The effects of nonlinear damping on degenerate parametric amplification
TThe effects of nonlinear damping on degenerate parametricamplification
Donghao Li ∗ and Steven W. Shaw † Department of Mechanical and Civil Engineering, Florida Institute of Technology, Florida32901, USANovember 1, 2020
Abstract
This paper considers the dynamic responseof a single degree of freedom system with nonlinearstiffness and nonlinear damping that is subjected toboth resonant direct excitation and resonant paramet-ric excitation, with a general phase between the two.This generalizes and expands on previous studies ofnonlinear effects on parametric amplification, notablyby including the effects of nonlinear damping, whichis commonly observed in a large variety of systems,including micro- and nano-scale resonators. Using themethod of averaging, a thorough parameter study iscarried out that describes the effects of the amplitudesand relative phase of the two forms of excitation. Theeffects of nonlinear damping on the parametric gainare first derived. The transitions among various topo-logical forms of the frequency response curves, whichcan include isolae, dual peaks, and loops, are deter-mined, and bifurcation analyses in parameter spacesof interest are carried out. In general, these resultsprovide a complete picture of the system response andallow one to select drive conditions of interest thatavoid bistability while providing maximum amplitudegain, maximum phase sensitivity, or a flat resonantpeak, in systems with nonlinear damping.
Keywords
Parametric amplification · Nonlineardamping · Bifurcation analysis · MEMS
Parametric amplification (PA) is the use of a resonantparametric excitation to enhance the response of a res-onantly driven oscillator. This approach allows one toalter the effective damping of the system, even to thelimit of zero damping at the point of parametric in-stability, thus bringing benefits of spectral narrowingand higher frequency selectivity to resonant systems ∗ lid2016@my.fit.edu † sshaw@fit.edu [1, 2]. Specifically, the amplification, deamplification,and thermal noise squeezing have been analyzed fora Josephson parametric amplifier (JPA) [3, 4]. In aclassic study of a mechanical device, PA and thermo-mechanical noise squeezing were observed in a vibrat-ing microcantilever and were analyzed using a linearmodel [5]. These studies were based on a linear model,which demonstrated the main effects. Studies on theimpacts of stiffness nonlinearity on PA and similar sys-tems have shown that the response can be quite rich[6, 7, 8]. The Duffing nonlinearity is especially of in-terest, as it is oftentimes exhibited in systems withlarge vibration amplitude, resulting in both opportu-nities for improved performance and challenges due tothe complexity in dynamic responses [9, 10].PA is used in a wide variety of applications, es-pecially in the realm of nano- and micro-electro-mechanical-systems (N/MEMS) [11]. Due to theirsmall size, N/MEMS devices have the advantages ofhigh sensitivity, high frequency range, low power con-sumption, low noise, and excellent integration withelectronics [12]. Specifically, PA is used for nano-scale applications including piezoelectrically pumpedparametric amplifiers [13, 14], carbon nanotubes [15],and graphene-based resonators [16]. Furthermore,it is also used in micro-scale applications includ-ing multi-analyte mass sensors [17], mass sensingarrays [18], Coriolis mass flow sensors [19], phase-modulated microscopy [20], parametric symmetrybreaking transducers [21], parametron (a resonator-based logic device) [22], and parametrically pumpedthermomechanical-noise-driven resonators [2]. In ad-dition, the effect of PA can also arise from mode cou-pling, such as 1 : 1 and 1 : 2 internal resonances. Asan example of 1 : 1 resonance, the combined effectof Coriolis force and nonlinear resonance coupling isobserved in MEMS vibratory gyroscopes, such as vi-brating ring gyroscopes (VRG) [23] and disk resonatorgyroscopes (DRG) [24], from which PA arises [25, 26].1 a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov any different kinds of mechanical structures havebeen considered to achieve PA, including torsionaloscillators [27], metallized silicon–nitride diaphragms[28], silicon disk oscillators [29], single-crystal siliconin-plane arch microbeams [30], stoichiometric siliconnitride (SiN) membranes [31], and double-clampedcantilever beams with a concentrated mass at the cen-ter [32]. Several analyses have been carried out inexperiments, including parametric noise squeezing fora microcantilever [33], effects of geometric nonlinear-ity for a integrated piezoelectric actuation and sens-ing system [34], closed-loop stability in the presenceof nonlinearity [35], and distortion of the actuationwaveform by the displacement-dependent electrostaticnonlinearity [36].Moreover, PA can also be employed in macroscopicdevices. This can be achieved by using base excitedcantilever beams [37], wherein the effects of cubic non-linearity [38], cubic parametric stiffness [39], and para-metric bistability [40] have been demonstrated. Othermacroscopic systems include sheet metal plates [8],horizontal wind turbine blades in steady rotation en-during cyclic transverse loading [41], dual-frequencyparametric amplifiers (DFPA) with macroscopic mod-ular mass and linear voice-coil actuator [42], thinstretched strings carrying an alternating electric cur-rent in a non-uniform magnetic field [43], and doublyclamped strings [44].Physics applications also constitute an importantfield for parametric amplifiers. Josephson paramet-ric amplifiers are of particular interest [3, 4], whichhave oftentimes been used for superconducting quan-tum interference devices (SQUID) [45, 46]. Forexample, terahertz Josephson plasma waves ampli-fied in a cuprate superconductor [47] and quasi-particles flowing through a superconductor-insulator-superconductor junction [48] are analyzed. PA hasalso been used to study the reheating of an inflation-ary Universe [49].Parametric suppression, also known as deamplifica-tion, attenuation, or splitting in the spectrum, whichcan be achieved by modulating the relative phase be-tween the direct and parametric excitations, has alsogained interest in both theory [9, 37] and experiments[1, 27, 29, 31, 38, 50]. It can be utilized to enhancethe phase response of an oscillator by increasing thesteepness of the phase slope near resonance [51]. Inaddition to the dual peaks that can be seen in thesesystems, it is also observed that loops may also ex-ist, similar to those previously observed in self-excitedsystems [52, 53].Recent theoretical analysis for PA include the ef-fects of quadratic and cubic nonlinearities [7], dual fre-quency parametric amplifiers with quadratic and cubic nonlinearities [54], regular and chaotic vibrations withtime delay [55], and frequency comb responses [56].In this present work, nonlinear damping, also knownas nonlinear dissipation or nonlinear friction, is takeninto account, in