Local analysis of co-dimension-one and

co-dimension-two grazing bifurcations in

impact microactuators

Harry Dankowicz and Xiaopeng Zhao

Department of Engineering Science and Mechanics,Virginia Polytechnic Institute and State University,

Blacksburg, VA 24061, USA

Abstract

Impact microactuators rely on repeated collisions to generate gross displacementsof a microelectromechanical machine element without the need for large appliedforces. Their design and control rely on an understanding of the critical transitionbetween non-impacting and impacting long-term system dynamics and the associ-ated changes in system behavior. In this paper, we present three co-dimension-one,characteristically distinct transition scenarios associated with grazing conditions fora periodic response of an impact microactuator: a discontinuous jump to an impact-ing periodic response (associated with parameter hysteresis), a continuous transitionto an impacting chaotic attractor, and a discontinuous jump to an impacting chaoticattractor. Using the concept of discontinuity mappings, a theoretical analysis is pre-sented that predicts the character of each transition from a set of quantities that arecomputable in terms of system properties at grazing. Specifically, we show how thisanalysis can be applied to predict the bifurcation behavior on neighborhoods of twoco-dimension-two bifurcation points that separate the co-dimension-one bifurcationscenarios. The predictions are validated against results from numerical simulationsof a model impact microactuator.

Key words: Impact microactuators, grazing bifurcations, local analysis,discontinuity mappingsPACS:

1 Introduction

Precise displacement control and manipulation is required in microscopes,optical devices, and nanoscale data storage as well as during microsurgery.Adhesive forces, such as electrostatic forces, van der Waals forces, and surface

Preprint submitted to Physica D 28 February 2005

tension become significant when the characteristic length scale is less than amillimeter [8,22]. Therefore, manual assembly of microcomponents is difficultdue to sticking, as this causes inaccuracy in positioning.

Microactuators and microsystems have been extensively studied to obtain pre-cise positioning [1,2,4,5,12–17,19,22]. Regular microactuators used to producemeasurable displacements would need large actuation forces and a long drivingdistance. In contrast, actuators based on impulsive forces provide a solutionto this problem, as repeated impacts can be used to generate relatively largemotion from the accumulation of many small displacements [21]. Recently, im-pact microactuators have attracted a lot of attention due to ease of fabrication,capability of batch processing, robustness to environment, high accuracy, andhigh power [5,15–17,19,22]. For example, in the impact microactuator stud-ied by Mita et al [17], controlled excitation is applied to the relative motionbetween a frame resting on a substrate and a movable block. For sufficientlylarge excitation, the relative motion results in repeated impacts between theblock and the frame and resultant brief episodes of sliding motion of the framealong the substrate. Although each such sliding event contributes only on theorder of hundredths of microns of displacement, periodic excitation may beused to generate controlled gross sliding.

Analysis and design of proper function of an impact microactuator, such asthat developed by Mita et al [17] rely on an understanding of the possiblydramatic changes in system response that originate at the onset of impactingmotions as a system parameter µ (for example, the amplitude or frequencyof the excitation) is varied. Specifically, we are concerned with bifurcationsin the long-term system response that occur as µ increases past some criticalvalue µgrazing, at which value there exists a periodic non-impacting oscilla-tion of the movable block that achieves zero-relative-velocity contact withthe frame. In a state-space description of the dynamics of the microactuator,such zero-relative-velocity contact corresponds to a grazing contact between astate-space trajectory and a discontinuity surface, here representing the sud-den changes in the velocities of the frame and the movable block that resultfrom an impact. In contrast to periodic trajectories in smooth systems, thelocal description in the vicinity of a grazing trajectory is well-known to benon-differentiable with dramatic implications to the stability of the grazingtrajectory and to its persistence under further parameter variations [3,9,23].Indeed, as there is no advance warning of this instability, any local descrip-tion must account for the nonsmooth character of the flow near the grazingtrajectory.

The local dynamics in the vicinity of a grazing trajectory for µ ≈ µgrazing can beanalyzed through the introduction of a discontinuity mapping that i) capturesthe local dynamics in the vicinity of the grazing contact including variations intime-of-flight to the discontinuity and the impact mapping; ii) can be entirely

2

characterized by conditions at the grazing contact; iii) is nonsmooth in thedeviation from the point of grazing contact; and iv) can be studied to arbitraryorder of accuracy [20]. Properly formulated, the discontinuity mapping thusintroduces the correction to the otherwise smooth dynamics that is due to thebrief interaction with the discontinuity [7,10,11,18,20].

In this paper, we revisit the dynamics of a piecewise smooth model of an exper-imental impact microactuator and analyze the transition between impactingand non-impacting system behavior using the the discontinuity-mapping ap-proach. In particular, we show how three previously identified transition sce-narios correspond to co-dimension-one bifurcations and derive local mappingsthat accurately describe the dynamics in the vicinity of the grazing trajec-tory. Moreover, we identify the boundaries of the regions corresponding tothe co-dimension-one bifurcation scenarios with co-dimension-two bifurcationpoints and apply the discontinuity-mapping approach to formulate approx-imate mappings that capture the local dynamics on open neighborhoods ofthese boundary points.

Specifically, Section 2 reviews the piecewise smooth formulation of the math-ematical model of the impact microactuator and the bifurcation scenarios. InSection 3, we formulate the discontinuity-mapping methodology in the con-text of the impact microactuator. The local mappings derived in the vicinityof the two co-dimension-two bifurcation points are given in Section 4 togetherwith numerical simulations showing the agreement between the local mappingand the results of numerical simulations of the piecewise smooth system. Aconcluding discussion is presented in Section 5.

2 A Model Impact Microactuator

2.1 Mathematical model

A schematic two-degree-of-freedom model of a micromachined impact actuatorfabricated by Mita et al [17] is shown in Figure 1. The actual dimensions ofthe actuator are 3mm×3mm×0.6mm. Here, impulsive forces are generated aselectrostatic actuation results in collisions between a silicon micromass and astopper.

In the mechanical model of the Mita actuator shown above, the silicon mi-cromass is represented by a movable block of mass m2 that is connected toa frame of mass m1 by a linear spring and damper. The stoppers and theelectrode are rigidly fixed to the frame. The movable mass acts as one of theelectrodes. The frame is assumed to rest on a horizontal substrate. Friction

3

stopper

stopperelectrodes

V q2

q1

m2

m1

d

k

c

δδ

Fig. 1. Schematic of the Mita et al [17] impact microactuator.

between the frame and the ground is modelled using Coulomb friction dur-ing slip and Amonton’s law during stick. We denote the coefficient of staticfriction by µs and that of dynamic friction by µd.

