) TPa. These are the values that we will use in our computations. Table 1 contains realistic values of the parameters that are employed to obtain the numerical results for the graphene-based MEMS model, when a voltage V(t) = V DC + V AC cos (ωt) is supplied (see equation (1)). Notwithstanding, we notice that V DC and V AC can be tuned to convenience almost like "control parameters" to explore the dynamical behavior of the system as long as the required conditions from main theorems hold. In section 4.2 we will consider V DC = 5 V and V AC = 3 V, and later in section 4.4 we will consider V DC = 3 V and V AC = 1.5 V.

Numerical continuation results
The theoretical results of previous section establish the existence of a stable and an unstable solution coexisting for the same value of the parameters provided some conditions are fulfilled (see Theorem 1). In this section we present numerical continuation results as one of the parameters is changed using AUTO which is a well tested and powerful program to analyze (among others) periodic solutions of dynamical systems [10]. Figure 5 shows the bifurcation diagram where the primary continuation parameter is the amplitude of the forcing V AC for a fixed value of the DC voltage V DC = 5V . The blue hollow square at V AC = 3 is the starting solution of the diagram which is stable and was computed numerically solving a boundary value problem with the parameters and boundary value inferred from Theorem 1. As the branch (solid line) moves to higher values of V AC the stability is preserved up to a critical value at which a limit point occurs (LP) (red asterisk). At this point the branch becomes unstable (dashed line) and the value of the continuation parameter diminishes until a vanishing value of the alternating voltage is reached. The vertical axis is the norm of the solution averaged over a period [10] and is just a a scalar measure for the bifurcation diagram. Note that the equilibria for V AC = 0 (one stable and another unstable) can be analytically computed for the autonomous case. The symmetric shape of the negative V AC part of the bifurcation diagram is to be expected since the sign of alternating forcing does not play a role in the bifurcation behavior.
Two solutions close to the LP (V AC = 19.9 (black squares)) has been selected and correspond to the stable and unstable solutions predicted by the theorem. They will also appear in the stroboscopic Poincaré plot of the next section. The linear stability around these periodic solution is measured by the corresponding Floquet multipliers and can be relevant for the applications.    We notice that the zoom close to the rightmost end of the curves shows a highly non symmetric and nonlinear behavior of the oscillation of the plate of the graphene MEMS. The solutions are always far enough from the singularity at x = 1.

Stroboscopic map
In this section we present some results to illustrate the stability and basins of attraction The dynamical behavior of the trajectories can be visualized using of two different color-maps with four colors each and two different time direction integrations. Color-bar A goes from green to blue and the trajectories remain one quarter of the time at each successive color in a forward time integration. The initial conditions for this case have been selected in a grid around to the stable equilibrium point. Color-bar B runs from yellow to red with the same convention (a quarter of the total time at each sub color) in a backwards time integration close to the unstable equilibrium point. and in the second case was to −3000T . Figure  7b is a close up to the map that shows the existence of a region containing the fixed point on the left, for which the trajectories have an asymptotic behavior to that fixed point. This region also seems to be bounded by some of the trajectories corresponding to the mesh of the fixed point on the right. We notice the existence of trajectories that escape towards the singularity, resulting in a possible blow-up behavior.
(a) Region of interest of the stroboscopic map with a typical fish-like figure. (b) Close up to the central region of the stroboscopic map. Fig. 7: Region of interest of the stroboscopic map and close up associated to the case V AC = 19.9 V. In 7a the desired experimental region is the green stable lobe around the stable periodic solution. A separatrix curve can also be observed separating the stable and unstable regimes. The unstable equilibrium point has two unstable and two stable directions.
A typical fish-like figure is formed around the stable-unstable equilibrium couple. As expected the green "fish" shrinks at the fold bifurcation and disappears. Note that from the applications point of view the region of interest is precisely the stable basin of attraction and the unstable initial conditions have to be avoided. Figure 8 shows the region of interest of the stroboscopic map when V AC = 4.5 V by using the same colour-maps as in the former case, a mesh grid of 10200 nodes with a perturbation parameter of 1.95 × 10 −2 for the fixed point on the left, and a mesh grid of 7675 nodes with a perturbation parameter of 3.46 × 10 −3 for the fixed point on the right. The conditions of Theorems 1 and 2 are satisfied, thus as the previous section shows, one of the positive Tperiodic solutions is asymptotically locally stable and the other is unstable. Thus, we obtain the existence of a region of initial conditions for which trajectories have an asymptotic behavior to the fixed point on the left, moreover, this region is completely bounded by some of the trajectories with initial conditions nearby the fixed point of the right. Note that for this case the basin of attraction has increased significantly. This fact could be anticipated from the numerical continuation diagram 5 by looking at the distance between the two equilibria.
In some sense the continuation and the stroboscopic diagrams provide complementary information about the dynamical behavior of the solutions.