addition to stiffness nonlinearity.This is of practical interest since nonlinear damp-ing is frequently observed in a large variety of struc-tures. For instance, nonlinear damping has com-monly been observed in NEMS resonators based oncarbon nanotubes [57], graphene [57, 58, 59], anddiamond [60]. Likewise, it has also been observedin micro-structures including non-contacting atomicforce microscope (AFM) microbeams [61] and MEMSclamped-clamped beams [62, 63]. In addition, non-linear damping has been observed in macroscopic me-chanical systems, such as large-amplitude ship rollingmotions [64], concrete structures [65], stainless steelrectangular plates, stainless steel circular cylindricalpanels, and zirconium alloy hollow rods [66]. Non-linear damping of a given mode can also result frommode interactions such as induced two-phonon pro-cesses [67] and internal resonances [59, 68, 69]. Inaddition to the experimental observations, many the-oretical works have also been completed, covering thetopics of the relaxation of nonlinear oscillators inter-acting with a medium [70], estimation using Melnikovtheory [71], estimation using analytic wavelet trans-form [72], dynamic response to harmonic drive [62],and characterization using the ringdown response [63].In this paper, a degenerate parametric amplifierwith nonlinearities in both stiffness and damping isconsidered. One quantity of particular interest is theparametric gain, defined as the ratio of the peak am-plitudes (near resonance) with and without the para-metric pump [5], and expressed as G = ¯ r peak | pump on ¯ r peak | pump off , (1)where ¯ r peak is the steady-state peak amplitude. When G >
1, the oscillator is amplified, and when
G < ψ , plays a pivotal rolein the nature of the system response. It affects theparametric gain of the system, an important quan-tity that will be defined in Sect. 2.2. It also has asignificant effect on the structure of the steady-stateresponse curves. Two special values of the relativephase, − π/ π/
4, will be considered in detail,as these values provide the system with maximum andminimum parametric gains, respectively.This paper is organized as follows. In Sect. 2, weformulate the problem, preview the general effects ofnonlinear damping on PA, and provide an analysis ofhow the PA gain is affected by nonlinear damping. In2ect. 3, we analyze the case where the relative phaseprovides maximum parametric gain ( ψ = − π/ ψ = + π/ We consider a single degree of freedom system con-sisting of a weakly nonlinear, weakly damped oscilla-tor with eigenfrequency ω that is excited by both anear resonant direct drive at frequency ω ≈ ω andby a parametric pump at frequency 2 ω , and a relativephase between the drive and the pump. Specifically,in addition to the usual Duffing nonlinearity, we as-sume that the oscillator is also subjected to nonlineardamping.The equation of motion for this degenerate nonlin-ear parametric amplifier is given by¨ x + 2 (cid:0) Γ + Γ x (cid:1) ˙ x + ω [1 + λ cos (2 ωt )] x + γx = f cos ( ωt + ψ ) , (2)where Γ and Γ represent the linear and nonlineardamping coefficients and are both assumed to be pos-itive, ω and γ denote the linear and nonlinear stiffnesscoefficients, λ indicates the amplitude of the paramet-ric pump, f specifies the direct drive amplitude, ω dictates the drive frequency, and ψ describes the rel-ative phase of the two drives. The effects of damping,nonlinearities, and drives are assumed to be small, inthe sense that when both the amplitude and the eigen-frequency are normalized to O (1), all other coefficientsare O ( ε ), where 0 < ε (cid:28) Since only primary and principal parametric reso-nances are of interest, it is assumed that ω is close to ω . Thus, a nondimensional frequency detuningcan be introduced as σ = ω − ω ω , (3)which is used to illustrate how the drive frequency de-viates from the natural frequency, for example, duringa frequency sweep.Under the stated assumptions, the amplitude andphase of x ( t ) will be slowly varying functions of time.The method of averaging is employed to obtain thetime-invariant equations that govern these quantities.To obtain equations suitable for averaging, we firstapply the van der Pol transformation x ( t ) = r ( t ) cos [ ωt + φ ( t )] , (4)˙ x ( t ) = − ωr ( t ) sin [ ωt + φ ( t )] , (5)where r ( t ) and φ ( t ) are the slowly-varying polar co-ordinates representing the amplitude and the phase of x ( t ). We next average time over one period, 2 π/ω ,and implement the detuning definition above, whichconverts Eq. (2) into to the following approximateaveraged equations˙ r = − Γ r −
14 Γ r + λω r sin (2 φ )4 ω − f sin ( φ − ψ )2 ω , (6)˙ φ = − σω ω + 3 γr ω + λω cos (2 φ )4 ω − f cos ( φ − ψ )2 ωr . (7)Under the stated assumptions, all terms on the righthand side are small so that r ( t ) and φ ( t ) vary slowlyin time, consistent with the physical assumptions.All the subsequent analyses will be based on the au-tonomous dynamical system governed by Eqs. (6)-(7).The steady-state condition is obtained by solving˙ r = 0 and ˙ φ = 0 simultaneously, which provides so-lutions for the steady-state amplitude ¯ r and phase ¯ φ ,representing a fixed point of the averaged dynamicalsystem. The stability of a fixed point can be deter-mined by the local Jacobian matrix. To facilitatefurther analytical calculations, the steady-state phase¯ φ is first eliminated from the steady-state condition,leaving a single equation for the steady-state ampli-tude ¯ r . With only the leading-order terms kept, thisequation is given by16 (cid:104) λ ω + 4 (cid:0) + Γ ¯ r (cid:1) ω + (cid:0) σω − γ ¯ r (cid:1) + 4 (cid:0) σω − γ ¯ r (cid:1) λω cos (2 ψ ) − (cid:0) + Γ ¯ r (cid:1) λω sin (2 ψ ) (cid:105) f ¯ r = (cid:104) λ ω − (cid:0) + Γ ¯ r (cid:1) ω − (cid:0) σω − γ ¯ r (cid:1) (cid:105) ¯ r . (8)3ince Eq. (8) is quintic in ¯ r when ignoring thetrivial solutions, which have no physical meaning, itsuggests that there can be a maximum of five fixedpoints for a given frequency. From Eq. (8), it can beseen that nonlinear stiffness and detuning appear onlyin the collective term (4 σω − γ ¯ r ), which impliesthat in the amplitude-frequency space, the nonlinearstiffness has the effect of bending the response curveshorizontally, yet has no effect on the amplitude or thetopological structure of the curves, other than thoseeffects associated with the bending of the curves. Thisterm can be rewritten as the backbone curve equation σ b = 3 γ ¯ r ω , (9) which quantifies the amount that the curves bend hor-izontally as a function of the amplitude. This is, ofcourse, the same backbone curve as that for the usualDuffing equation. The amplitude on the backbonecurve can then be written as ¯ r b = ω (cid:112) σ/ (3 γ ). Forsufficiently prominent effects of nonlinear stiffness, theusual Duffing-type bistability can be exhibited (plusmore, as will be seen subsequently).For the special cases of maximum parametric gain( ψ = − π/