When a driving voltage V (t) is applied between the electrodes, the movableblock is accelerated toward the stopper until an impact occurs with the stop-per. In the analysis below, we assume that the impact impulse is large enoughto overcome the static friction between the frame and the ground for all impactvelocities. As a result, an impact produces a small displacement of the frame.When a periodically varying voltage is applied, there are repeated impacts,thereby producing the needed displacement over some period of time.

The dynamics of the microactuator can be decomposed into distinct phasesseparated by the occurrence of impacts and the associated onset of slip aswell as the subsequent cessation of slip through an instantaneous transitionto stick. Specifically, we introduce the state vector

x =(

q1 u1 q2 u2 θ

)T

, (1)

where q1 and u1 are the displacement and velocity, respectively, of the framerelative to the ground; q2 and u2 are the displacement and velocity, respec-tively, of the movable block relative to the frame; and θ = ωt is the phase ofthe sinusoidally varying driving voltage V (t) = Vamp cos(ωt).

4

During stick, the equations of motion can then be written as

dx

dt= fstick (x) =

0

0

u2

1m2

(αV 2

amp cos2 θ

(d−q2)2− kq2 − cu2

)

ω

. (2)

Here, α = 12ε0 A, where ε0 is the permittivity of free space, A is the overlap

area, and d is the zero-voltage gap between the electrodes. These equations ofmotion are valid as long as

hfront (x) = q2 − δ < 0, (3)

hback (x) = q2 + δ > 0, (4)

hstick+ (x) = kq2 + cu2 −αV 2

amp cos2 θ

(d− q2)2− µsN < 0, (5)

and

hstick− (x) = kq2 + cu2 −αV 2

amp cos2 θ

(d− q2)2+ µsN > 0, (6)

where N is the normal reaction from the ground. We assume that gravity isthe only external force, in which case N = (m1 + m2)g, where g ≈ 9.81 m/s2

is the acceleration of gravity.

During slip, the equations of motion can be written as

dx

dt= fslip± (x) =

u1

1m1

F

u2

1m2

(αV 2

amp cos2 θ

(d−q2)2− kq2 − cu2

)− 1

m1F

ω

, (7)

where F =(kq2 + cu2 − αV 2

amp cos2 θ

(d−q2)2∓ µd N

)and we use the upper sign when

u1 > 0 and the lower sign when u1 < 0. Again, the corresponding equationsof motion are valid as long as

hfront (x) = q2 − δ < 0, (8)

hback (x) = q2 + δ > 0, (9)

andhslip (x) = u1 6= 0. (10)

5

At the moment that contact is established between the movable block and thestopper, hfront (x) or hback (x) equal zero. Assuming an inelastic collision with acoefficient of restitution e and using conservation of momentum, the functionthat maps the state immediately prior to impact to the state immediatelyafter impact is given by the jump map

gimpact (x) =(

q1 u1 + (1+e)m2

m1+m2u2 q2 −eu2 θ

)T

. (11)

The transition from slip to stick occurs as the velocity of the frame relativeto the ground becomes zero, i.e., as hslip (x) equals zero. Although there is adiscontinuous change in the vector field as a result of this transition, there isno associated instantaneous change of state, i.e.,

gstick (x) = x. (12)

In the numerical results reported below, we have normalized mass, length,

time, and voltage by m2, d,√

m2/k and V0, respectively, where V0 is somecharacteristic voltage. The nondimensional system parameter values used fornumerical computations are m1 = 5, m2 = 1, k = 1, c = 0.04, d = 1, δ = 0.5,e = 0.8, µs = 0.4, µd = 0.27, and α = 1. As shown in the authors’ previouspaper [24], these parameters are chosen to fit the experimental data reportedby Mita et al [17].

2.2 Transition Scenarios

The piecewise smooth dynamical system formulated above possesses two typesof recurrent solutions, namely, those with impacts (and thus sliding episodes)and those without impacts (and no sliding episodes). In a previous paper (Zhaoet al [24]), we have numerically investigated representative transitions betweenimpacting and non-impacting recurrent solutions that occur under changes insystem parameters. Here, we review the different transition scenarios foundpreviously as well as describe additional scenarios that occur under parametervariations through isolated points in the two-dimensional parameter spacegiven by Vamp and ω.

In the absence of forcing, the only recurrent motion available to the impact mi-croactuator is the trivial equilibrium state. For sufficiently small, but nonzero,values of Vamp the movable mass exhibits a periodic oscillation of the same pe-riod as that of the forcing term (note that the voltage input appears squaredin the forcing term, i.e., the angular frequency of the forcing term equals 2ω)but without impacting the front or back stopper. Indeed, as the forcing termis always in the positive direction, the oscillatory motion of the movable mass

6

is shifted in the direction of positive values of q2. It follows that, under in-creasing values of Vamp, impacting motions that impact with q2 = δ will occurbefore impacting motions that impact with q2 = −δ. In this and the previouspaper, we therefore focus on recurrent motions that only impact with q2 = δand, consequently, for which the relative velocity between the frame and thesubstrate is positive during sliding episodes.

For notational convenience, consider the following notation

D ={x

∣∣∣ hD (x)def= hfront (x) = 0

}(13)

D+ ={x ∈ D

∣∣∣ fstick (x) • hDx (x) = u2 > 0}

(14)

D0 ={x ∈ D

∣∣∣ fstick (x) • hDx (x) = u2 = 0}

(15)

D− ={x ∈ D

∣∣∣ fstick (x) • hDx (x) = u2 < 0}

. (16)

As per the previous discussion, for sufficiently small values of Vamp, the recur-rent motion is contained within the region hD (x) < 0. Since gimpact maps D+

to D−, trajectories that reach D+ experience an instantaneous jump to D− (asthe incoming velocity u2 > 0 is changed to an outgoing velocity −eu2 < 0).By a grazing periodic trajectory, we refer to a periodic trajectory on whichthere exists a locally unique point x∗ ∈ D0, since such a trajectory experi-ences no jump in state upon reaching D. As discussed in Zhao et al [24], wecan use a Newton method (see appendix) to numerically locate parametervalues for ω and Vamp for which grazing periodic trajectories exist. Indeed,over the region of interest in this paper (0.46 ≤ ω ≤ 0.51), we find a curve

Γ ={(ω, Vamp)

∣∣∣ Vamp = V ∗amp (ω)

}in parameter space, such that there exists

a grazing periodic trajectory of the fundamental period of the forcing term forevery choice of parameters on Γ (see Figure 4, below).

