Frequency unlocking-based MEMS bifurcation sensors

MEMS resonators exhibit rich dynamic behaviors under the internal resonance regime. In this work, we present a novel MEMS bifurcation sensor that exploits frequency unlocking due to a 1:3 internal resonance between two electrostatically coupled micro-resonators. The proposed detection mechanism allows the sensor to operate in binary (digital) and analog modes, depending on whether the sensor merely detects a significant jump event in the peak frequency upon unlocking or measures the shift in the peak frequency after unlocking and uses it in conjunction with a calibration curve to estimate the corresponding change in stimulus. We validate the success of this sensor paradigm by experimentally demonstrating charge detection. High charge resolutions are achieved in binary mode, up to 0.137 fC, and in analog mode, up to 0.01 fC. The proposed binary sensor enables extraordinarily high detection resolutions due to the excellent frequency stability under internal resonance and the high signal-to-noise ratio of the shift in peak frequency. Our findings offer new opportunities for high-performance ultrasensitive sensors.


Introduction
MEMS-resonant sensors have demonstrated their tremendous potential in both fundamental science and industrial applications. They can be used in weak force, mass, and field sensing, due to significant advantages including ultrasmall mass and high resonant frequencies, that enables ultralow detection limits and superior sensitivities 1,2 . The conventional sensing approach is based on quantitative variations in the linear response that relates a measurable shift in the resonant frequency of the sensors to the bonding of a target stimulus 3 . However, some limits, including the limited signal-to-noise ratio (SNR), the frequency uncertainty, or the effective nonlinearity, inevitably degrade the detection resolution and sensitivity of these sensors that operate in linear regimes [4][5][6] .
Resonant sensors operating in the nonlinear regime have attracted increasing attention due to their larger output signals increasing the SNR, as well as intriguing nonlinear dynamic features, such as multivalued responses 7 , bifurcations [8][9][10] , and various types of nonlinear resonances [11][12][13] . In this regime, various detection principles have been proposed to enhance the performance of resonant sensors, such as tracking the shift in the nonlinear resonant frequency 14 or exploring the bifurcation phenomena [15][16][17] . In particular, bifurcation-based sensors have attracted significant interest in recent years. The underlying mechanism of bifurcation sensors is that the target stimulus causes the sensor to cross a bifurcation point, creating a sudden (discontinuous) jump in the response amplitude 16 or a qualitative change in the phase angle 10 . This paradigm allows the sensor to operate in either binary (detector) or analog modes, depending on the bifurcation detection approaches. The analog mode can be achieved by sweeping the excitation frequency toward the bifurcation to trigger a qualitative change event and then measuring the quantitative change in the bifurcation frequency 18 . On the other hand, a binary sensor operates at a constant excitation frequency just below the bifurcation point while monitoring the qualitative change event of the sensor amplitude as a detector of a single event that stimulus change exceeds a threshold 16 . Unfortunately, because the resonant frequency has a strong dependence on the oscillation amplitude (amplitude-frequency (a-f) effect) in the nonlinear regime 19 , this unavoidably results in the transformation of amplitude fluctuations into frequency fluctuations. Therefore, it will increase the frequency noise of the resonator, and as a consequence, noise-activated stochastic switching between qualitatively different responses (the sensor detection metric) is triggered 20,21 , which represents a false-positive 22 .
Another remarkable nonlinear phenomenon is internal resonance (IR), which arises due to strong modal interactions and intermodal energy transfer [23][24][25] . It occurs among different vibration modes of a specially designed structure 26 or between coupled multi-resonator structures (a system) 27 when the following conditions are satisfied: (i) the natural frequencies of distinct vibrational modes in a single resonator or across coupled resonators are commensurate or nearly commensurate with each other and (ii) a proper type of nonlinearity in accordance with the frequency ratio exists in the resonators. A strong intermodal coupling in IR results in an effective and fast energy transfer between the two modes. As a result, the undriven mode drains mechanical energy from the externally excited mode and vibrates at its own frequency, thereby causing the amplitude of the externally excited mode to drop to a lower level. IR provides an alternative strategy to steer energy in micromechanical resonators and, thus, can be used as negative feedback to stabilize frequency fluctuations 28 . Antonio et al. 26 demonstrated that the frequency of the fundamental vibrational mode can be stabilized by coupling it to a higher mode through IR, as the energy exchange between two modes causes the resonance of the higher mode to absorb the amplitude and frequency fluctuations of the first vibrational mode.
