Noise-induced chaos in the electrostatically actuated MEMS resonators

Physics Letters A 375 (2011) 2903–2910

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Physics Letters A www.elsevier.com/locate/pla

Noise-induced chaos in the electrostatically actuated MEMS resonators Wen-Ming Zhang a,b,∗ , Osamu Tabata b , Toshiyuki Tsuchiya b , Guang Meng a a b

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Department of Microengineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan

a r t i c l e

i n f o

Article history: Received 10 March 2011 Received in revised form 18 May 2011 Accepted 10 June 2011 Available online 23 June 2011 Communicated by A.R. Bishop Keywords: MEMS Resonator Electrostatic actuation Bounded noise Chaos

a b s t r a c t In this Letter, nonlinear dynamic and chaotic behaviors of electrostatically actuated MEMS resonators subjected to random disturbance are investigated analytically and numerically. A reduced-order model, which includes nonlinear geometric and electrostatic effects as well as random disturbance, for the resonator is developed. The necessary conditions for the rising of chaos in the stochastic system are obtained using random Melnikov approach. The results indicate that very rich random quasi-periodic and chaotic motions occur in the resonator system. The threshold of bounded noise amplitude for the onset of chaos in the resonator system increases as the noise intensity increases. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Micro-electromechanical resonators are one of the most commonly used components for various communication and signal processing applications [1]. Electrostatically actuated resonators have the advantages of simple structures that allow easy batch fabrications and they form a major component in many MEMS devices [2,3]. However, the resonators have to driven close to or even into nonlinear regimes to store enough energy at micro scale [1,4]. When a resonator is used as a frequency source in MEMS devices, noise plays numerous effects, such as limiting dynamic range and selectivity, causing loss of lock, limiting acquisition and reacquisition capability in phase-locked-loop systems, on their performance [5]. Various nonlinearities will occur and dominate in the resonator dynamic behavior due to the specific resonator design and the operating conditions. In order to predict the influence of resonator nonlinearities on the performance of MEMS devices, the nonlinear dynamics of the resonators has to be investigated. Carr et al. [6] and Zalalutdinov et al. [7] studied the parametric amplification of the motion of resonators through electrostatic and optical actuation. However, the reported methods should be used to discuss the instability and control strategies. Abdel-Rahman and Nayfeh [8] applied the

*

Corresponding author at: State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. Tel./fax: +86 21 3420 6813 818. E-mail address: [email protected] (W.-M. Zhang). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.020

super-harmonic and combination parametric resonances as suitable excitation methods to minimize electrical “feedthrough”. The study on the array of parametrically excited strings were carried out both experimentally [9] and theoretically [10]. Hu et al. [11] discussed the resonances of electrostatically actuated microcantilevers, while Baskaran and Turner [12] demonstrated the coupled modes parametric resonance. Fu et al. [13] analyzed the nonlinear dynamic stability for an electrically actuated viscoelastic microbeam. Krylov et al. [14] investigated the possibility of parametric stabilization of electrostatically actuated microstructures under the effects of AC component and dc component voltages. Younis and Nayfeh [15] and Abdel-Rahman and Nayfeh [8] studied the dynamic responses of an electrostatically resonator to a primaryresonance excitation [15], a superharmonic-resonance excitation of order two [8], and a subharmonic-resonance excitation of order one-half [8]. It can be concluded that these models gave accurate results for small AC amplitudes and hence small motions. Kacem et al. [16] studied the nonlinear dynamics of nanomechanical beam resonators to improve the performance of MEMS-based sensors. Alsaleem et al. [17] investigated the nonlinear phenomena, including primary resonance, superharmonic and subharmonic resonances, in electrostatically actuated resonators both experimentally and theoretically. Zhang and Meng [18] analyzed the nonlinear dynamics of the electrostatically actuated resonant MEMS sensors under parametric excitation. Mestrom et al. [1] modeled the dynamics of a MEMS resonator numerically and experimentally considering the effect of thermal noise. Haghighi and Markazi [19] predicted the chaos in MEMS resonators and presented a robust adaptive fuzzy control method to control the chaotic motion. The