We now consider recurrent dynamics of the impact microactuator on someneighborhood in parameter space of arbitrary points on the grazing curveΓ. To visualize the results of the numerical study, we introduce a Poincaresection P corresponding to the zero-level surface of the event function hP (x) =fstick (x)• hDx (x) = u2 for u2 decreasing. (Note that P ∩ D is contained in D0.)In the absence of impacts, points on P correspond to local maxima in the valueof q2 along system trajectories (since q2 = u2). In the presence of impacts,trajectories that reach D+ jump across P without intersecting P . We representsuch crossings by the virtual point of intersection with P of the correspondingforward trajectory segment in the absence of the jump in velocity, as suggestedin Figure 2.

Figures 3a)-c) show three distinct bifurcation scenarios corresponding to transver-sal one-parameter variations across some selected points on Γ. Further suchnumerical studies establish the existence of three distinct regions on Γ, re-ferred to here as ΓI , ΓII , and ΓIII , for which the bifurcation scenarios in the

7

D+

D¡

D0

P

gimpact

Fig. 2. Intersections of nonimpacting and impacting trajectories with the Poincaresection P. The nonimpacting trajectory reaches its local maximum in q2 at the in-tersection. The impacting trajectory reaches P virtually by neglecting the existenceof the discontinuity surface D.

different panels of Figure 3 are representative.

As shown in Figure 3a), variations in Vamp across ΓI are associated with adiscontinuous transition of the asymptotic motion from a nonimpacting to animpacting periodic trajectory as Vamp is increased above the grazing parametervalue V ∗

amp (ω); and from an impacting to a nonimpacting periodic trajectoryas Vamp is decreased below V sn1

amp (ω) < V ∗amp (ω) corresponding to a cyclic fold

bifurcation. For Vamp ∈(V sn1

amp, V∗amp

), three distinct periodic trajectories (one

nonimpacting and two impacting) coexist. Of these, the nonimpacting tra-jectory and the larger-impact-velocity impacting trajectory are stable, whilethe lower-impact-velocity impacting trajectory is unstable. The coexistenceof multiple attractors implies the possibility of parameter hysteresis in thelong-term response of the impact actuator.

Variations in Vamp across ΓII are associated with a continuous transition of theasymptotic dynamics from a nonimpacting periodic trajectory to an impact-ing chaotic attractor and back as Vamp is increased above and decreased belowV ∗

amp (ω). In contrast to the transition across ΓI , the nonimpacting trajectorypersists beyond the grazing contact (albeit with impacts and sliding episodes),but experiences a discontinuous change in stability as one of its eigenvaluesjumps to −∞. Under further increases in Vamp, this unstable impacting pe-riodic trajectory becomes stable again in a period-doubling bifurcation atV pd

amp (ω). Moreover, as suggested in the magnified portion of Figure 3 d), theimpacting chaotic attractor undergoes an inverse period-doubling cascade toan impacting period-two trajectory that connects to the branch of period-twotrajectories born at the period-doubling bifurcation through a grazing bifur-

8

V¤

amp Vamp

SN

G

penetration

0

V¤

amp Vamp

PDG

penetration

0

V¤

amp Vamp

penetration

0G

SN

PD

C

a) b) c)

Vamp

penetration

0

PD

V¤

amp Vamp

penetration

0

C

G

d) e)

Fig. 3. Schematic bifurcation scenarios associated with the switching between im-pacting motions (black) and non-impacting motions (gray) (G=grazing contact,SN=saddle-node bifurcation, PD=period-doubling bifurcation, C=global crisis).Here, solid curves correspond to stable periodic motions and dashed curves to unsta-ble periodic motions. The black regions correspond to impacting chaotic attractors.Panel a), b), and c) describe the transitions under variations in Vamp across ΓI , ΓII ,and ΓIII , respectively. Panel d) and e) show enlargement of the corresponding areasin b) and c), respectively.

cation. The continuous nature of the transitions between system attractorsin the case of variations across ΓII implies that parameter hysteresis cannotoccur.

Finally, variations in Vamp across ΓIII are associated with a discontinuous tran-sition of the asymptotic dynamics from a nonimpacting periodic trajectory toan impacting chaotic attractor as Vamp is increased above V sn2

amp (ω) > V ∗amp (ω) ;

and from an impacting chaotic attractor to a nonimpacting periodic trajectoryas Vamp is decreased below V cr

amp (ω) < V sn2amp (ω), where V ∗

amp (ω) < V cramp (ω) ,

corresponding to the disappearance of the basin of attraction of the chaoticattractor in a global crisis bifurcation. For Vamp ∈

(V ∗

amp (ω) , V sn2amp (ω)

), three

distinct periodic trajectories (two nonimpacting and one impacting) coexist.Of these, the unstable impacting trajectory originates in the grazing contactand persists as Vamp is increased beyond V sn2

amp (ω) . It eventually becomes stablein a period-doubling bifurcation at V pd

amp (ω). Of the nonimpacting trajectories,the small-amplitude motion is stable, while the large-amplitude motion (thatexperiences grazing contact at V ∗

amp (ω) and that collides with the chaotic at-tractor at V cr

amp (ω)) is unstable. Again, the coexistence of multiple attractorsimplies the possibility of parameter hysteresis in the long-term response of theimpact actuator.

From these scenarios, we find it useful to collect in a diagram of parameter

9

space, the grazing curve Γ, the locus of the saddle-node bifurcations that oc-cur when crossing ΓI , the locus of the period-doubling bifurcations that occurwhen crossing ΓII and ΓIII , the locus of the saddle-node bifurcations thatoccur when crossing ΓIII , and the locus of the global crisis of the chaoticattractor that occurs when crossing ΓIII . As can be seen from Figure 4, thesaddle-node bifurcation curve associated with ΓI and the period-doubling bi-furcation curve appear to terminate at the boundary between ΓI and ΓII andto be tangent to Γ at this point. Similarly, the crisis and saddle-node bifurca-tion curves associated with ΓIII terminate at the boundary between ΓII andΓIII and are tangent to Γ at that point.

0:34

0:18

¡II

¡II

¡I¡I

¡III

V cramp (!)

V pdamp (!)

V pdamp

V sn1amp (!)

V sn2amp (!)

0.22

0.26

0.30

0.480.46 0.50

Vamp

a)V sn1amp

!