Moreover, IR can also be employed for sensor applications, such as introducing an additional sensing channel to detect two different physical quantities simultaneously 29 , sensing the angular rate 30 or the mass 31,32 , and detecting a single electron 33 . Zhang et al. 34 reported that the resonant frequency shift of the fundamental mode could be magnified integer times in the higher modes via IR, leading to an enhanced sensitivity in frequency-shiftbased sensors.
Internal resonances in electrostatically or mechanically coupled multi-resonators have unique advantages compared to those of a single resonator. It does not involve different vibrational modes, making the measurement of each element easier and the IR more controllable and flexible for practical applications. On the other hand, the use of coupled multi-resonators also has the benefit of improving common mode rejection capabilities, thereby being less susceptible to environmental noise 35 . Wang et al. 27 investigated the IR between two electrostatically coupled micro-resonators with a natural frequency ratio of 1:3 and the resulting phenomena of frequency locking and amplitude saturation, resulting in a sevenfold improvement in frequency stability. Shi et al. 36 also studied the 1:3 IR-induced amplitude and frequency locking in two mechanically coupled nonlinear resonators, with nearly an order of magnitude improvement in amplitude stability. Therefore, it is suitable as an amplitude-based sensor.
In this work, we explore the potential for the use of IR in two electrostatically coupled micro-resonators with a natural frequency ratio of nearly 1:3 to design a novel bifurcation sensor. The sensor can function as a binary detector to detect the target stimuli in excess of a given (safe) value by exploiting the transition (jump) in the nonlinear resonance frequency from locking to unlocking. It can also serve as an analog sensor to quantify the stimuli change explicitly by measuring the changes in the peak frequency after unlocking. We present experimental measurements of the IR response of the coupled resonators to interpret the frequency locking and unlocking behaviors. We derive a closed-form expression for the size of the peak frequency shift after unlocking using the multiple-scale method. Then, we experimentally demonstrate an electrometer to validate the proposed detection principle. Finally, we report the sensor's outcomes and limitations.

Device and characterization
The micro-resonators under test were fabricated through a commercial silicon-on-insulator (SOI)-MUMPs micromachining process, presented in Fig. S1. The two resonators labeled R1 and R3 were made of double-ended tuning forks (DETFs), as shown in Fig. S2, with dimensions of 550 7 25μm and 292 7 25μm, respectively, for the length, width, and height. The endings of the tuning fork are anchored in gold electrodes to connect with external electrical sources and circuits. The gold electrodes in the mid-span are utilized for electrostatic actuation and detection across a capacitive gap of 3μm. The parallel plates between the two tuning forks are utilized to produce an electrostatic coupling force by exerting different voltages on two plates separately across a capacitive gap of 3μm. A summary of the dimensions and design parameters assigned for the resonators is given in Fig. S3 and Table S1.
The experimental setup is shown in Fig. S4. The deviceunder-test (DUT) was placed inside a vacuum chamber under 10 −2 Pa of pressure at room temperature. A power supply was used to apply the desired DC voltage signal. The Zurich Instruments lock-in amplifier (LIA-HF2LI) was used to apply the desired AC voltage of the drive signal and extract the output signal in combination with an external measurement circuit, as shown in Fig. 1a.
The basic features of the uncoupled two resonators were tested first. Each resonator was electrostatically excited by applying an electric voltage of V ðtÞ ¼ V dc þ V ac sinðωtÞ to the side electrode in the mid-span, where V dc is the DC bias and V ac is the amplitude of the applied AC voltage. The motion of each resonator was detected via piezoresistive transduction. Resonators made of monocrystalline silicon intrinsically have a piezoresistive effect and can serve as piezoresistors. The schematic of the piezoresistive detection is shown in Supplementary  S4. The original current through the resonator was generated by applying a DC voltage to one of the anchor pads of the resonator. The vibration amplitudes of the resonators were measured by monitoring the dynamic current. Amplifiers were used to convert the dynamic current to voltage output signals. The parasitic feedthrough signal 37 intrinsic to all electrically interfaced MEMS resonators can lead to a distortion of the output signal. However, it can be eliminated via an equivalent capacitance generated by adding an adjustable capacitor and a differential operation in the amplifier, as shown in the green block in Fig. 1a.