Basin of Attraction Comparison
In this section, we compare the basins of attraction (see [22]), of the two MEMS models, paying special attention to the robustness properties.
In order to tackle the comparison between the two models we need to ensure equivalent operating conditions. Therefore we will use in both models the same gap, area of the plates, damping coefficient, and the geometry of a parallel plates device. Additionally, we will take into account for the Nathanson's model a linear restoration force as follows.
where A c , L and x are defined as in section 1, and E denotes the Young's modulus of some material, for example, the silicon. Hence we consider E = 100 GPa. On the other hand, to achieve the comparison we ensure the existence of two periodic solutions for the linear response model through the application of lower and upper solutions theorems, moreover, one of them is asymptotically locally stable. Before we present the results of this section we notice that the stroboscopic map associated to the linear response model has been shifted so that fixed points corresponding to the asymptotically locally periodic solutions for both models match.
We notice that for a periodic input voltage with V DC = 3 V and V AC = 1.  The size of the basin of attraction for the graphenebased MEMS is considerably larger than for the classical silicon based MEMS (please note the different scales). Figure 10 gives an insight into the comparison between the two models considering dimensional variables. We can observe that the set of initial conditions that leads to a good operation of the device, i.e., the safe operation region, of the model with nonlinear response is greater than the safe operation region of the model with linear response.