4) and minimum parametric gain ( ψ =+ π/ r . These expressions,given here, are convenient for plotting the steady-stateresponse in the amplitude-frequency space, σ (cid:16) ψ = − π (cid:17) = 3 γ ¯ r ω ± ω ¯ r (cid:114) λ ω ¯ r + 2 f − (4Γ + Γ ¯ r ) ω ¯ r ± f (cid:113) f + 2 ( λω + 4Γ + Γ ¯ r ) λω ¯ r , (10) σ (cid:16) ψ = + π (cid:17) = 3 γ ¯ r ω ± ω ¯ r (cid:114) λ ω ¯ r + 2 f − (4Γ + Γ ¯ r ) ω ¯ r ± f (cid:113) f + 2 ( λω − − Γ ¯ r ) λω ¯ r . (11)The subscripts of the two plus-minus signs in eachexpression indicate their mutual independence, imply-ing that there can be up to four different drive frequen-cies resulting in the same amplitude. Furthermore, thestructure of this equation indicates that the frequencyresponse can have up to four saddle-node bifurcations.The role of the backbone curve, given by Eq. (9), is evident in the conditions for the response curves givenby Eqs. (10)-(11).The points at which the response curves intersectthe backbone curve can be expressed in closed form.To calculate these amplitudes, let Eq. (9) apply toEq. (10) and Eq. (11), respectively. Each can besimplified into a factored form, given by (cid:2) Γ ω ¯ r − ( λω − ) ω ¯ r b − f (cid:3) (cid:2) Γ ω ¯ r − ( λω − ) ω ¯ r b + 2 f (cid:3) (cid:2) Γ ¯ r b + ( λω + 4Γ ) (cid:3) ω ¯ r = 0 , (12) (cid:2) Γ ω ¯ r + ( λω + 4Γ ) ω ¯ r b − f (cid:3) (cid:2) Γ ω ¯ r + ( λω + 4Γ ) ω ¯ r b + 2 f (cid:3) (cid:2) Γ ¯ r b − ( λω − ) (cid:3) ω ¯ r = 0 . (13)Note that there are three factors and, in fact, atmost three such points, in contrast to the usual Duff-ing equation with direct drive which has only a singlesuch point. Some of these factors do not have physi-cal meaning. These two equations will be discussed indetail later in Sect. 3.1 and Sect. 4.1, respectively. We include nonlinear damping in the model for threeimportant reasons: (i) it is commonly observed in ex-periments across many fields [57, 58, 59, 60, 61, 62, 63,64, 65, 66, 67, 68, 69], (ii) it arises from fundamentalmicroscopic considerations in micro/nano resonators[59, 70], and (iii) it allows for the closure of the non-trivial response branches in parametric resonance bysaddle-node bifurcations that can occur even near res-4nance (in fact, it can limit the response even in theabsence of the Duffing nonlinearity). In order to makethe third point, consider an oscillator being excitedonly parametrically, in which case the parametric in-stability threshold can be observed from Eq. (8) byconsidering ¯ r = 0, which reduces to λ AT = 2 (cid:115) ω + σ . (14)This is the condition for the well-known Arnoldtongue, which is a pitchfork bifurcation condition inthe equations and corresponds to a period doubling inthe original system. This stability condition is inde-pendent of nonlinearities since it relates to the linearstability of the trivial response. Note that the para-metric instability threshold at zero detuning has thelowest value, denoted here as λ AT , = 4Γ /ω . If theoscillator with only linear damping is driven paramet-rically above λ AT , , the frequency response, as pre-dicted by first order perturbation methods, has fourbranches that do not merge by saddle-node bifurca-tions at any frequency, even far from resonance wherethe perturbation analysis is not valid. To see the effectof nonlinear damping, consider removing the nonlin-ear stiffness terms 3 γ ¯ r from Eq. (8), as they do notdisplay any effect pertinent to the range of possibleamplitudes, but only to the frequencies at which spe-cific amplitudes occur. In this case, the presence ofnonlinear damping, Γ >
0, always results in a finiteamplitude, even at resonance, for Γ is the coefficientof the highest order term in ¯ r .Another important effect of nonlinear damping isthat it competes with nonlinear stiffness in terms ofthe system exhibiting Duffing bistability. To clarifythis, we consider a simple Duffing oscillator with non-linear damping, for which the oscillator will not un-dergo a cusp bifurcation (i.e., the condition for theonset of bistability [73]) if the nonlinear damping isgreater than the following value [62, 74]Γ ∗ = √ | γ | ω . (15)For a Duffing oscillator with only direct drive, nonlin-ear damping with Γ > Γ ∗ eliminates any possibilityof the system exhibiting Duffing bistability. For thesystem with both direct and parametric drives, how-ever, the bistability may still be observed under con-ditions to be demonstrated in the following. However,as will be shown, the system can undergo a cusp bi-furcation which terminates the Duffing bistability forsufficiently large direct drive. To examine how the parametric gain, defined in Eq.(1), is affected by nonlinear damping, it is of inter-est to compare this gain with the gain of the linearsystem (Γ = 0). For λ < λ AT , , where the paramet-ric pump level is below the instability threshold, η isintroduced to denote the ratio of the gains with andwithout nonlinear damping at the peak, defined by η = G (Γ ) G linear , (16)where G linear is the linear gain given in [5]. For ψ = ± π/
4, this ratio can be obtained from Eqs. (12)-(13)(which are subsequently elucidated by Eq. (21) andEq. (27), respectively), and written in the form of thedepressed cubic equation ξη + 2 η − , (17)whose real root is given by η = 12 ξ / (cid:104) (1 + χ ) / + (1 − χ ) / (cid:105) , (18)where χ = (cid:114) ξ , (19) ξ (cid:16) ψ = ± π (cid:17) = Γ f (4Γ ± λω ) ω . (20)From Eqs. (18)-(20), it can be seen that the ratio η is characterized by a single variable ξ , and this depen-dence is shown in Fig. 1a. As ξ increases, η decreasesmonotonically, indicated by the linear-log plot.In a similar manner, as shown in Fig. 1b, theparametric gain attenuates as the nonlinear damping(nondimensionalized in the figure) increases. The im-pact of nonlinear damping on the ψ = − π/ ψ = + π/ = 0 strongly depends on the system parameters,as suggested by Eq. (20). For instance, if the para-metric pump level is close to the threshold, that is,4Γ − λω ≈
0, then η declines rapidly, indicating thatnonlinear damping has a very strong impact on theparametric gain when operating near the threshold.5 a ) - - ξη ( b ) ψ = + π / ψ = - π / - - Γ f /( Γ ω ) η Figure 1: Ratio of the parametric gains for ψ = ± π/
4. (a) Gain ratio η versus ξ . (b) η versus nonlineardamping for ψ = ± π/ for three levels of the parametric pump below theinstability threshold λ AT , . The solid lines represent the maximum gain case ( ψ = − pi/ ψ = + pi/ ψ = − π/ ) For the majority of applications related to PA, thisvalue of the relative phase is of primary interest since itgenerates the highest effective quality factor (as mea-sured by the width of the resonance peak in linearsystems with a parametric pump) [1]. This makesthe oscillator optimally energy-efficient and frequency-selective when compared to other values of relativephases. One interesting phenomenon that can be ob-served in the frequency domain is that, when nonlineardamping is present, for sufficiently large levels of theparametric pump, an isolated response branch, calledan isola, appears in the frequency response curve, asdemonstrated below.