The natural frequencies of resonators R1 and R3 were evaluated from their linear frequency-response curves and found to be ω 1 = 136 kHz and ω 3 = 361 kHz, respectively. However, to experimentally obtain a 1:3 IR, we need 3ω 1 <ω 3 ; thus, by driving R1 with an appropriate force, it can be made to resonate at a frequency equal to (1/3) ω 3 . Therefore, we use electrothermal actuation by passing an electrical current generated by loading the constant DC voltages ± V th ¼ ± 23:7V on the ends of R1, as shown in Fig. 1a. This heats the resonator through Joule heating and induces an axial stress inside it 38 , tuning (softening) the natural frequency of R1 to 119.78 kHz, Fig. 1d, which is slightly lower than one-third of the natural frequency of R3. The frequency-response curves of R1 (Fig. 1b) and R3 (Fig. 1c) show that R1 exhibits a strong Duffing-like nonlinearity, while R3 is essentially linear within their respective motion domains. The quality factors of R1 and R3 are measured as 36 296 and 11 541, respectively, from their linear frequency response under a relatively small-amplitude excitation, using the half-power bandwidth method, as shown in Fig. 1d, e.

Frequency locking and unlocking
Next, we investigate the phenomena of frequency locking and unlocking under a 1:3 IR between the two coupled resonators. A DC voltage of V C ¼ 8V is loaded on one of the anchor pads of resonator R3 to produce a coupling voltage V C between R1 and R3, where the coupling force is proportional to V 2 C . Resonator R1 is electrostatically excited by a biased waveform with V dc ¼ 20V and various AC voltages, while no excitation is applied to resonator R3. The motion of R1 is detected via piezoresistive transduction, while that of R3 is detected capacitively, as shown in Fig. 1a.
The frequency-response curves in the vicinity of the primary resonance of resonator R1 were obtained by performing a forward frequency sweep under varied excitation amplitudes, as shown in Fig. 2a. The curves bend to the right due to the dominance of the mechanical hardening nonlinearity. As the excitation amplitude V ac varies from 50 mV to 180 mV, the peak amplitude and frequency increase, as expected in a Duffing resonator. When V ac increases from 180 mV to 230 mV, the response falls to the lower branch at the same frequency f ir ¼ 120:335 kHz (f ir ¼ f 3 =3), indicating the activation of IR. At this point, a strong interaction between R1 and R3 is achieved and, as a result, R3 drains more energy from R1, reaching its maximum vibration amplitude, as shown in the inset of Fig. 2a, which locks both the amplitude and frequency of oscillation of R1. Fig. 2b provides a graphical interpretation of this relationship.
When V ac is further increased to 240 mV, the end of the IR condition is reached and the response of R1 unlocks and jumps back to the values expected for the Duffing resonator while R3 almost stops vibrating. Although both the peak frequency and amplitude of R1 undergo a large and discrete surge from locking to unlocking, the former is more remarkable, in which a shift of 475 Hz is observed. The dependence of the peak frequency on the excitation amplitude V ac , extracted from Fig. 2a, is shown in Fig. 2c, where a frequency locking region and a bifurcation point A, corresponding to the locked frequency and the critical (unlocking) excitation amplitude, are observed. This suggests the possibility of implementing a bifurcation sensor that exploits the marked jump in the peak frequency across the bifurcation point as a detection mechanism.
In Fig. 2d, we show the effect of the electrothermal voltage loaded on the ends of the resonator R1 on frequency locking and unlocking as V th is changed from 23:7V to 23:8V . In this case, the natural frequency of R1 is reduced, and thus, the critical excitation amplitudes V ac needed to activate and closure the internal resonance are increased. Consequently, the shift in the peak frequency from locking to unlocking is observed to be 1415 Hz at an excitation amplitude of 10 mV above the critical value. We conclude that the peak frequency shift beyond the bifurcation increases with the initial frequency offset σ between the natural frequency f 1 of R1 and the locked frequency f ir (σ ¼ f ir À f 1 ).