¢!0

0

4

8

-0.8 0.8

£10¡5

£10¡4

b)(!)

Vamp¡V¤ amp

¡II¡III

V cramp

V sn2amp

(!)

¢!

0

1

2

0-0.4 0.4

£10¡5

c)

Vamp¡V¤ amp

£10¡3

Fig. 4. Collection of bifurcation curves in the parameter space, where Γ representsthe grazing curve; V sn1

amp represents the locus of saddle-node bifurcations when cross-ing ΓI ; V pd

amp represents the locus of period-doubling bifurcations when crossing ΓII

and ΓIII ; V sn2amp represents the locus of saddle-node bifurcations when crossing ΓIII ;

V cramp represents the locus of the global crisis of the chaotic attractor when crossing

ΓIII . Panels b) and c) show enlargements of neighborhoods of the boundary pointbetween ΓI and ΓII and the boundary point between ΓII and ΓIII , respectively.

Additional transition scenarios may be found by considering transversal one-parameter variations across the boundary points between ΓI and ΓII and be-tween ΓII and ΓIII , respectively, see Figure 5a)-b). As seen in Figure 5a) andin contrast to the transitions across ΓI or ΓII , crossing the boundary point be-tween ΓI and ΓII is associated with a continuous transition of the asymptoticdynamics from a nonimpacting periodic trajectory to an impacting periodictrajectory and back as Vamp is increased above and decreased below V ∗

amp (ω).The transition in Figure 5b), on the other hand, shows significant similar-ities with the transition across ΓII , although here the grazing trajectory isnonhyperbolic. As the additional transition scenarios occur only near isolatedpoints in the ω−Vamp parameter space, we refer to the corresponding grazingbifurcations as co-dimension two, thereby distinguishing them from the co-dimension-one bifurcations associated with passage through ΓI , ΓII , or ΓIII .

10

V¤

amp Vamp

penetration

0G

V¤

amp Vamp

penetration

0G

PD

a) ω = 0.4878 b) ω = 0.4803

Fig. 5. Schematic bifurcation scenarios associated with the switching between im-pacting motions (black) and non-impacting motions (gray), at the boundary be-tween ΓI and ΓII in Panel a) and those at the boundary between ΓII and ΓIII inPanel b) (G=grazing contact, PD=period-doubling bifurcation). Here, solid curvescorrespond to stable periodic motions and dashed curves to unstable periodic mo-tions. The black regions correspond to impacting chaotic attractors.

We note that bifurcation scenarios similar to those observed here in the case ofΓI and ΓII have previously been documented in piecewise smooth dynamicalsystems, for example, in linear impact oscillators (Foale & Bishop [9]) andin friction oscillators (Dankowicz & Nordmark [6]). The transition scenarioassociated with ΓIII is inherent in the nonlinearity of the electrostatic forcesand is a consequence of a dynamic pull-in phenomenon in electrostaticallyattracting microelectrodes.

The goal of the following sections is to develop closed-form maps that approx-imate the dynamics near the grazing trajectory in state and parameter spaceand that allow accurate predictions of the long-term response without the needto simulate the original piecewise smooth system of differential equations. In-deed, as simulations of recurrent motions that involve many near-zero-velocityimpacts require very high numerical accuracy and long integration times, suchclosed-form maps can be enormously beneficial.

3 Discontinuity mappings

We wish to associate a Poincare mapping P with the Poincare section Pintroduced above.

Ignore, for a moment, the jump map associated with the discontinuity D andassume that the dynamics are governed entirely by the vector field fstick andconstrained to a constant-q1 slice of the submanifold S corresponding to thezero-level surface of the event function hS (x) = u1. Suppose that for somevalues ω = ωref and Vamp = V ref

amp, the forward trajectory based at a point

11

xref ∈ P intersects P transversally after some time tref, i.e., that

hP(Φstick

(xref, tref; ωref, V ref

amp

))= 0 (17)

and

hP,i(Φstick

(xref, tref; ωref, V ref

amp

))· f i

stick

(Φstick

(xref, tref; ωref, V ref

amp

))< 0. (18)

where Φstick is the smooth flow corresponding to the vector field fstick. Here,the latter condition corresponds to the requirement that u2 be decreasing, i.e.,that the acceleration of the movable mass relative to the frame be negative.

Now introduce the function

F (x, t, ω, Vamp) = hP (Φstick (x, t; ω, Vamp)) . (19)

It follows thatF

(xref, tref, ωref, V ref

amp

)= 0 (20)

andFt

(xref, tref, ωref, V ref

amp

)< 0. (21)

The implicit function theorem implies that there exists a unique smooth func-tion τ (x, ω, Vamp) defined on a neighborhood of

(xref, ωref, V ref

amp

), such that

τ(xref, ωref, V ref

amp

)= tref (22)

andF (x, τ (x, ω, Vamp) , ω, Vamp) ≡ 0, (23)

i.e., τ (x, ω, Vamp) is the time of flight from x back to P . A smooth Poincare

mapping P can now be defined on a neighborhood of(xref, ωref, V ref

amp

)by the

expression

P (x, ω, Vamp) = Φstick (x, τ (x, ω, Vamp) ; ω, Vamp) . (24)

If we reintroduce the nontrivial jump map gimpact associated with D, the aboveexpression is still valid as long as hD (x) ≤ 0. If, instead, hD (x) > 0, werecognize that the point x corresponds to a virtual point of intersection asper the definition in the previous section that cannot actually be reached bythe piecewise smooth dynamical system. It would, nevertheless, be convenientif we could again define the Poincare mapping P by the above formula, butpossibly including an initial correction to account for the virtual nature of theinitial point x. To this end, consider the Poincare mapping P defined by

P (x, ω, Vamp) = Φstick (D (x, ω, Vamp) , τ (D (x, ω, Vamp) , ω, Vamp) ; ω, Vamp) ,(25)

12

where the discontinuity mapping D maps x to some point on P in such away that the subsequent dynamics respect those of the corresponding actualtrajectory (For more discussion of the concept of discontinuity mappings andtheir derivation, see also [6,7,10,18,20]).