The equation for the steady-state amplitude is givenby Eq. (10). Fig. 2 shows the transition ofthe steady-state response curves in the amplitude-frequency space as the parametric pump level is var-ied for systems without and with nonlinear stiffness.As the pump level increases, an isola emerges underthe main response curve, consisting of two branchesbounded by a pair of saddle node bifurcations, asshown in Fig. 2b for a system with linear stiffness.Since the nonlinearity in stiffness bends the curveshorizontally, when it is sufficiently large, the Duff-ing nonlinearity can lead to bistability in the main response branch, whose frequency range is bounded byanother pair of saddle-node bifurcations, as shown inFig. 2c, d. When the pump level leads to an isola andthe nonlinear stiffness leads to bistability, four saddle-node bifurcations occur, as depicted in Fig. 2d. Therehave been several experiments where the observationof the isola is reported [21, 22, 40, 44].From Fig. 2, it is known that all the three possibleamplitude local extrema occur on the backbone curve,given by Eq. (12). It can be seen that, in this equa-tion, the first factor always has exactly one positivesolution, which corresponds to the peak amplitude onthe main response branch given by the depressed cubicequationΓ ω ¯ r − ( λω − ) ω ¯ r peak − f = 0 . (21)The second factor in Eq. (12) may have up to twopositive solutions, depending on the system parame-ters, which corresponds to the highest and the lowestamplitudes on the isolaΓ ω ¯ r − ( λω − ) ω ¯ r isola + 2 f = 0 . (22)When there is no nonlinear stiffness ( γ = 0), asshown in Fig. 3a, b, the dynamic response at zerodetuning is symmetric about φ = π/ nπ . The peakamplitude has the steady-state phase of ¯ φ = 5 π/ φ = π/ ∂/∂ ¯ r of Eq. (22), and eliminating6 a ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = γ = - - σ r ( b ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲ λ = γ = - - σ r ( c ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ λ = γ = - - σ r ( d ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ λ = γ = - - σ r Figure 2: Frequency response curves for different values of the parametric pump level λ and nonlinear stiffness γ , with ψ = − π/
4, Γ = 0 . = 0 . ω = 1, and f = 0 .
01. The solid curves represent stable responses,and the dashed curves represent unstable responses, and the triangles represent direct numerical simulationresults. (a) A case with no bistability. (b) A case with an isola. (c) A case with Duffing bistability. (d) Acase with both an isola and Duffing bistability, which exhibits four saddle-node bifurcations and up to threestable steady states¯ r , which yields f isola = (cid:115) ( λω − ) ω . (23)When f > f isola , there is no isola in frequency re-sponse, as shown in Fig. 2a, c. When f < f isola , theremay exist an isola in frequency response, as shown inFig. 2b, d. Note that Eq. (23) suggests that the pumplevel satisfies λ > /ω . Therefore, the conditionsfor an isola to exist are λ > λ AT , and f < f isola ,independent of the Duffing nonlinearity. It is impor-tant to point out that these conditions provide usefulinformation regarding the selection of the drive levelsneeded to achieve maximum gain without encounter-ing an isola.Fig. 3 demonstrates how these aforementioned am-plitudes change by varying the two drive levels. Theblue and green curves show the peak amplitudes, theorange curves show the extrema on the isola, and thegray curves show the cases of pure parametric reso-nance or pure direct excitation. The saddle-node bi-furcations seen from the orange curves signify the ap-pearance or the disappearance of the isola in the fre-quency response curve, which demonstrate the isola condition described by Eq. (23). Additionally, bycomparing the blue and green curves ( λ >
0) to thegray curve ( λ = 0), the effect of PA can be seen. More-over, the green curve in Fig. 3b has a steeper slopethan the gray curve, suggesting that it can be used tosignificantly improve the sensitivity in force detectionand signal enhancement.Local bifurcation conditions are also of interest, asthey provide information regarding regions of multi-stability in various parameter spaces. Bifurcations ofboth codimension one and codimension two occur inparameter planes of interest. Obtaining the saddle-node bifurcation conditions in the frequency domainis similar to that of the isola condition. They canbe calculated numerically by simultaneously solvingEq. (10), ∂/∂ ¯ r of Eq. (10), and eliminating ¯ r . Thecusp bifurcation condition, on the other hand, can beobtained by utilizing a third equation, ∂/∂ ¯ r of Eq.(10), and further eliminating σ . Obtaining these con-ditions allows for a detailed analysis of multistabilityin various parameter spaces.Fig. 4 provides examples of bifurcation diagrams insome of the parameter planes of interest, with γ > γ <
0, the figuresare simply reflected about σ = 0. The orange curves7 a ) f = = =
0, peak λ r ( b ) λ = λ = λ = λ =
0, peak
Figure 3: Changes in the amplitudes on the backbone curve by varying the drive levels, with ψ = − π/ = 0 . = 0 . ω = 1, and γ = 0. The solid curves represent stable fixed points, and the dashedcurves represent saddle points, where a saddle-node bifurcation can be seen. (a) The blue curve demonstratesthe amplification on the main response branch. (b) The parametric pump levels of the blue and orange curvesare above λ AT , , while those of the green curve are below λ AT , ; for both cases, amplification can be observed ( a ) - σ f ( b )
12 23SN SNCP - σ f ( c )
12 2 3SN SNCP - σλ ( d ) isola γ = γ = γ = γ = γ = λ Figure 4: Bifurcation conditions shown in some parameter planes of interest, with ψ = − π/
4, Γ = 0 . ω = 1, and γ = 0 .