Concluding remarks
We provide sufficient conditions that give an insight into the existence and the stability properties of constant sign periodic oscillations for the graphene-based MEMS when a non constant periodic voltage is supplied. Specifically, under some conditions we obtain the existence of exactly three constant sign periodic oscillations, one of them is negative and the others are positive in sign. Furthermore, the positive one in the middle is asymptotically locally stable and the others are unstable. These results could be an approach to a design principle for stabilizing the microelectronic device without an external controller, since a proper adjustment of the input voltage leads to a required operation whenever the damping coefficient is under a certain quantity.
We notice that numerical continuation results provide a region of values for which the amplitude of the alternating forcing can range, ensuring that we can still obtain a stable operating regime of the device. Moreover, these results give numerical evidence of a fold bifurcation of positive periodic solutions when a periodic input voltage is considered. On the other hand, numerical simulations show one of the advantages granted by the graphene when it is considered as a material for MEMS. Specifically, from a lumped-mass model approach, we do observe that under proper conditions on the control parameters and ensuring same operating conditions, the safe operation region of the graphene-based MEMS is greater than the safe operation region of a canonical MEMS for a parallel plates capacitor.
We point out that our results comprehend constant sign periodic solutions and thus we do not know if there exist variable sign periodic solutions that correspond to oscillations where the movable plate passes through its relaxed position, i.e., when distance between both fixed and movable plate is the gap d. Therefore, a future work is to provide a global multiplicity result that reveals if there exist more periodic solutions besides those of constant sign studied here by following the ideas in [7]. Then we expect to provide an analytical proof for the existence of a fold bifurcation of periodic solutions for the graphene-based MEMS model when a parameter, e.g., the gap d, is changed.
Application to the dimensional graphenebased MEMS model: Now we discuss our results for the graphene-based electrostatic parallel plates micro actuator which has associated the following dimensional equation where V (t) is a T -periodic input voltage such that V (t) and V min := min and the parameters E, D, ǫ, A, A c , and the variable x are defined as in section 1.
Thus the hypothesis 1.a) is equivalent to the following restriction The hypothesis 1.b) is equivalent to the following restriction over the period T where denotes the natural frequency of the system, as the nondimensionalization process reveals. Then, the input voltage frequency denoted by ω should satisfy Finally, hypothesis 1.c) implies a restriction over the damping coefficient, such that To sum up, we obtain under conditions (13), (14) and (15) that high enough AC voltage frequencies lead to stabilization of the microelectromechanical system with a periodic oscillation of the movable plate. Indeed there will be at least three lateral T -periodic oscillations, and at least two of them will be positive. Imposing the following additional condition we obtain the existence of exactly two positive oscillations which will be equivalent to the restriction Finally, we notice that the condition (16) implies also that the positive oscillation nearest to the fixed plate won't be observable since it will be unstable. Moreover, we know the existence of a negative and unstable oscillation, nevertheless, we do not know if it is "always" physically possible since in this case the material seems to have other regime of elastic response. Theory Now we introduce some classical results about Lower and Upper solutions Method. Consider the second order differential equation of the form where f :D → IR and f ∈ C 1 (D) withD = It is well known fact that the problem of finding periodic solutions of (17) is equivalent to solve the periodic boundary problem x + f (t, x,ẋ) = 0, A particular case of Theorem 3.2 in chapter 5 of [4], about lower and upper solutions in reversed order, is easily obtained for constant lower and upper solutions and f regular enough.
Theorem 4 Let L and U be constant lower and upper solutions of the periodic problem (18) such that U < L, definē and assume that f :Ē → IR is continuous with continuous partials derivatives onĒ. Suppose that there exists M ∈ 0, ( π T ) 2 such that ∂ x f (t, x, v) ≤ M for all (t, x, v) ∈Ē, that there exists N > 0 so that and let ℓ ∈ M, ( π T ) 2 such that N ≤ G(ℓ), with G the auxiliary function defined in (4). Then there exists at least one solution x(t) of the periodic problem (18) such that for all t ∈ [0, T ] U ≤ x(t) ≤ L.
Next, we introduce an important result due to R. Ortega and N. Dancer [3], that relates the Lower and Upper Solutions Method and the instability.

Proposition 1 ([3]
, Proposition 3.1 and its remark) Assume that (18) has strict lower and upper solutions, L and U , respectively, satisfying L(t) < U (t) for all t ∈ IR. In addition, assume that f satisfies the Nagumo condition (for a definition see [4], chapter 1, section 4) and that the number of T -periodic solutions satisfying L(t) < x(t) < U (t) for all t ∈ IR, is finite. Then at least one of them is unstable.
A necessary Lemma in order to get multiplicity results is given in this section. (17) and assume that f is a function inD = IR×]x 1 , x 2 [× IR, with second partial derivatives continuous inD. If additionally f depends linearly on v such that f (t, x, v) = bv + g(t, x) where ∂ x g(t, x) < π T 2 + b 2 4 and ∂ 2 x g(t, x) < 0, on D = IR × ]x 1 , x 2 [, then 1. T -periodic solutions of (17) are ordered for all t ∈ IR. 2. Equation (17) has at most two T -periodic solutions in ]x 1 , x 2 [.

Appendix B: Some auxiliary results
Next we present some auxiliary results about linear second order differential equations with periodic coefficients.

If Hill's equation
Lemma 3 Consider (19) and assume that a(t) < 0 for all t ∈ IR. Then (19) can not admit nontrivial T -periodic solutions.