To arrive at an expression for D consider the trajectory segments shown inFigure 6. Here, an incoming trajectory (solid) in S governed by the vectorfield fstick reaches the discontinuity surface D at a point xin ∈ D+, experiencesa jump to a point gimpact (xin) ∈ D−, flows under the vector field fslip+ untilreaching the stick manifold S at a point xout, and then continues to flow inS under the vector field fstick. The dashed trajectory segments correspond toa flow in S governed by the vector field fstick from xin forward in time untilreaching P at a point x0; and from xout backward in time until reaching P ata point x1. The sought correction to the smooth flow given by Φstick is thenobtained by mapping x0 to x1, as this correctly accounts for the effects of thejump map and the subsequent sliding episode.

S

D

P

xin

xout

x0x1

gimpact

Fig. 6. Trajectories associated with the discontinuity mapping D. Here S stands forthe stick manifold, D is the discontinuity surface, and P is the Poincare section.We note that S, D, and P are all 4 dimensional hyper surfaces in a 5 dimensionalspace.

Thus, given an initial point x on P , such that hD (x) > 0, we define D as thecomposition of the following steps:

(1) Flow for a time t1 < 0 with the vector field fstick until reaching D;(2) Apply the jump map gimpact;(3) Flow for a time t2 > 0 with the vector field fslip+ until reaching S;(4) Flow for a time t3 < 0 with the vector field fstick until reaching P .

To arrive at a functional expression for D, we seek to express the flow timesin terms of the corresponding initial conditions in state space and the corre-sponding parameter values.

13

Suppose, in particular, that for ω = ω∗ and Vamp = V ∗amp, there exists a point

x∗, such that

hD (x∗) = 0, (26)

hP (x∗) = hD,i (x∗) · f istick

(x∗, ω∗, V ∗

amp

)= 0, (27)

hS (x∗) = 0, (28)

hS,i (x∗) · f islip+

(x∗, ω∗, V ∗

amp

)< 0, (29)

and

a∗(ω∗, V ∗

amp

)def= hP,j (x∗) · f j

stick

(x∗, ω∗, V ∗

amp

)

=

hD,ij (x∗) · f istick

(x∗, ω∗, V ∗

amp

)

+hD,i (x∗) · f istick,j

(x∗, ω∗, V ∗

amp

)

· f j

stick

(x∗, ω∗, V ∗

amp

)< 0.

Clearly, x∗ ∈ P ∩S is a point of simple grazing contact with D correspondingto a local maximum in hD along a trajectory segment of the vector field fstickbased at x∗.

Step 1. Suppose that hD (x) > 0 and consider the function

E(1) (x, y, t, ω, Vamp) = t

√hD (x)− hD (Φstick (x,−t; ω, Vamp))− thP (x)

t2−y.

(30)Since

hD (x)− hD (Φstick (x,−t; ω, Vamp))− thP (x)

t2(31)

is bounded in t for t ≈ 0 (as seen by Taylor expanding the numerator), itfollows that

E(1)(x∗, 0, 0, ω∗, V ∗

amp

)= 0 (32)

and

E(1)t

(x∗, 0, 0, ω∗, V ∗

amp

)=

√√√√−a∗

(ω∗, V ∗

amp

)

2. (33)

The implicit function theorem now implies the existence of a unique smoothfunction τ (1) (x, y, ω, Vamp) on an open neighborhood of

(x∗, 0, ω∗, V ∗

amp

),

such that

τ (1)(x∗, 0, ω∗, V ∗

amp

)= 0 (34)

and

E(1)(x, y, τ (1) (x, y, ω, Vamp) , ω, Vamp

)≡ 0. (35)

It follows from the definition of E(1) that t1 = −τ (1)(x,

√hD (x), ω, Vamp

).

As the smooth function τ (1) (x, y, ω, Vamp) is implicitly defined by Eq. (35),

14

we can compute arbitrary partial derivatives of τ (1) at(x∗, 0, ω∗, V ∗

amp

)us-

ing implicit differentiation and demanding that all partial derivatives of E(1)

must vanish at(x∗, 0, ω∗, V ∗

amp

). Now let

D1 (x, y, ω, Vamp) = Φstick

(x,−τ (1) (x, y, ω, Vamp) ; ω, Vamp

). (36)

Since Φstick is smooth, we can compute arbitrary partial derivatives of D1

at(x∗, 0, ω∗, V ∗

amp

). Step 1 is completed by expanding D1 (x, y, ω, Vamp)

to desired order in the deviation from(x∗, 0, ω∗, V ∗

amp

)and substituting

y =√

hD (x).

Step 2. Since gimpact is smooth, it can be expanded to desired order in the

deviation from(x∗, ω∗, V ∗

amp

), after which the result of step 1 may be sub-

stituted for x thus completing step 2.Step 3. Now consider the function

E(2) (x, t, ω, Vamp) = hS (Φslip+ (x, t; ω, Vamp)) , (37)

where Φslip+ is the flow corresponding to the vector field fslip+. Then,

E(2)(x∗, 0, ω∗, V ∗

amp

)= hS (x∗) = 0 (38)

andE

(2),t

(x∗, 0, ω∗, V ∗

amp

)= hS,i (x∗) · f i

slip+

(x∗, ω∗, V ∗

amp

)< 0. (39)

The implicit function theorem again implies the existence of a unique smoothfunction τ (2) (x, ω, Vamp) on an open neighborhood of

(x∗, ω∗, V ∗

amp

), such

thatτ (2)

(x∗, ω∗, V ∗

amp

)= 0 (40)

andE(2)

(x, τ (2) (x, ω, Vamp) , ω, Vamp

)≡ 0. (41)

It follows from the definition of E(2) that t2 = τ (2) (x, ω, Vamp).

As the smooth function τ (2) (x, ω, Vamp) is implicitly defined by Eq. (41),

we can compute arbitrary partial derivatives of τ (2) at(x∗, ω∗, V ∗

amp

)using

implicit differentiation and demanding that all partial derivatives of E(2)

must vanish at(x∗, ω∗, V ∗

amp

). Now let

D2 (x, ω, Vamp) = Φslip+

(x, τ (2) (x, ω, Vamp) ; ω, Vamp

). (42)

Since Φslip+ is smooth, we can compute arbitrary partial derivatives of D2 at(x∗, ω∗, V ∗

amp

). Step 3 is completed by expanding D2 (x, ω, Vamp) to desired

order in the deviation from(x∗, ω∗, V ∗

amp

)and substituting the result of step

2 for x.