01. Bifurcation types are indicated: SN is for saddle-node and CP is for cusp. Orange andblue curves correspond to SN conditions associated with the isola and the Duffing bistability, respectively. (a)Parameter plane ( σ, f ) for Γ = 0 .
005 and λ = 0 .
05. (b) Parameter plane ( σ, f ) for Γ = 0 .
012 and λ = 0 . σ, λ ) for Γ = 0 .
005 and f = 0 .
01. (d) Parameter plane ( f, λ ) for Γ = 0 . < Γ ∗ , while in the latter case,Γ > Γ ∗ . For Γ < Γ ∗ , increasing the direct drive am-plitude leads to the appearance and expansion of theDuffing bistability, while for Γ > Γ ∗ , increasing thedirect drive amplitude leads to the contraction anddisappearance of the Duffing bistability.Furthermore, it is also of interest to explore theparameter space pertinent to the parametric pumplevel and the direct drive amplitude, that is, ( f, λ ),as shown in Fig. 4d. The orange, dashed curve repre-sents the isola onset condition, give by Eq. (23). Theblue, solid curves represent the onset of the cusp bi-furcation for five values of γ , which imply the appear-ance/disappearance of the Duffing bistability. The in-tercepts of the blue curve corresponds to the criticaldrive values for a simple Mathieu or Duffing oscillator,given by λ cr = 4Γ (cid:112) ω + 9 γ | γ | ω , (24)for the blue curves at f = 0 in Fig. 4d, and f cr = 16 (cid:118)(cid:117)(cid:117)(cid:116) (4Γ ω + 9 γ ) (cid:34) Γ ω (cid:0) √ | γ | − ω (cid:1) (cid:35) , (25)for the blue curves at λ = 0 in Fig. 4d. From Eq. (25),it is clear that its denominator verifies the expressionfor Γ ∗ given by Eq. (15). In Fig. 4d, for Γ > Γ ∗ ,instead of eventually reaching the abscissa as f is in-creased, the curve of the cusp bifurcation conditionincreases monotonically, which is consistent with thephenomenon shown in Fig. 4b. The transient dynamics of the averaged system aregoverned by Eqs. (6)-(7), with ψ = − π/
4. From Fig.2d, it is known that the system can have either one,two, or three stable steady states. Samples of the threegeneric cases of the possible phase portraits are shownin Fig. 5. The gray curve shows a sample trajectory,the orange curves show the stable manifolds of thesaddle points, and the blue curves show the unstable manifolds of the saddle points. The cyan arrows depictthe vector fields. The transient process and the phaseportraits for the system with other relative phases arevery similar to these. Note that the system is globallybounded, due to positive damping, which is apparentfrom Eq. (6).Fig. 5 shows that for generic cases, there are onlythree distinct possibilities. The first case is to havea single fixed point, as shown in Fig. 5a. This equi-librium is globally, asymptotically stable. Fig. 5bevolves from Fig. 5a by adding a pair of fixed points,one stable and one unstable, via a saddle-node bifur-cation; this is the appearance of the isola in the fre-quency response. In a similar manner, the case ofFig. 5c emerges from a second saddle-node bifurca-tion, resulting in three stable equilibria and two sad-dle points; this is the first saddle-node on the Duff-ing response branch. The stable (orange) and unsta-ble (blue) manifolds of the saddle points are shown inthe figure. The stable manifolds serve as separatrices,which form boundaries among the basins of attractionof the stable fixed points. The unstable manifolds ofsaddle points, on the other hand, are asymptotic tothe stable fixed points. As the detuning is continu-ally increased, the isola fixed points will remerge inanother saddle-node bifurcation and then two of theremaining fixed points will undergo the typical Duff-ing saddle-node bifurcation. This full transition, froma single stable response through the four saddle-nodebifurcations to another single stable response, may re-quire a global bifurcation, as considered next.For the situation in which there are two saddlepoints, it is worth noting that a global bifurcationwill occur, as shown in Fig. 6, where the green curvedemonstrates the saddle connection. This global bi-furcation is topologically required in order for thephase portraits to transition as they do as parame-ters are varied. The bifurcation is a standard planarsaddle connection, and the transition across this bi-furcation alters the basins of attraction of the stablesteady states. ψ = + π/ ) The minimum parametric gain occurs for the relativephase of ψ = + π/
4. In certain contexts, this is knownas parametric suppression because the parametric gaincan be less than unity. This special value of the rel-ative phase is also of interest in certain applications,for example, to enhance the phase sensitivity near res-onance, due to the strong impact that the parametricpump has on the steepness of the phase slope at reso-9 a ) sample flow π / π π / π ϕ r ( b ) W s of saddle pointW u of saddle point π / π π / π ϕ r ( c ) W s of saddle pointW u of saddle point π / π π / π ϕ r Figure 5: Phase portraits at three representative values of detuning σ , with ψ = − π/
4, Γ = 0 . = 0 . ω = 1, γ = 0 . λ = 0 .
05, and f = 0 .
01, showing the transition from one to five steady states. (a) σ = 0.(b) σ = 0 . σ = 0 . ( a ) W s of saddle pointW u of saddle point - π - π / π / ϕ r ( b ) W s of saddle pointW u of saddle pointsaddle connection - π - π / π / ϕ r ( c ) W s of saddle pointW u of saddle point - π - π / π / ϕ r Figure 6: Phase portraits showing a global bifurcation that occurs as detuning σ is varied, with ψ = − π/ = 0 . = 0 . ω = 1, γ = 0 . λ = 0 .