Next, we further examine the impact of the interaction between the initial frequency offset σ and coupling voltage V C on the peak frequency shift Δf of R1 from locking to unlocking. First, we hold the coupling voltage fixed at V C ¼8V while varying the initial frequency offset σ of R1 by increasing the electrothermal voltage V th applied to it. The procedure followed to generate Fig. 2d was adopted at different initial frequency offsets σ. The peak frequency-shift Δf at an excitation amplitude of 10 mV above the critical value was recorded and shown as blue circles in Fig. 2e. The frequency-shift value increases as the initial frequency offset increases. Alternatively, we hold the initial frequency offset constant at 500 Hz and vary the coupling voltage between two resonators to examine the peak frequency-shift, Fig. 2f  To further interpret those relationships, we derive a closed-form expression for the size of the peak frequency-shift after unlocking (see Eq. (13) in the "Materials and methods" section) from the mathematical model using the multiple-scale method. The theoretical predictions of Eq. (13) are shown in Fig. 2d, e as solid green lines, which are in good agreement with the experimental results. These results suggest that the size of the peak frequency-shift of R1 from locking to unlocking can be elevated by increasing the initial frequency offset or the coupling strength between two resonators. This would contribute to a high SNR and thus high detection resolution for frequency unlocking-based bifurcation binary sensors.

Detection mechanisms
We propose a novel bifurcation sensor that exploits the sudden transition from locking to unlocking, resulting in a large increase in the peak frequency in coupled resonators under an IR regime. The primary principle of this sensor is to relate a detectable shift in the peak frequency of resonator R1 from locked to unlocked to a target stimulus change. Since frequency-locking arises due to IR between two coupled resonators, the onset of frequency unlocking is triggered by breaking that IR. In addition to increasing the excitation amplitude, two other approaches can be deployed to achieve this purpose. The first is to increase the frequency ratio of resonators R1 and R3 beyond 1:3, by increasing the frequency of R1 or reducing the frequency of R3. This approach suggests the possibility of detecting stimuli that affect the stiffness or mass of R1 or R3, such as changes in a potential field force or the addition of mass. The second approach is to reduce the coupling voltage to shrink the frequency locking region such that the resonator R1, initially operating at the critical point (Point A in Fig. 2c), will exit that region and unlock. This approach allows the sensor to act as an electrometer for charge detection.
The proposed detection mechanism allows the sensor to operate in binary or analog mode. This is achieved by observing the jump in the peak frequency and using it to detect a target stimulus in excess of a given (safe) threshold or by creating a calibration curve to relate the peak frequency to the amount of change in stimulus: • A binary detector where the OFF (0) signal would correspond to the locked peak frequency while the ON (1) signal would correspond to a discrete and significant shift in peak frequency after unlocking, as shown in Fig. 3a. This operation mode is more robust due to the high SNR provided by the frequency jump.
• An analog sensor where a one-to-one relationship between the peak frequency after unlocking and the target stimulus magnitude is established and used as a calibration curve to quantify continuous stimulus changes, as shown in Fig. 3b. This operation mode requires a precise readout circuit for the quantification of the measurand.

Charge detection
To validate the proposed sensor detection paradigm, we conducted an experimental demonstration of charge detection using the coupled MEMS resonators. The schematic of the proposed electrometer is shown in Fig. 4a. The external charge input ΔQ collected at the coupling parallel plates will cause a change in the coupling voltage V C , as per ΔQ ¼ CΔV C (where C ¼ 0:0137707 pF is the capacitance between two resonators' coupling parallel plates, described in Supplementary S6). Injecting a positive charge to the negatively charged plate, and vice versa, reduces the total charge between the plates, thereby deprecating the voltage across them and, consequently, decreasing the coupling strength between the resonators. This will trigger the onset of the frequency unlocking of R1, initially operating near the critical bifurcation point. Frequency Frequency Δf f f Fig. 3 Detection mechanism for unlocking-based bifurcation sensors. a The binary mode exploits the qualitative difference between the peak frequency before and after unlocking, reporting On-state for target stimuli above the threshold and OFF-state for stimuli below the threshold. b The analog mode exploits the peak frequency shift after unlocking The resulting significant shift in the peak frequency from locking to unlocking can be used to create discrete output states, realizing a binary charge detector. On the other hand, the quantitative changes in peak frequency after unlocking due to coupling voltage changes can be exploited to realize an analog electrometer.