15

Step 4. Finally, consider the function

E(3) (x, t, ω, Vamp) = hP (Φstick (x,−t; ω, Vamp)) , (43)

Then,E(3)

(x∗, 0, ω∗, V ∗

amp

)= hP (x∗) = 0 (44)

andE

(3),t

(x∗, 0, ω∗, V ∗

amp

)= −a∗

(ω∗, V ∗

amp

)> 0. (45)

The implicit function theorem again implies the existence of a unique smoothfunction τ (3) (x, ω, Vamp) on an open neighborhood of

(x∗, ω∗, V ∗

amp

), such

thatτ (3)

(x∗, ω∗, V ∗

amp

)= 0 (46)

andE(3)

(x, τ (3) (x, ω, Vamp) , ω, Vamp

)≡ 0. (47)

It follows from the definition of E(3) that t3 = −τ (3) (x, ω, Vamp).

As the smooth function τ (3) (x, ω, Vamp) is implicitly defined by Eq. (47),

we can compute arbitrary partial derivatives of τ (3) at(x∗, ω∗, V ∗

amp

)using

implicit differentiation and demanding that all partial derivatives of E(3)

must vanish at(x∗, ω∗, V ∗

amp

). Now let

D3 (x, ω, Vamp) = Φstick

(x,−τ (3) (x, ω, Vamp) ; ω, Vamp

). (48)

Since Φstick is smooth, we can compute arbitrary partial derivatives of D3 at(x∗, ω∗, V ∗

amp

). Step 4 is completed by expanding D3 (x, ω, Vamp) to desired

order in the deviation from(x∗, ω∗, V ∗

amp

)and substituting the result of step

3 for x.

Now consider the compositions

D (x, y, ω, Vamp) =

=

Id when hD (x) ≤ 0

D3 (D2 (gimpact (D1 (x, y, ω, Vamp)) , ω, Vamp) , ω, Vamp) when hD (x) > 0

andP (x, ω, Vamp) = Φstick (x, τ (x, ω, Vamp) ; ω, Vamp) . (49)

It follows that

D (x, ω, Vamp) = D(x,

√hD (x), ω, Vamp

)(50)

andP (x, ω, Vamp) = P (D (x, ω, Vamp) , ω, Vamp) . (51)

16

We note that although D and P are smooth on an open neighborhood of(x∗, 0, ω∗, V ∗

amp

)and

(x∗, ω∗, V ∗

amp

), respectively, it is not true that D and P

are smooth on an open neighborhood of(x∗, ω∗, V ∗

amp

)due to the substitution

y =√

hD (x).

While P results in no change in q1, the inclusion of the jump mapping gimpact

and the associated sliding episode yields a discrete change in q1 containedwithin the discontinuity mapping D and, consequently, in the composite Poincaremapping P. As the system is invariant under variations in q1 and as, for low-velocity impacts P (x) ∈ P ∩ S, we consider the reduced system obtained byrestricting attention to the third and fifth components of P.

4 Numerical results

We now apply the discontinuity-mapping approach to yield explicit numericalexpressions for the composite Poincare mapping P (x, ω, Vamp) on neighbor-hoods of selected grazing periodic trajectories as found numerically using theNewton approach outlined in the appendix and described previously. For con-venience, we let x = x∗ + ∆x, ω = ω∗ + ∆ω, and Vamp = V ∗

amp + ∆Vamp, anduse (∆x, ∆ω, ∆Vamp) as local coordinates.

The grazing periodic trajectory at the co-dimension-two point at the boundarybetween ΓI and ΓII corresponds to

ω∗ ≈ 0.487809, V ∗amp ≈ 0.171067 and x∗ ≈

(· 0 0.5 0 5.496544

)T

. (52)

Following the methodology in the previous section and the comments made

17

in the last paragraph of the previous section, we find

P (x, ω, Vamp)− x∗ =

=

0.936621∆q2 + 0.775788 ∆Vamp − 0.174538 ∆ω

+ (0.111825 ∆q2 − 0.0103004 ∆θ + 0.960382 ∆Vamp − 0.627897 ∆ω) ∆q2

− (0.103166 ∆θ + 0.14153 ∆Vamp + 0.669646 ∆ω) ∆θ

+3.45355 ∆V 2amp − 1.52077 ∆ω2 − 3.21475 ∆Vamp ∆ω

0.142633 ∆q2 + 0.825198 ∆θ + 0.98176 ∆Vamp + 5.82623 ∆ω

+ (0.394304 ∆q2 − 0.148787 ∆θ + 2.22656 ∆Vamp − 0.441188 ∆ω) ∆q2

+ (0.0138705 ∆θ − 2.25913 ∆Vamp − 0.53266 ∆ω) ∆θ

+4.23226 ∆V 2amp − 1.82448 ∆ω2 − 7.41318 ∆Vamp ∆ω

(53)

and

D (x, ω, Vamp)− x∗ =

=

Id when ∆q2 ≤ 0

0.237147 ∆q2

−1.55767√

∆q2 − 0.189248 ∆q2 + ∆θ

+ (0.206574 ∆θ − 1.21057 ∆Vamp − 3.1932 ∆ω)√

∆q2

when ∆q2 > 0,

(54)

where we have omitted terms of order three or higher in√

∆q2, ∆θ, ∆ω, and∆Vamp.

Figures 7, 8, and 9 show the predictions of the local map P = P ◦D on thebifurcations associated with variations in ∆Vamp for select values of ∆ω corre-sponding to the schematic transition scenarios across ΓI , ΓII , and the bound-ary point between ΓI and ΓII as described in Section 2. While the predictedresults are in excellent agreement with the simulated data for the original dy-namical system, we recognize that the former were obtained at a fraction ofthe computing time required to generate the latter.

The local map shown above is valid on an open neighborhood of the co-dimension-two point at the boundary between ΓI and ΓII . It follows that itshould be able to capture the saddle-node bifurcation curve associated withΓI and the period-doubling bifurcation curve associated with ΓII . In fact, theexplicit nature of the local map allows the determination of the bifurcationcurves as Taylor expansions in the perturbation ∆ω. Specifically, we find that

18

0 0.5

¢q 2

-0.4

0

¢Vamp

£10¡3

£10¡3

0 0.5

0.4

0

imaginary part

real parteigen

valu

es

£10¡3

¢Vamp

a) b)

Fig. 7. Comparison of the bifurcation results predicted from the local map withthose obtained through numerical integration, when ω is at the boundary betweenΓI and ΓII . Panel a) shows variation of ∆q2 with the applied voltage, where theimpacting motions are in black and the nonimpacting motions are in gray. Wenote that the predicted results are indistinguishable from the numerical results.Panel b) shows the dependence of the real and imaginary parts of the eigenvaluesof the periodic impacting orbit on the applied voltage as obtained from numericalsimulations (solid) and as predicted by the local mapping (dashed).