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chaotic motion was suppressed and came into a periodic motion with a robust fuzzy sliding mode controller when the MEMS resonators had system uncertainties [20]. Rhoads et al. [21] reviewed nonlinear dynamics and its applications in micro- and nanoresonators, and focused on the simple lumped-mass models for individual resonators throughout the review paper. However, when the resonator operates, small fluctuations in the gap (Brownian motion, thermal-mechanical noise, random vibrations and oil-canning of the package) will cause frequency fluctuations [1,5]. The dynamic response of the electrostatically actuated MEMS resonator subjected to random disturbance due to fluctuations and other uncertainties has not been paid more attention in literature. In this Letter, based on the random Melnikov approach, which is widely used by most researchers [22–26], the nonlinear dynamic behavior of a clamped–clamped microbeam loaded by a symmetric combined voltage, which is composed of a direct current bias voltage and an alternating current (AC) voltage, is investigated analytically and numerically. Our aim is to predict the analytical shape of the most generic dynamics equation (Duffing-like equation) for an ideally elastic nonlinear micromechanical resonator that has been reduced to a single-degree-of-freedom. This equation could contain a number of nonlinear coefficient, such as nonlinear cubic stiffness, dc bias voltage and AC voltage, and random intensity, each of which having its well definition. This Letter is organized as follows. Section 2 presents the mathematical model of the clamped–clamped resonator. Section 3 describes the brief introduction of the bounded noise. The threshold of bounded noise amplitude for the onset of chaos in the system is determined by using the random Melnikov method in Section 4. Numerical results and discussion of the nonlinear dynamics and chaos under random disturbance are provided in Section 5, and Section 6 concludes the Letter.

Fig. 1. A schematic diagram of the electrostatically actuated MEMS resonator.

Given the state y = x˙ , the non-dimensional equation of motion for the dynamic system can be written as

⎧ x˙ = y   ⎪ ⎪ ⎪ 1 1 ⎨ 3 y˙ = −α x − β x − η y + γ − (1 − x)2 (1 + x)2 ⎪ ⎪ A ⎪ ⎩ + sin ω1 τ + F (τ ) (1 − x)2

(4)

3. Bounded noise Bounded noise is a harmonic function with constant amplitude and random frequency and phases. Stratonovich [24] firstly presented the bounded noise process and defined it as a slowly varying random process. The random process f (t ) is governed by the following form [22–24,26]

2. Mathematic model

f (t ) = σ cos(Ω2 t + Φ),

For the electrostatically actuated clamped–clamped beam resonator [1], as shown in Fig. 1, is subject to a random disturbance, and using the dynamic model for the MEMS resonator derived by Mestrom et al. [1] and Haghighi and Markazi [19], the nondimensional governing equation of motion can be given by

where σ and Ω2 are the amplitude and frequency of the random excitation, respectively, and Ω2 , σ , and δ are the positive constants, B (t ) is a standard Wiener process, Γ is a random variable uniformly distribution in [0, 2π ). f (t ) is a non-Gaussian distributed stationary stochastic process in wide sense with zero mean and its covariance function and spectral density are

m z¨ + c z˙ + k1 z + k3 z3 = F elec ( z, t ) + f (t )

(1)

where z is the vertical displacement of the microbeam, m, c , k1 and k3 are effective lumped mass, damping coefficient, linear mechanical stiffness and cubic nonlinear stiffness of the system, respectively. f (t ) is a random process to be described in Section 3. The electrostatic force F elec ( z, t ) can be expressed as [1]

F elec ( z, t ) =

1

C0

( V dc + V AC sin Ω1 t )2 −

2 (d − z)2

1

C0

2 (d + z)2

2 V dc (2)

where C 0 is the capacitance of the parallel-plate actuator over the gap when z = 0, d is the initial gap width, V dc is the dc bias voltage, and V AC and Ω1 are the amplitude and frequency of the applied alternating current voltage, respectively. Introducing the non-dimensional variables in the following forms:

τ = ω0t , α=

k1 mω02

A = 2γ where

ω1 = β=

,

V AC V dc

,

Ω1

ω0

x=

,

k3 d2 mω02

F=

,

z d

,

γ=

η=

c mω0

2 C 0 V dc

σ2 2



exp −

and

S f (ω) =

(σ δ)2 2π



δ 2 |τ |

(5)



cos(Ω2 τ )

2

1 4(ω − Ω2 )2 + δ 4

+

(6)

1 4(ω + Ω2 )2 + δ 4

 (7)

The variance of the bounded noise is

C f (0) =

σ2 2

(8)

It indicates that the bounded noise has finite power. The shape of spectral density depends on Ω2 and δ , while the bandwidth of the bounded noise depends mainly on δ . It is a narrow-band process when δ is small and approaches to white noise when δ → ∞. It can be found that the sample functions of the bounded noise are continuous and bounded which are required to obtain the Melnikov function [25]. 4. Random Melnikov analysis

2mω02 d3

f 2 0

C f (τ ) =

Φ = δ B (t ) + Γ

(3)

mω



ω0 is the purely elastic natural frequency and ω0 = k1 /m.

On the assumption that a MEMS resonator has high quality factor and AC voltage is much smaller than the dc bias voltage, which is verified to be valid [19], the governing equation of the dynamic system (4) can be rewritten as

W.-M. Zhang et al. / Physics Letters A 375 (2011) 2903–2910

⎧ x˙ = y ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ y˙ = −α x − β x3 + γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 + ε −η˜ y +

1

(1 − x)2

˜ A

(1 − x)2

−

∞



1

−η˜ y 20 (τ ) dτ

M (τ0 ) =

(1 + x)2

 sin ω1 τ + ϑ˜ cos(ω2 τ + Φ)

−∞

(9)

∞



3 ⎩ y˙ = −α x − β x + γ

1

(1 − x)2

−

(10)

(1 + x)2

−∞ ∞

1 2

 y 2 + α x2 +

1 2

 β x4 − 2γ −2 +

1 1−x

+

1



1+x (11)

 1 1 4 2 V (x) = α x + β x − 2γ −2 + 2

2

where I = − 23 η˜

1 1−x

+

√ x0 (τ ) = ±Λ κ sech( κτ ) √ √ √ y 0 (τ ) = ∓Λ κ sech( κτ ) · tanh( κτ )

√

1 1+x

(14)

κ Λ2 + A˜ Λ √2π λ

1−Λ2



cos(λτ0 ), in

2 √ 3

η˜ κ Λ2

+ A˜ Λ √

(12)

2π λ

sinh[−λ(arccos Λ)]

1 − Λ2

sinh(λπ )

cos(λτ0 )

(15)

where E [•] is an expectation operator. Eq. (15) is a constant and it means that in mean sense chaos never occurs in dynamic system (9). When the random Melnikov process (14) has simple zeros in mean-square sense, the impulse response function can be given by

√

h(τ ) = x0 (τ ) · y 0 (τ ) = −Λ2 κ sech2 (

√

sinh[−λ(arccos Λ)] sinh(λπ )

∞

which λ = ω1 / κ , and Z (τ0 ) = −∞ ϑ˜ y 0 (τ ) cos[ω2 (τ + τ0 ) + Φ] dτ . The first two integrals in Eq. (14) represent the mean of the Melnikov process due to damping force and periodic parametric excitation, and the last integral denotes the random part of the Melnikov process with the bounded noise F (t ). Consider the mean of random process, on can obtain

E [ M (t 0 ) = −



It can be found that the homoclinic trajectory of (9) has no approximated formulation. When the unperturbed term (1/(1 − x)2 − 1/(1 + x)2 ) in the right side of Eq. (9) is expanded to (4x + 8x3 + O(x5 )) in Taylor series up to the forth power of x [19], one can obtain that there are three equilibrium points. P 0 (0, 0) is the hyperbolic saddle, which be connected by two homoclinic orbits