05, and f = 0 .
01. (a) σ = 0 .
05. (b) σ ≈ . σ = 0 . The equation for the steady-state amplitude is givenby Eq. (11). Fig. 7 shows the transition of the fre-quency response curves as the parametric pump levelis varied for a system with zero nonlinear stiffness. Asthe pump level is initially increased, the peak trans- forms into a dimple which is between a pair of peakswith equal amplitudes. When the pump level furtherincreases, the dimple becomes more acute and trans-forms into a cusp. (This is not to be confused witha cusp bifurcation.) Beyond this, the cusp transformsinto a loop. It should be pointed out that the twosteady states at the crossing point in the loop havehave distinct phases. When introducing the nonlinearstiffness to the system, these response curves may ex-hibit Duffing bistabilities, yet neither the amplitudesnor the general structure change, which is similar towhat has been shown in Fig. 2.When there are dual maxima, as shown in Fig. 7b-d, a condition for the amplitude of the dual peaks canbe obtained by setting the inner radicand of Eq. (11)equal to zero, given byΓ λω ¯ r . peak − λω − ) λω ¯ r . peak − f = 0 . (26)10 a ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = - - σ r ( b ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = - - σ r ( c ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ ≈ - - σ r ( d ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲ λ = - - σ r Figure 7: Frequency response curves as the parametric pump level is varied, with ψ = + π/
4, Γ = 0 . = 0 . ω = 1, γ = 0, and f = 0 .
01. The solid curves represent stable responses, the dashed curvesrepresent unstable responses, and the triangles represent direct numerical simulation results. (a) A case withonly a single peak. (b) A case with a dimple between the dual peaks. (c) A case where the dimple is a cusp.(d) A case with a loopIt is apparent that all other possible amplitude ex-trema are on the backbone curve. It can be seen thatthe first factor of Eq. (13) always has exactly one pos-itive solution, which corresponds to the amplitude ofthe peak or the dimple on the response curve, asΓ ω ¯ r + ( λω + 4Γ ) ω ¯ r peak − f = 0 , (27)Γ ω ¯ r + ( λω + 4Γ ) ω ¯ r dimple − f = 0 . (28)The third factor of Eq. (13) can have up to two equalpositive solutions, depending on the system parame-ters, which corresponds to the amplitude of the cusppoint shown in Fig. 7c or the amplitude of the crossingpoint shown in Fig. 7d, given by¯ r cusp = ¯ r cross = (cid:114) λω − Γ . (29)It can be seen that the amplitude of the crossing pointis independent of the direct drive amplitude.When the stiffness nonlinearity is zero, as shownin Fig. 7, the dynamic response at zero detuning issymmetric about φ = 3 π/ nπ . The amplitude ex-tremum on the backbone curve has the steady-statephase of ¯ φ = 7 π/
4. The steady-state phase of the twofixed points at the crossing point of the loop, however,are symmetric about ¯ φ = 7 π/
4. The condition for which the response curve transi-tions from a single peak to dual peaks and a dimple iscalculated by setting both radicands of Eq. (13) equalto zero, yielding f flat = 2 λω (cid:114) λω − Γ . (30)This condition is interesting since it provides a rela-tively flat frequency response at resonance, that is, aminimal sensitivity to the amplitude to variations inthe frequency.The cusp condition can be obtained in a mannersimilar to that used for determining the isola con-dition. Specifically, it is obtained by simultaneouslysolving Eq. (13), ∂/∂ ¯ r of Eq. (13), and eliminating ¯ r ,which yields f cusp = λω (cid:114) λω − Γ . (31)For f ≥ f flat , the frequency response has only a singlepeak, as shown in Fig 7a. For f cusp < f < f flat , ithas a dimple and dual peaks, as shown in Fig 7b. For f = f cusp , the dimple has become a cusp, as shownin Fig 7c. Finally, for f < f cusp , it has is a loop, asshown in Fig 7d. Similar as before, in order for the11esponse curve to form a cusp or a loop, it is impliedthat λ > λ AT , .Fig. 8 demonstrates how these aforementioned am-plitudes change by varying the drive levels. The bluecurves show the amplitudes of the peaks, the orangecurves show the amplitude of the dimple, the purplecurves show the amplitude at the crossing point ofthe loop, and the gray curves show cases with oneof the drive levels at zero. The blue double lines in-dicate the dual peaks, while the orange double linesindicate that the crossing point consists of two fixedpoints with equal amplitude but different phases. InFig. 8 (a), as the parametric pump level is increased,the response curve transitions from a single peak toa dimple and dual peaks, and finally a loop can beclearly seen. The intersection between the orange andpurple curves indicates a supercritical pitchfork bifur-cation, which corresponds to the dimple transforminginto a loop via a cusp. In Fig. 8 (b), it can be seenthat ¯ r cross is independent of the direct drive ampli-tude. Additionally, by comparing to the gray curve( λ = 0), parametric suppression is also shown fromthe blue and orange solid curves.The process of obtaining the local bifurcation con-ditions is also very similar to that described in Sect.3.1. Fig. 9 provides examples of bifurcation dia-grams in some of the parameter planes of interest, with γ > < Γ ∗ ,while the latter case corresponds to Γ > Γ ∗ , whereΓ ∗ is given in Eq. (15). For Γ < Γ ∗ , increasing thedirect drive amplitude leads to the appearance and ex-pansion of the Duffing bistability, while for Γ > Γ ∗ ,increasing the direct drive amplitude leads to the con-traction and disappearance of the Duffing bistability.The bifurcations in the parameter space involvingthe parametric pump and the direct drive are shown in Fig. 9d. The orange curve represents the cusp bi-furcation condition associated with the dimple or loop,the blue curve represents the SN bifurcation conditionassociated with the Duffing bistability, the magentadashed curve indicates the appearance of the dimple,which is a condition described by Eq. (30), and thegreen dashed curve indicates the appearance of theloop, which is a condition described by Eq. (31). Theintercepts of the blue curve are the same as those pro-vided in Sect. 3.1, which are given by Eqs. (24)-(25). In this section, we consider the case where the systemparameters satisfy the condition given by Eq. (31) andwithout nonlinear stiffness, which is shown in Fig. 7c.For the present analysis, we assume that the stiffnessnonlinearity is zero. The conditions at the cusp pointare given in Sect. 4.1 as ¯ r cusp = (cid:112) ( λω − ) / Γ and ¯ φ cusp = − π/