Binary mode
In the binary mode, the initial coupling voltage between the two resonators is set to 8 V. The locked frequency of resonator R1 is remeasured as f ir ¼ 122:898 kHz due to the frequency drift of R3 under different room temperatures. A manual forward sweep starting from V ac ¼ 360 mV was carried out with an excitation voltage step of 1 mV to obtain a better estimate of the critical excitation amplitude V crit under ambient external disturbances. We note that all the excitation amplitudes within the frequency-locking region can be selected as the operating point, the detection resolution, however, increases as the operating amplitude of the excitation force is brought closer to the critical bifurcation point. This closeness is bounded by the uncertainty of the critical excitation amplitude, which may inadvertently activate the frequency-unlocking event, thereby causing false-positives. Therefore, the closest operating excitation amplitude of the sensor should exclude this uncertainty region to protect against false-positives. We define an operational setoff excitation amplitude as δV ac = V crit À V op , where V op is the operating excitation amplitude. A stability study was carried out to determine the closest operating point by increasing the set-off excitation amplitude δV ac in steps of 1 mV. A set-off excitation amplitude was declared stable if it can provide consistent response in 10 independent tests. We found the minimum realizable set-off excitation amplitude for the experimental setup to be δV ac ¼ 3mV.
As a safety margin and to consider a higher confidence level, we set δV ac ¼ 5mV, yielding an operating excitation amplitude V op ¼ 380mV. The external charge input is imitated by decreasing the coupling voltage. A forward frequency sweep starting from f ¼ 122 kHz is carried out to track the peak frequency until it finds a qualitative (sharp) drop in the amplitude and then recalls that frequency as the peak frequency. As the coupling voltage drops by 10 mV, from 8 V to 7.99 V, the peak frequency unlocks and undergoes a sudden surge of Δf ¼ 1177 Hz, as shown in Fig. 4b. This indicates the detection threshold (minimum detectable charge) of the sensor at the set-off excitation amplitude of 5 mV to be 1:37 10 À16 C, as per ΔQ ¼ CΔV C . Any charge change in excess of this threshold is detected.
The uncertainty of the critical excitation amplitude arises from the fluctuations of the excitation voltage, the locked frequency and the natural frequency of R1. The latter is dominant and is caused by the additional effect of Joule heating introduced to tune the natural frequency of R1, such that 3ω 1 <ω 3 , in our experiment. It could be mitigated by introducing eligible resonator designs. In that case, the minimum detectable charge would be mainly determined by the locked frequency stability. We evaluate the frequency stability in the locking region via Allan deviation in Fig. 4c. To this end, we use a LIA-based phase locked loop (PLL) to track a time series of the locked frequency using a fixed sampling rate of 7Sa/sec for a duration of 300 s. The minimum Allan deviation of the locking frequency is observed to be 13.13 ppb, yielding extremely small frequency fluctuations of 0.0016 Hz. This indicates that the proposed binary electrometer would achieve an extraordinarily high resolution, provided that the heating induced natural frequency fluctuation in the frequency sweep process can be eliminated.

Analog mode
In the analog mode, a closed-loop circuit-based PLL is set to track the peak frequency for real-time charge quantitative detection. Details of the closed-loop procedures are described in Supplementary S7. The peak frequencies of R1 after unlocking are shown in Fig. 5a as the coupling voltage is varied in steps of 0.1 V. The peak frequency increases linearly as the coupling voltage decreases with a fitting coefficient of R 2 = 0.9904 and a dynamic range of 4 V, which can serve as the calibration curves to quantitatively detect charge changes. Figure 5(b) shows the real-time detection ladder diagram as the coupling voltage varies in steps of 0.1 V, which is equivalent to the charge variation of 1:37 10 À15 C. The dotted black line corresponds to the average peak frequency. The detection time for a voltage step of 0.1 V is only 0.047 s, as shown in the inset of Fig. 5b. The real-time detection ladder diagram with steps of 0.01 V is also shown in Supplementary S8. The sensitivity S (scaling factor), defined as the ratio of the shift in peak frequency to the change in coupling voltage, was found to be 33 Hz/V.