−∞



and the potential function is



y 0 (τ ) sin ω1 (τ + τ0 ) dτ

 ϑ˜ y 0 (τ ) cos ω2 (τ + τ0 ) + Φ dτ

+

√

The system (10) is a Hamiltonian system with the Hamiltonian function

H (x, y ) =

(1 − x0 (τ ))2

= I + Z (τ0 )



1

˜ A

+

˜ = A /ε and ϑ˜ = ϑ/ε , and ε is a small parameter, where η˜ = η/ε , A ω2 = Ω2 /ω0 and ϑ = σ /(mω02 ). For ε = 0, Eq. (9) can be regarded as an unperturbed system and written as

⎧ ⎨ x˙ = y

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√

κτ ) · tanh( κτ )

(16)

and the associated frequency response function is

(13)

√ 2/(β − 8γ ) and κ = 4γ − α . P 1 (Λ κ , 0) and

where √ Λ = P 2 (−Λ κ , 0) are the two centers. Fig. 2 shows the potential function of the system (10) for different γ (a) and different β (b) with α = 1.0 in the region |x|  1. The region ∞ > |x| > 1 is impossible for the system due to the largest region is |x| = 1, which means that the upper and bottom plates collide to each other. It can also found from Fig. 2 that the number of equilibrium points changes when applied voltages and nonlinear cubic stiffness vary. When the dc bias voltage is not applied, i.e. γ = 0, as shown in Fig. 2(a), there is only one stable state and the equilibrium point is a stable center one at P 0 (0, 0). When the dc bias voltage is applied up to γ = 0.2, three equilibrium points display, one is the stable center point√at P 0 (0, 0) and the √ other two unstable saddle points at P 1 (Λ κ , 0) and P 2 (−Λ κ , 0). It indicates that the resonator becomes unstable and deflects against one of the stationary electrodes. When the dc bias voltage increases to be large enough, such as γ = 0.8, the equilibrium point at x = 0 becomes unstable. In this case, the resonator enters into complete unstable state. When the dc bias voltage and nonlinear cubic stiffness are not considered, the resonator stays at the stable state, as illustrated in Fig. 2(b). As the cubic stiffness β increases from β = 0 to β = 6, the equilibrium point changes from an unstable center point to one unstable center point together with two unstable saddle points. The larger the cubic stiffness is, the more obvious this phenomenon displays. Therefore, the dc bias voltage and nonlinear cubic stiffness play important roles in the dynamic characteristics of MEMS resonators. The random Melnikov process for the dynamic system (9) can be given by using Wiggins’ method [27] as follows:

∞ H (ω) =

h(τ )e − j ωτ dτ

−∞

√ √ = j πω2 Λ2 csch(πω/4 κ ) sech(πω/4 κ )/4

(17)

The variance of Z (τ0 ) as the output of the system can be obtained in frequency domain as

∞ 2 Z

σ =



H (ω) 2 S f (ω) dω

(18)

−∞

It is noted that f (t ) has zero mean and I 2 = σ Z2 , one can obtain

 2 2 √ 2π λ sinh[−λ(arccos Λ)] − η˜ κ Λ2 + A˜ Λ √ cos(λτ0 ) 3 sinh(λπ ) 1 − Λ2 ∞

√ √

j πω2 Λ2 csch(πω/4 κ ) sech(πω/4 κ )/4 2 = −∞

×

(σ δ)2 2π



1 4(ω − Ω2 )2 + δ 4

+

1 4(ω + Ω2 )2 + δ 4

 dω

(19)

The integral in Eq. (19) can be completed numerically, and the threshold of the bounded noise amplitude σ for the onset of chaos in system (10) is illustrated in Fig. 3. It can be seen that the threshold of the noise amplitude σ for onset of chaos by numerical simulation decreases when the noise intensity δ increases in the range of its smaller value. However, in contrast to that the threshold of the noise amplitude σ obtained by the random Melnikov method