4. This is a case of special inter-est because it provides infinite phase slope versus thefrequency detuning and therefore infinite sensitivityto fluctuations in phase. This is shown in Fig. 10a,which can be used to maximize phase sensitivity atresonance. This is also the supercritical pitchfork bi-furcation condition, as demonstrated in Fig. 8.In order to analyze the dynamics at the cusp point,it is convenient to use an alternative formulation forthe averaged equations, in particular, a Cartesianform, in contrast to the polar form in Eqs. (4)-(5).This formulation, which leads to simpler calculationsin this case, is given by x ( t ) = a ( t ) cos ( ω t ) + b ( t ) sin ( ω t ) , (32)˙ x ( t ) = − ω a ( t ) sin ( ω t ) + ω b ( t ) cos ( ω t ) . (33)where a ( t ) and b ( t ) are the slowly-varying Cartesiancoordinates, or quadratures. The two coordinate sys-tems are related in the usual manner: r = √ a + b and tan φ = − b/a . With the cusp condition givenby Eq. (31) satisfied, the averaged equations are asfollows˙ a = − Γ a −
14 Γ (cid:0) a + b (cid:1) a + λω (cid:16) √ r cusp − b (cid:17) , (34)˙ b = − Γ b −
14 Γ (cid:0) a + b (cid:1) b + λω (cid:16) √ r cusp − a (cid:17) . (35)The fixed point is ¯ a = ¯ b = (cid:112) ( λω − ) / (2Γ ), andits Jacobian matrix is J = (cid:20) Γ − λω Γ − λω Γ − λω Γ − λω (cid:21) , (36)with eigenvalues − ( λω − ) and 0. The eigenvec-tors are (cid:0) √ / , √ / (cid:1) T and (cid:0) √ / , −√ / (cid:1) T , which12 a ) f = = = =
0, peak λ r ( b ) λ = λ = λ = λ =
0, peak
Figure 8: Changes in the amplitudes on the backbone curve by varying the drive levels, with ψ = + π/ = 0 . = 0 . ω = 1, and γ = 0. The solid curves represent stable fixed points, and the dashedcurves represent saddle points, where a supercritical pitchfork bifurcation can be seen. (a) The transitionsfrom a single peak to dual peaks and then to a loop are shown. (b) Parametric suppression is demonstratedby comparing the gray curve with the others ( a ) - σ f ( b ) - σ f ( c )
12 2 3SN SNCP CP - σλ ( d ) CP CP0 0.05 0.1 0.15 0.200.020.040.06 f λ Figure 9: Bifurcation conditions shown in some parameter planes of interest, with ψ = + π/
4, Γ = 0 . ω = 1, and γ = 0 .
01. Orange and blue curves correspond to SN conditions associated with the isola and theDuffing bistability, respectively. The magenta and green dashed curves correspond to the appearance of thedimple and the loop, respectively. (a) Parameter plane ( σ, f ) for Γ = 0 .
005 and λ = 0 . σ, f ) for Γ = 0 .
009 and λ = 0 . σ, λ ) for Γ = 0 .
005 and f = 0 .
03. (d)Parameter plane ( f, λ ) for Γ = 0 . E s and E c ,respectively.To analyze the dynamics on the center manifold, a further coordinate transformation to local eigencoor-13 a ) - - - π - π π π σϕ ( b ) - - - - Figure 10: Infinite phase slope condition (cusp condition), with ψ = + π/
4, Γ = 0 . = 0 . ω = 1, λ ≈ . f = 0 .
03. (a) Infinite phase slope at resonance. (b) Dynamics in the corresponding phaseplane, showing the center manifolddinates is considered, given by (cid:20) ab (cid:21) = (cid:34) √ √ − √ √ (cid:35) (cid:20) uv + ¯ r (cid:21) . (37)This yields ˙ u = ˙ a/ √ − ˙ b/ √ v = ˙ a/ √ b/ √ u and v coordinates as follows˙ u = −
14 Γ (cid:0) u + v (cid:1) u − (cid:112) ( λω − ) Γ uv, (38)˙ v = − ( λω − ) v −
14 Γ (cid:0) u + v (cid:1) v − (cid:112) ( λω − ) Γ (cid:0) u + 3 v (cid:1) . (39)This dynamical system governed by Eqs. (38)-(39)can be written in the standard form of ˙ u = Au + f ( u, v ) and ˙ v = Bu + g ( u, v ), where A = 0, B = − ( λω − ), and f (0 ,
0) = Df (0 ,
0) = g (0 ,
0) = Dg (0 ,
0) = 0 for center manifold reduction [75].The phase plane is shown in Fig. 10b. With thestandard center manifold approach, v = h ( u ) is com-puted to the leading-order term h ( u ) = − (cid:112) ( λω − ) Γ λω − ) u + O (cid:0) u (cid:1) , (40)where the coefficients for all odd-order terms are iden-tically zero due to symmetry. The dynamics on thecenter manifold is then governed by˙ u = − Γ λω λω − ) u + O (cid:0) u (cid:1) , (41) where all the even-order terms vanish due to symme-try. Because the condition given by Eq. (31) mustsatisfy λ > λ AT , , the sign of ˙ u near u = 0 is negative,proving that the dynamics of the system on the slowmanifold is stable near this equilibrium. Therefore,when operating at this special condition with infinitephase slope, the system is weakly, that is, nonlinearly,dynamically stable. The relative phase between the direct drive and theparametric pump ( ψ ) is of vital importance. Not onlydoes it affect the structure of the frequency responsecurves and bifurcation diagrams, it also determinesthe parametric gain of the system. The parametricgain, as described in Eq. (1), reflects how much theparametric pump can amplify or suppress the peakresponse amplitude.Fig. 11 shows frequency response curves for sam-ple values of the parametric pump level and relativephase. Generally, for ψ (cid:54) = π/ nπ/
2, isolae, loops,and dimples can be off-centered and, as parameters arevaried, the loops can pinch off to form isolae. Whilethe structure of the response curves in the amplitude-frequency space can be diverse, there are still limita-tions on the possibilities. Specifically, there can onlybe up to one isola and one dimple or loop in the re-sponse curve. In addition, as mentioned in Sect. 2.1,the maximum number of fixed points is five, in whichcase three are stable equilibria and two are saddlepoints. Fig. 11a, b illustrate how an the isola becomes14 a ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲ λ = ψ = π - - σ r ( b ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲ λ = ψ = π - - σ r ▲ ▲ ▲▲▲▲ ( c ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = ψ = π - - σ r ▲ ▲ ▲▲▲▲ ( d ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = ψ = π - - σ r ▲▲▲▲▲ ( e ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = ψ = π - - σ r ▲ ▲ ▲ ▲ ( f ) ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ λ = ψ = π - - σ r Figure 11: Frequency response curves for various parametric pump levels and relative phases, with Γ = 0 . = 0 . ω = 1, γ = 0, and f = 0 .