To examine the impact of temperature drift on the stability of the peak frequency in a closed-loop circuit, we measured the time series of a specific peak frequency over 5 minutes, Fig. 5c. The colored markers are the average frequencies over 200 adjacent points. A frequency fluctuation of 0.4 Hz is observed within 5 minutes, which indicates that the temperature effect is negligible in our real-time detection.
The limit of detection (LOD) or charge resolution R can be obtained as the ratio of the minimum detectable frequency shift δf to the sensitivity S, such as R ¼ C Á δf =S. The minimum detectable frequency change is dependent on the frequency fluctuation and can be determined by evaluating the Allan deviation σ A ðτÞ of the characteristic frequency f p as δf ¼ f p Á σ A ðτÞ. Figure 5(d) shows the Allan deviation σ A of the tracked peak frequency of R1 as a function of the integration time τ. The calculated minimum Allan deviation is 190.83 ppb, yielding a charge resolution of 101:48 10 À19 C equivalent to 63 electrons. Table S1 in Supplementary S9 lists comparisons of the state-of-the-art resonant electrometers 33,39,40 in terms of the charge resolution, response time, and power consumption. In comparison, the electrometer operating in analog mode from this work achieves a moderate charge resolution and relatively shorter response time. However, the potential of our detection mechanism is not fully realized due to the large size of the resonators 4 and the limitations of the real-time frequency tracking PLL circuit 41 and measurement circuit we used in the experiment. If a smaller-scale resonator was used under an experimental condition with a higher-end PLL and measurement circuits, the proposed detection mechanism would have a better charge resolution in the analog mode.

Conclusions
We report a new bifurcation-based sensor paradigm exploiting the frequency unlocking associated with IR between two electrostatically coupled micro-resonators. This sensor can function as a binary detector of a target stimulus in excess of a given (safe) value by monitoring the jump in peak frequency from locking to unlocking. It can also serve as an analog sensor to quantify the stimulus change explicitly by measuring the continuous change in peak frequency beyond unlocking. The proposed sensor enables the detection of stimuli that affect the stiffness or mass of the resonators, such as changes in a potential field force or the addition of a mass. It can also act as an electrometer for charge sensing.
The frequency locking and unlocking behaviors under IR regime were investigated experimentally. A closedform formula of the size of the peak frequency shift from locking to unlocking, upon crossing the bifurcation point, was derived and validated experimentally. We found that a larger frequency-shift from locking to unlocking and, therefore, a higher SNR and resolution for the binary detector can be attained by designing resonators with a larger initial frequency offset or by increasing the coupling strength between them.
We demonstrated the feasibility of the proposed detection mechanism for charge detection. Sweep-based peak tracking was utilized to estimate the frequency-jump event in binary mode, while a closed-loop circuit was built for real-time charge detection in analog mode. The realized minimum detectable charge in binary mode is 1:37 10 À16 C and in analog mode is 1 10 À17 C. While the binary sensor had no upper limit for its dynamic range, the upper limit of the analog sensor was 5:48 10 À14 C.
Compared to traditional bifurcation sensing mechanisms, the proposed mechanism has the advantages of (i) a remarkable SNR in binary mode because of the large jump across the bifurcation in peak frequency, (ii) excellent frequency stability in both binary and analog modes provided by IR, and (iii) the flexibility to operate the sensor in either binary or analog mode. The detection resolution in binary mode is dependent on the minimum realizable set-off excitation amplitude, which is affected by the uncertainty of the excitation voltage and R1's natural frequency. In our experiment, Joule heating, which is introduced to tune the natural frequency of R1 for IR, was the main driver of this uncertainty. The detection resolution in binary mode could be significantly improved by designing the resonators to satisfy the condition 3ω 1 <ω 3 or by an active temperature compensation circuit. We plan to investigate those approaches in future work.