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Fig. 2. Potentials of the system (10) for different

Fig. 3. The threshold

σ as a function of the noise intensity δ .

increases. On the other hand, for larger noise intensity, both the analytic and numerical results show the same variation trend. The threshold increases with the increase of the noise intensity, which represents that the diffusion of the frequency reduces the effect of bounded noise on triggering chaos in the dynamic system. 5. Numerical results and discussion Eq. (9) is a nonlinear one with time varying coefficient, disturbed with a random noise process. Monte Carlo method can be used to numerically analyze the dynamic response of MEMS resonator system subject to random excitation. Eq. (5) can be rewritten as



F(τ ) = ϑ cos



ϕ (τ ) , ϕ˙ (τ ) = ω2 + δζ (τ ), ζ (τ ) = B˙ (τ )

The formal derivative ζ (τ ) of the unite Wiener process B (τ ) is a Gaussian white noise, which has the power spectrum of a constant and is unrealized in the physical view. The pseudorandom signal can be used for the numerical simulations [28] and is given by

 ζ (τ ) =

N 4ω2 

N

k =1

 cos

ω2 N

 (2k − 1)τ + φk

(20)

γ (a) and different β (b) with α = 1.0.

where φk ’s are independent and uniformly distributed in (0, 2π ], and N is a larger integer number. The largest Lyapunov exponent is quantitatively characterized the average rate of separation of infinitesimally close trajectories in the phase space for a dynamic system. It can be used to determine how sensitive a dynamic system is to the initial conditions. If the largest Lyapunov exponent is less than or equal to zero, the dynamic system can be regarded as quasi-periodic or periodic. If the largest Lyapunov exponent is positive, the system is chaotic and will have a strange attractor. In the following analysis, the largest Lyapunov exponents are calculated to identify the noise-induced chaos according to the Melnikov condition presented in the above section. Fig. 4 shows the largest Lyapunov exponent as a function of γ with three noise intensities for the dynamic system (9). It can be seen from Fig. 4(a) that the largest Lyapunov exponent is mainly negative when γ has a smaller value. When γ increases, as shown in Fig. 4(b) and (c), the largest Lyapunov exponent changes from negative value to positive one. It implies that the dynamic system comes into chaotic motion with larger noise intensity. Fig. 5 shows the bifurcation diagram of the AC amplitude V AC on the response of MEMS resonator system with α = 1, β = 12, η = 0.01, γ = 0.26, V dc = 3.8 V, 2ω1 = ω2 = 1. Two cases of the noise-free system at ϑ = 0 and δ = 0 and noisy system at ϑ = 0.001 and δ = 0.05 are compared. Fig. 5(a) shows the bifurcation diagram of the AC amplitude V AC at the interval of 0  V AC  0.47 V for noise-free system with ϑ = 0 and δ = 0. The response of the coupled nonlinear system undergoes a complete process from chaotic motion through period-4 and period-2 motions to steady-state motion with period-1 by the form period-doubling bifurcations. When the intensity of the random disturbance changes from δ = 0 to δ = 0.05, the response of the coupled dynamic system undergoes the process of chaotic and quasi-periodic motions alternatively. It can be seen from Fig. 5(b) that chaotic motion components increase gradually with the increase of the density of the bounded noise random disturbance. The bifurcation curve becomes thicker than that shown in Fig. 5(a). It is indicated that the noise intensity can result in more and more chaotic components in the system response. The Poincaré maps, phase portraits, and response time histories of dynamic system (9) are shown in Fig. 6 in the cases of noisefree system (ϑ = 0, δ = 0) and noisy system (ϑ = 0.001, δ = 0.05). It can be seen from Fig. 6(upper) that the response is a period-4 motion while the phase trajectory is a limit cycle with four pe-

W.-M. Zhang et al. / Physics Letters A 375 (2011) 2903–2910

Fig. 4. The largest Lyapunov exponent maps of

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γ for the noisy system with ϑ = 0.01 at different noise intensity values: (a) δ = 0, (b) δ = 0.001, and (c) δ = 0.01.