01. The stability of the equilibria is not shown, but can be inferredfrom the aforementioned discussions. The triangles represent direct numerical simulation resultstangent to the main response branch and forms a loop.Fig. 11c, d illustrate how a dimple transforms into aloop via a cusp. Fig. 11e, f illustrate how a dimple isformed via an inflection point. Considered together,these transitions describe the complete evolutionarypossibilities for the frequency response curves.Fig. 12a provides an overview of the topology ofthe frequency responses that occur as one varies therelative phase ψ and how they change from one formto another. This figure shows the amplitudes of thelocal extrema that exist over the entire resonant fre-quency range for various parametric pump levels. Inthe figure, these amplitudes are normalized by the cor-responding amplitudes that would occur without thepump.In order to describe the possible transitions in thefrequency response curves, we start our discussion fo- cusing on the blue curve, corresponding to λ = 0 .
04, aportion of which is shown in the inset of Fig. 12a,and relate these to structures in the frequency re-sponse. This case shows all possible transitions andsets the stage for more special cases. First, for ψ near − π/ a ) AB C λ = λ = λ = λ = λ = λ = - π / π / π / π / ψ r e x t / r e x t λ = ( b ) λ = λ = λ = λ = - π / π / π / π / ψ r e x t / r e x t λ = Figure 12: Ratio of the amplitude local extrema with various parametric pump levels to those values ofextrema without the pump versus the relative phase ψ , with Γ = 0 . ω = 1, and f = 0 .
01. The highestratio in each color is considered to represent the parametric gain of the system. The stability is not indicated.The green curves in both diagrams have the parametric pump level of λ AT , , the instability threshold. (a)Γ = 0 . = 0different parameter values), which then develops intoa dimple. Such a dimple, without the isola, is shownin Fig. 11f. This dimple then forms into a cusp, asshown in the inset of Fig. 11c, which then transitionsinto a loop. (Note that this cusp transition, at thecondition given by Eq. (31), does not alter the num-ber of extrema and thereby is not indicated in Fig.11.) Such a loop, without the isola, is shown in Fig.11d. Therefore, between points B and C there existfive extrema. For this pump level, three of these arevery close to one another, as shown in the inset of Fig.11b. Here two extrema are associated with the isola(including the rightmost and uppermost of the threeshown in the inset) and two are from the loop (whichis very small in the inset). At point C, the two right-most and uppermost extrema shown in the inset ofFig. 11b merge, the outcome of which is a large loopthat replaces the isola. This process is reversed as ψ isincreased, due to the symmetry of the diagram about ψ = + π/
4. Note that for λ = 0 .
04, the transitionsnear points B and C occur very rapidly as ψ varies.For larger pump levels, for example, λ = 0 .
05 shownin Fig. 12a, point A no longer exists and the main isolaor associated loop exists for all values of ψ . As thepump level increases from 0 .
04 to 0 .
05, point A movestowards ψ = − π/
4, eventually reaching it at whichpoint it merges with its symmetric counterpart at thecondition given by Eq. (23), and then disappears.In this case, there are three extrema for all values of ψ , corresponding to either an isola or a dimple/loop.Points B and C still exist, resulting a dimple/loop thatexists around ψ = + π/
4, replacing the isola over a range of phases.For smaller pump levels, for example, λ = 0 . ψ = − π/ ψ is increased an inflection point is encoun-tered, resulting in a dimple/loop that exists around ψ = + π/
4, corresponding to three extrema. Herethere is a swallowtail structure [76] in the diagram.As the pump level is further decreased, the swallow-tail and the attendant range of ψ over which the dim-ple/loop exists continues to shrink until it eventuallydisappears at the inflection point condition given byEq. (30), in which case the swallowtail no longer ex-ists and there is a smooth response curve with a singlemaximum for all values of ψ .An additional feature of Fig. 12a is that it demon-strates that ψ = − π/ ψ = + π/ ψ = − π/ ψ = + π/ This paper describes in detail the frequency responseof systems with nonlinear damping and stiffness sub-jected to both direct and parametric near-resonantdriving. This generalizes and expands on previouswork in the area to include the effects of nonlineardamping, which is relevant to PA, for example, inmicro/nano-scale resonators.The results provide a thorough description of thepossible types of frequency responses that can be en-countered and how these depend on system and driveparameters. Of particular interest are how isolae,which were known to occur in these systems [9] andhave been experimentally observed [21, 22, 40, 44], areformed as the drive parameters, such as the relativephase ψ , are varied. New features for these systemsare also described herein, including loops and dim-ples that are closely related to the isolae, and ampli-tude response curves with degenerate flat resonancepeaks. The analysis provides a complete descriptionof how these features are related via transitions in-volving cusps, inflection points, and tangencies. Theseresults are useful, for example, by allowing one to se-lect drive conditions that provide maximum amplitudegain without bistability, maximum phase sensitivityat the desired vibration amplitude, or flat resonancepeaks with potential application for reducing ampli-tude noise, for given device parameters.It is expected that the features described in thispaper can be experimentally observed in any lightlydamped resonator with the described characteristics,which are quite generic. It is also of interest to con-sider the closed-loop, self-oscillating version of thissystem, with noise included in the model. Of particu-lar interest is the behavior of noise near the transitionpoints, for example, the flat resonance peak, where theeffects of certain noises on the system response mightbe amplified or attenuated. Acknowledgements
This work has been supported in part by NSF grantsNo. CMMI-1662619 and No. CMMI-1561829 and byFlorida Tech.
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