Theoretical analysis
We adopt a lumped-mass model that describes the transverse displacements of two electrostatically coupled micro-resonators by the governing nondimensional equations of motion 42,43 : ð1Þ where x i , ω i and γ i ði ¼ 1; 3Þ are the transverse displacements, natural frequency, and damping ratio of resonators R1 and R3, respectively. β 1 and f 1 are the cubic stiffness and driving amplitude of R1, respectively, while g 1 and g 3 represent the coupling coefficients which are proportional to the square of the coupling voltage V C . The derivation of Eq. (1) is presented in Supplementary S5.
We use the perturbation method of multiple scales 27 to solve Eq. (1). The general approximate solution of Eq. (1) is given by: where τ n ¼ ε n t. We define τ 0 ¼ t as the fast variable while τ 1 ¼ εt as the slow variable. A i and θ i ði ¼ 1; 3Þ are the motional amplitude and phase difference of R1 and R3, respectively. Considering the case of primary resonance, we define the excitation frequency Ω and the natural frequency ω 3 of R3 in terms of the small detuning parameters δ 1 and δ 3 , respectively, with respect to the Next, we expand the equations of motion, Eq. (1), in powers of ε. Then, the equations describing the amplitude and phase modulations are obtained by eliminating the secular terms. The first-order approximation of the solutions yields 45 where the prime symbol denotes the derivative with respect to the slow time τ 1 , and the phase angles ϕ 1 and ϕ 2 are defined as To obtain the steady-state response, we set the slow time derivatives to vanish. Setting ϕ 0 1 ¼0, ϕ 0 3 ¼0 yields θ 0 1 ¼ δ 1 , θ 0 3 ¼ 3δ 1 À δ 3 , while setting A 0 1 ¼ A 0 3 ¼ 0 gives the following algebraic equations for the stationary solutions: 8r 1 A 1 þ 3g 1 A 2 1 A 3 sin ϕ 3 À 4f 1 sin ϕ 1 ¼ 0 8ω 1 A 1 ðΩ À 1Þ À 3g 1 A 2 1 A 3 cos ϕ 3 À 3β 1 A 3 1 þ 4f 1 cos ϕ 1 ¼ 0 8r 3 ω 3 A 3 À g 3 A 3 1 sin ϕ 3 ¼ 0 8ω 3 A 3 ð3Ω À ω 3 Þ À g 3 A 3 1 cos ϕ 3 ¼ 0 Substituting the last two equations of Eq. (5) into its first two equations yields the algebraic equations describing the stationary response of R1, only with respect to A 1 and ϕ 1 8r 1 A 1 þ 3r 3 ω 3 g 1 g 3 A 5 1 8ω 2 3 ½ð3ΩÀω 3 Þ 2 þr 2 3 À 4f 1 sin ϕ 1 ¼ 0 8A 1 ðΩ À 1Þ À 3g 1 g 3 ð3ΩÀω 3 ÞA 5 1 8ω 2 3 ½ð3ΩÀω 3 Þ 2 þr 2 3 À 3β 1 A 3 1 þ 4f 1 cos ϕ 1 ¼ 0 ð6Þ IR occurs when 3Ω % ω 3 , where a frequency gap occurs because Eq. (6) does not provide a (real) solution for ϕ 1 44 . This can be seen in: sin ϕ 1 j j¼ ð8r 1 A 1 þ 3r 1 ω 3 g 1 g 3 A 5 1 8ω 2 3 ½ð3Ω À ω 3 Þ 2 þ r 2 3 As the excitation amplitude f 1 is further increased, the right-hand side of Eq. (7) decreases gradually until its solution becomes real. The critical excitation amplitudef c required to eliminate this frequency gap and unlock the frequency can be calculated by setting sin ϕ 1 ¼ 1: Under this critical excitation amplitude, the nonlinear resonance (peak) frequency Ω p of R1 after unlocking can be evaluated using the amplitude-frequency relation in the Duffing oscillator equation 13 as: where A p is the amplitude of R1 after unlocking. Substituting Eq. (8) into Eq. (9) and retaining terms up to the sixth-order in A 1 yields where the amplitude A 1 of resonator R1 can be assumed as where σ ¼ ω 3 3 À 1 is the initial frequency offset of R1 between its natural frequency and the locked frequency. Substituting Eq. (11) into Eq. (10) yields Then, the frequency shift Δω from locking to unlocking is given by: Equation (13) captures the dependence of the peak frequency-jump on the initial frequency offset, coupling strength and material properties.