Fig. 5. Bifurcation diagram of the AC amplitude V AC at the interval of 0  V AC  0.47 V with

α = 1, β = 12, η = 0.01, γ = 0.26, V dc = 3.8 V, 2ω1 = ω2 = 1.

Fig. 6. Poincaré maps, phase portraits, and time histories with V AC = 0.16 V for noise-free system (upper, ϑ = 0, δ = 0) and noisy system (lower, ϑ = 0.001, δ = 0.05).

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Fig. 7. Bifurcation diagram of the cubic nonlinear stiffness β at the interval of 10  β  25 with

riods. In comparison with Fig. 6(upper), it can be observed from Fig. 6(lower) that random noise f (t ) will change the steady-state response of system (9) from a periodic solution to a chaotic one such that change the phase portrait from a limit cycle to a series of diffused limit cycle, and change the Poincaré map from four points to many diffused ones. Further numerical simulation illustrates that the width of the diffused limit cycle will be large when the intensity of the random noise increases. Meanwhile, the corresponding time history also changes from regular type to disarray. The larger the intensity of random disturbance is, the greater the various phenomena are occurring. It is indicated that the nonlinear dynamic responses of the MEMS resonator system can be controlled to reduce chaos by decreasing the intensity of the random disturbance. Fig. 7 illustrates the bifurcation diagram of the cubic nonlinear stiffness β on the response of MEMS resonator system at the interval of 10  β  25 with α = 1, η = 0.01, γ = 0.3, V dc = 3.8 V, V AC = 0.2 V, 2ω1 = ω2 = 1. Comparisons between the noise-free system at ϑ = 0 and δ = 0 and noisy system at ϑ = 0.02 and δ = 0.05 are simulated. The response of the system for noisefree system with ϑ = 0 and δ = 0 undergoes a complete process from period-2 and period-4 motions through period-6 and chaotic motions to steady-state motion with period-2 by the form period-doubling and anti-period-doubling bifurcations, as shown in Fig. 7(a). When the intensity of the random disturbance changes from δ = 0 to δ = 0.05, the response of the system undergoes the process of chaotic and random quasi-periodic motions alternatively. It can be observed from Fig. 7(b) that chaotic motion components increase gradually with the increase of the density of the bounded noise random disturbance. The bifurcation curve becomes thicker than that shown in Fig. 7(a). It also implies that the noise intensity can lead to random chaos in the system response. Figs. 8 and 9 show the influence of ϑ on the response of the MEMS resonator system subject to a white noise process, when α = 1, η = 0.01, γ = 0.26, V dc = 3.8 V, V AC = 0.2 V, 2ω1 = ω2 = 1. From Fig. 8(b)–(d), one knows that the noise excitation plays a dispersive role to the system responses. Due to the randomness of the excitation, all the system responses vary randomly too. Bifurcation diagrams corresponding to a gradual increase of the threshold of the bounded noise amplitude ϑ of the system response are illustrated in Fig. 8(a) (for the intensity of the random disturbance δ = 0), Fig. 8(b) (for δ = 0.001), Fig. 8(c) (for δ = 0.01), and Fig. 8(d) (for δ = 0.1), respectively. Fig. 9 shows Poincaré maps and phase portraits of the system response corresponding to ϑ = 0.07.

α = 1, η = 0.01, γ = 0.3, V dc = 3.8 V, V AC = 0.2 V, 2ω1 = ω2 = 1.