£10¡4

£10¡4

eigenvalues

eigenvalues

¢Vamp

¢q 2

0

0.8

0.4

-2 -1£10¡4

¢Vamp

0

20

50

0.4

0.2

0

0-2 -110

a)

b)

1.0

1.6

0.4

c)

Fig. 8. Comparison of the bifurcation results predicted from the local map withthose obtained through numerical simulation, when ω = 0.4888. Panel a) showsvariation of ∆q2 with the applied voltage, where the impacting motions are in black,the nonimpacting motions are in gray, solid curves correspond to stable periodicmotions, and dashed curves correspond to unstable periodic motions. Panels b) andc) show the dependence of the eigenvalues of the periodic impacting orbit on theapplied voltage as obtained from numerical simulations (dots for stable and circlesfor unstable) and as predicted by the local mapping (solid for stable and dashed forunstable).

the saddle-node bifurcation curve is given by

∆Vamp = 0.224982 ∆ω + 17.7774 ∆ω2 +O(

∆ω3), ∆ω > 0 (55)

19

0 0.02 0.04 0.06

-0.5

0.5

0

1

¢q 2

£10¡3

p¢Vamp

0 0.02 0.04 0.06

-0.5

0.5

0

1

¢q 2

£10¡3

p¢Vamp

a) b)

p¢Vamp

0 0.060.040.02

-0.2

0

eigenvalues

p¢Vamp

0 0.02 0.04 0.06

-20

-40

0

eigenvalues

c) d)

Fig. 9. Comparison of the bifurcation results obtained through numerical simulation(Panel a)) with those predicted from the local map (Panel b)), when ω = 0.485.The impacting motions are in black, the nonimpacting motions are in gray, solidcurves correspond to stable periodic motions, dashed curves correspond to unstableperiodic motions, and black regions correspond to chaotic attractors. Panels c) andd) show the dependence of the eigenvalues of the periodic impacting orbit on theapplied voltage as obtained from numerical simulations (dots for stable and circlesfor unstable) and as predicted by the local mapping (solid for stable and dashed forunstable).

and the period-doubling curve is given by

∆Vamp = 0.224982 ∆ω + 1557.35 ∆ω2 +O(

∆ω3), ∆ω < 0. (56)

Figures 10 a) and b) compare the predicted bifurcation points using the localmap and the predicted bifurcation curves using the truncated Taylor expan-sions, respectively, with the actual bifurcation points obtained from directnumerical simulation of the original dynamical system. Again, it is clear thatthe local map provides a very accurate description of the dynamics in thevicinity of the grazing periodic trajectory in state and parameter space.

The grazing periodic trajectory at the co-dimension-two point at the boundarybetween ΓII and ΓIII corresponds to

ω∗ ≈ 0.480271, V ∗amp ≈ 0.176796 and x∗ ≈

(· 0 0.5 0 5.286519

)T

. (57)

20

¢!

0£10¡3

1-1

0

0.4

0.8

£10¡3

Vamp¡V¤ amp(!)

¢!

0£10¡3

1-1

0

0.4

0.8

£10¡3

Vamp¡V¤ amp(!)

a) b)

Fig. 10. Bifurcation curves obtained numerically (Panels a) and b)), compared tothose predicted from the local map a)), and those predicted from the Taylor expan-sions b)). In both panels, triangles, circles, and ”+” correspond to the numericallyobtained grazing, period doubling, and saddle node bifurcation points, respectively;the solid lines, dashed lines, and dotted lines correspond to the predicted grazing,period doubling, and saddle node bifurcation points, respectively.

Again, following the methodology in the previous section and the commentsmade in the last paragraph of the previous section, we find

P (x, ω, Vamp)− x∗ =

=

0.94297 ∆q2 + 0.0512257 ∆θ + 0.809064 ∆Vamp + 0.00715561 ∆ω

+ (0.138282 ∆q2 + 0.0544217 ∆θ + 1.24226 ∆V amp − 0.431478 ∆ω) ∆q2

− (0.105538 ∆θ − 0.426753 ∆Vamp + 0.888627 ∆ω) ∆θ

+4.24501 ∆V 2amp − 2.28863 ∆ω2 − 1.16325 ∆Vamp ∆ω

0.192815 ∆q2 + 0.826809 ∆θ + 1.5879 ∆Vamp + 6.00327 ∆ω

+ (0.724366 ∆q2 + 3.29973 ∆Vamp − 0.342789 ∆ω − 0.136173 ∆θ) ∆q2

− (0.0565448 ∆θ + 2.32779 ∆V amp + 1.15147 ∆ω) ∆θ

+7.2304971806356110 ∆V 2amp − 3.61189 ∆ω2 − 6.14775 ∆Vamp ∆ω

(58)

21

and

D (x, ω, Vamp)− x∗ =

=

Id when ∆q2 ≤ 0

0.238028 ∆q2

−1.58753√

∆q2 − 0.204934 ∆q2 + ∆θ

+ (0.219773 ∆θ − 1.92188 ∆Vamp − 3.30549 ∆ω)√

∆q2

when ∆q2 > 0,

(59)

where we have again omitted terms of order three or higher in√

∆q2, ∆θ, ∆ω,and ∆Vamp.

Figures 11 and 12 show the excellent agreement between the predictions ofthe local map P = P ◦D and the result of direct numerical simulations ofthe original dynamical system. Again, we may choose to use the local mapto predict the locus of the saddle-node and global crisis bifurcations. Figure13 shows predicted bifurcation points and corresponding bifurcation pointsobtained using the simulated data. The observation made previously regardingcomputation time applies again. Indeed, while the predicted bifurcation valueswere obtained in a matter of minutes, the simulated data required severaldays of computations. Using the predicted data as initial conditions for thenumerical simulation, the computation time was reduced to a couple of hours.

0 0.04 0.08 0.12

¢q 2

-2

2

0

4

£10¡3

p¢Vamp

0 0.04 0.08 0.12

¢q 2

£10¡3

-2

2

0

4

p¢Vamp

a) b)

Fig. 11. Comparison of the bifurcation results obtained through numerical simula-tion (Panel a)) with those predicted from the local map (Panel b)), when ω = 0.4803.The impacting motions are in black, the nonimpacting motions are in gray, solidcurves correspond to stable periodic motions, dashed curves correspond to unstableperiodic motions, and black regions correspond to chaotic attractors.