It can be seen from the bifurcation diagram, Fig. 8(a), that the bifurcation curves still contain periodic, quasi-periodic and chaotic components when the intensity of the random disturbance δ = 0. The reason is that the random parametric excitation on the dynamic system can be regarded as harmonic excitation at δ = 0. When the noise intensity δ increases and the bifurcation value of parameter ϑ is almost invariant, the bifurcation curves are thicker than those in Fig. 8(a). The bifurcation process cannot be distinguished clearly when 0 < ϑ < 0.002. It implies that the influence of ϑ on the response of the MEMS resonator system is very weak when the noise intensity δ has a small value. And such influence grows intensively as δ increases, as shown in Fig. 8(c) and (d), the bifurcation curves become thicker and the chaotic components occur frequently. From Fig. 9(a), it can be found that the phase portrait corresponding to ϑ = 0.07 turns to a family of closed curve and the Poincaré map comes to be four points. The profile of the closed curve is similar to the corresponding periodical solution curves of undisturbance system. Such solution can be defined as the random disturbed periodical solution. As the noise intensity δ increases, as shown in Fig. 9(b), the phase portrait becomes a family of unclosed curve. The Poincaré map comes to be a series of discrete points distributing round a closed curve due to the effect of random disturbance, which represents quasi-periodic solutions of the undisturbed system. The phase portrait represents the response is similar to period-4 solution and has some pervasion, one can call it quasi-periodic-4 solution. Along with the further increase of δ , the distribution of those discrete points change to be more dispersed, as shown in Fig. 9(c) and (d), the pervasion of the phase portraits will strength and even destroy the topological property of the phase trajectory. One can regard such responses as stochastic chaos. It indicates that the random disturbance could make the MEMS resonator system chaotic and uncertain. 6. Conclusions In this Letter, noise-induced chaos in the electrostatically actuated MEMS resonators investigated analytically and numerically. The random disturbance is described by a random bounded noise process. The random Melnikov process is derived and used to establish the threshold of bounded noise amplitude for the occurrence of chaos. Effects of the random disturbance, dc bias voltage, AC voltage and cubic nonlinear stiffness on the dynamic responses of the resonator system are investigated. The results are verified by the numerical simulations in terms of the largest Lyapunov

W.-M. Zhang et al. / Physics Letters A 375 (2011) 2903–2910

Fig. 8. Bifurcation diagrams of the noise amplitude ϑ at the interval of 0  ϑ  0.1 with noise intensities: (a) δ = 0, (b) δ = 0.001, (c) δ = 0.01, and (d) δ = 0.1.

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α = 1, η = 0.01, γ = 0.26, V dc = 3.8 V, V AC = 0.2 V, 2ω1 = ω2 = 1 for different

Fig. 9. Poincaré maps (upper) and phase portraits (lower) of the dynamic noisy system at ϑ = 0.07 for different noise intensities: (a) δ = 0, (b) δ = 0.001, (c) δ = 0.01, and (d) δ = 0.1.

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exponent, Poincaré map, phase portrait, and time history of the dynamic system. It shows that the bounded noise can enhance the largest Lyapunov exponent and diffuse the chaotic attractor. It can also be found that for larger noise intensity, noise perturbation increases the threshold. It is demonstrated that when the intensity of bounded noise increases, the random quasi-periodic and chaotic motions are enriched gradually, which lead the resonator system to the chaotic regions. Acknowledgements This work was supported by Shanghai Rising-Star Program under Grant No. 11QA1403400, and the National Science Foundation of China under Grant No. 11072147, and was grateful for the support by Japan Society for the Promotion of Science. References [1] R.M.C. Mestrom, R.H.B. Fey, J.T.M. van Beek, K.L. Phan, H. Nijmeijer, Sensor. Actuat. A: Phys. 142 (2008) 306. [2] A.R. Bahrampour, M. Vahedi, M. Abdi, R. Ghobadi, Phys. Lett. A 372 (2008) 6298. [3] W. Zhang, R. Baskaran, K.L. Turner, Sensor. Actuat. A: Phys. 102 (2002) 139. [4] R.M.C. Mestrom, R.H.B. Fey, K.L. Phan, H. Nijmeijer, Sensor. Actuat. A: Phys. 162 (2010) 225. [5] J.R. Vig, Y. Kim, IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control 46 (1999) 1558.

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