22

0 0.04 0.08 0.12

¢q 2

£10¡3

-0.4

0.4

0

p¢Vamp

0 0.04 0.08 0.12

¢q 2

£10¡3

-0.4

0.4

0

p¢Vamp

a) b)

Fig. 12. Comparison of the bifurcation results obtained through numerical simula-tion (Panel a)) with those predicted from the local map (Panel b)), when ω = 0.4793.The impacting motions are in black, the nonimpacting motions are in gray, solidcurves correspond to stable periodic motions, dashed curves correspond to unstableperiodic motions, and black regions correspond to chaotic attractors.

-5 0 5x10

-4¢!

0

0.5

1

1.5

2

x10 -4

Vamp¡V¤ amp(!

)

Fig. 13. Comparison of predicted bifurcation points with those obtained using thesimulated data (solid line - numerically obtained grazing bifurcation points, dots -predicted grazing bifurcation points, + signs - numerically obtained crisis points,circles - predicted crisis points, * signs - numerically obtained saddle node bifurca-tion points, triangles - predicted saddle node bifurcation points.

5 Conclusion

In the previous sections, we formulated discrete maps that capture the dynam-ics of a model microelectromechanical impact actuator in the vicinity in stateand parameter space of the transition between nonimpacting and impactingasymptotic dynamics. Specifically, using a discontinuity-mapping-based ap-proach, we were able to achieve excellent agreement between the results ofcomputationally-expensive direct numerical simulations of the impacting dy-namics and the computationally cheap predictions from the associated dis-

23

crete maps. Indeed, the discrete maps faithfully describe the local dynamicswhile reducing the computation time by several orders of magnitude. As suchthe methodology provides an efficient tool in the analysis and design of properfunction in impact microactuators. While the analysis was applied to the Mitaactuator, only nominal modifications would be required to derive similar localmaps to other impact microactuators.

Although we have only shown explicit formulae for the local map on neigh-borhoods of the two co-dimension-two points, similar maps can easily be de-rived on neighborhoods of the co-dimension-one points. It is interesting toobserve that the predictions of such local maps when truncated at first orderin√

∆q2, ∆θ, ∆ω, and ∆Vamp agree qualitatively with the numerical simula-tions for small perturbations near the co-dimension-one points, but that thesecond-order terms are necessary to capture the bifurcation behavior near theco-dimension-two points.

Following Zhao et al [24], the above discussion focused on the interval 0.46 <ω < 0.51, which contains the primary resonance (recall that the frequencyof the forcing term is twice that of the voltage oscillations). A study of thesystem behavior outside of this interval is currently underway. Preliminaryresults show the existence of additional co-dimension-two grazing bifurcations,e.g., involving the coincidence of a smooth period-doubling bifurcation withgrazing contact and associated transition scenarios. These will be the subjectof a separate discussion.

6 Acknowledgements

This material is based upon work supported by the National Science Founda-tion under Grant No. 0237370

7 Appendix

To map out the grazing curve Γ in the ω − Vamp parameter space, we desirea numerical method for locating points of grazing contact x∗ ∈ D ∩ P , suchthat P (x∗, ω, Vamp) = x∗, for the Poincare map P associated with P .

Denote by Φ the flow corresponding to motion in the submanifold S withvector field f = f stick. Consider the function

g (t,x,ω, Vamp) = hD (Φ (x, t; ω, Vamp)) (60)

24

for some x. Then,

gt (t,x,ω, Vamp) = hD,i (y) · f i (y,ω, Vamp)∣∣∣y=Φ(x,t;ω,Vamp)

(61)

gtt (t,x,ω, Vamp) =

hD,ij (y) · f i (y,ω, Vamp)

+hD,i (y) · f i,j (y,ω, Vamp)

· f j (y,ω, Vamp)

∣∣∣∣∣∣∣y=Φ(x,t;ω,Vamp)

(62)

Specifically, suppose that for some choice of parameters ω = ω∗ and Vamp =V ∗

amp, the point x∗ ∈ P lies on a periodic trajectory with period T of thenonimpacting system and that t = 0 is a simple local maximum of the functionhD

(Φ

(x∗, t; ω∗, V ∗

amp

)). It follows that

gt

(0,x∗,ω∗, V ∗

amp

)= hD,i (x∗) · f i

(x∗, ω∗, V ∗

amp

)= 0 (63)

and

gtt

(0,x∗,ω∗, V ∗

amp

)=

hD,ij (x∗) · f i(x∗, ω∗, V ∗

amp

)

+hD,i (x∗) · f i,j

(x∗, ω∗, V ∗

amp

)

· f j

(x∗, ω∗, V ∗

amp

)< 0.

(64)Now consider the function

F (x, t, ω, Vamp) = hP (Φ (x, t; ω, Vamp)) = hD,i (Φ (x, t; ω, Vamp))·f i (Φ (x, t; ω, Vamp)) .(65)

Then,

F(x∗, T ,ω∗, V ∗

amp

)= gt

(0,x∗,ω∗, V ∗

amp

)= 0 (66)

and

Ft (x∗, T ) = gtt

(0,x∗,ω∗, V ∗

amp

)< 0. (67)

The implicit function theorem implies the existence of a unique smooth func-tion τ (x) for x ≈ x∗, ω ≈ ω∗, and Vamp ≈ V ∗

amp, such that

τ(x∗, ω∗, V ∗

amp

)= T (68)

and

hP (Φ (x, τ (x, ω, Vamp) ; ω, Vamp)) = F (x, τ (x, ω, Vamp) , ω, Vamp) ≡ 0. (69)

Using τ (x), we can now uniquely define the Poincare mapping P correspond-ing to P by

P (x,ω, Vamp) = Φ (x, τ (x, ω, Vamp) ; ω, Vamp) (70)

for x ≈ x∗, ω ≈ ω∗, and Vamp ≈ V ∗amp.

25

Now consider the vector valued function

F (x, ω, Vamp) =

P (x, ω, Vamp)− x

hD (x)

. (71)

Zeros of this function clearly correspond to fixed points of the Poincare map-ping P that lie in D ∩ P , i.e., grazing periodic trajectories. Indeed, providedthat the matrix

F(x,Vamp) =

P,x (x, ω, Vamp)− Id P,ω (x, ω, Vamp)

hD,x (x) 0

(72)

is invertible, we may implement a Newton scheme to locate zeros of F and thusx∗ (ω) and V ∗

amp (ω). This is the method employed in generating the grazingcurve Γ in Figure 4.

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27

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28