Synchronization of clocks and metronomes:A perturbation analysis based on multiple timescales
aa r X i v : . [ n li n . AO ] A ug Antiphase versus in-phase synchronization of coupled pendulum clocksand metronomes
Guillermo H Goldsztein, a) Alice N Nadeau, b) and Steven H Strogatz c) School of Mathematics, Georgia Institute of Technology, Georgia 30332 Department of Mathematics, Cornell University, Ithaca, NY 14853 (Dated: 26 August 2020)
In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphaseafter hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform showthat they tend to synchronize in phase, not antiphase. Here, using a simple model of coupled clocks and metronomes,we calculate the regimes where antiphase and in-phase synchronization are stable. Unusual features of our approachinclude its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and a three-timescale asymptotic analysis.PACS numbers: 05.45.Xt,45.20.Da
There is still no fully satisfactory explanation for the “sym-pathy of clocks” that Huygens discovered more than 350years ago. Here we explore the roles played by the escape-ment mechanism and a pendulum’s amplitude-dependentfrequency, an otherwise well-known effect whose impor-tance in this context has been surprisingly overlooked.We show it explains why coupled pendulum clocks typ-ically synchronize in antiphase but coupled metronomessynchronize in phase. Given the historical significance ofHuygens’s work and the pervasiveness of synchronizationin nature and technology, we hope our work will bring thenonlinear science community closer to solving the ancientriddle of the sympathy of clocks.
I. INTRODUCTION
Synchronization occurs in diverse physical, biological, andchemical systems, from the coordinated beating of heart cellsto the coherent voltage oscillations of Josephson junction ar-rays . Historically, the study of synchronization began withHuygens’s discovery of the “sympathy of clocks,” an effecthe described as “marvelous” . While confined to his roomwith an ailment, Huygens noticed that two of his pendulumclocks were synchronized. Suspecting that they must be cou-pled somehow, perhaps through vibrations in their commonsupport, Huygens did a series of experiments to test the idea.In one experiment, he attached two clocks to a board sus-pended on the backs of two chairs (Fig. 1) and noticed, to hisamazement, that no matter how he started the clocks, withinabout thirty minutes their pendulums always synchronized inantiphase, repeatedly swinging toward each other and thenapart.Modern-day versions of a similar effect with metronomeshave attracted millions of views on YouTube. In these experi- a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
FIG. 1. Antiphase synchronization of two pendulum clocks.FIG. 2. In-phase synchronization of two metronomes. Themetronomes are drawn schematically, emphasizing the weight typ-ically hidden inside the case. ments, following the work of Pantaleone , anywhere from twoto 32 metronomes are placed on a light platform that is free tomove sideways, typically by rolling on empty soda cans orother light cylinders (Fig. 2). As with Huygens’s clocks, syn-chronization gradually occurs after several minutes. But inthis case, the mode of synchronization is in phase rather thanantiphase.In this article, we revisit these problems in hopes of shed-ding new light on the conditions that favor one form of syn-chronization over another. Notable features of our approachare: (1) the attention given to modeling the escapement mech-anism, (2) a small-angle approximation expanded past the lin-ear term, and (3) a scaling of the model equations that disen-tangles different physical effects through a two- and a three-time scale asymptotic analysis. While each of these ingredi-ents can be found in the literature , this is the first timethat all three have been considered simultaneously. Our re-sults reveal the importance of a nonlinear effect—the depen-dence of a pendulum’s frequency on its amplitude—that hasbeen neglected in previous studies. In our model, this non-linear effect selects for antiphase synchronization, in-phasesynchronization, or the bistability of both. II. THE ESCAPEMENT MECHANISM
We begin by describing the mechanics of the escapementmechanism . The left panel of Fig. 3 shows the compo-nents of a so-called deadbeat anchor escapement in clocks (theescapements for metronomes work similarly, except their en-ergy source is a spring that unwinds instead of a weight thatdescends). The main components are the axle, the escape-ment wheel, and the weight. The escapement wheel is a gearwith teeth. The axle extends in the direction perpendicularto the page and goes through the center of the escapementwheel. The escapement wheel and the axle rotate together.The weight provides energy to the system; it hangs from acord wound around the axle, and as it descends it applies atorque to the axle to turn the axle-escapement wheel systemin the clockwise direction.The right panel of Fig. 3 shows a pendulum rigidly attachedto an anchor. The sides of the anchor are known as pallets.The pendulum, anchor, and pallets all oscillate together abouttheir common pivot, as shown in Fig. 4, which in turn causesthe teeth of the wheel to interact with the pallets. Whenevera tooth strikes a pallet, it does so without recoil; this is wherethe “dead” in “deadbeat” comes from. Moreover, a tooth incontact with a pallet slides along the pallet face without ap-plying torque to the system. Torque is applied only when atooth reaches the end of a pallet. Note that the right and leftends of the pallets are differently shaped, as shown in the rightpanel of Fig. 3; this shape difference is crucial to obtain thedesired clock dynamics (but, for visual clarity, those shapedifferences are suppressed in Fig. 4).To see how energy is transferred from the escapement tothe pendulum, consider four key moments in a swing cy-cle (Fig. 4). At time ¯ t , the pendulum is swinging counter-clockwise, and the green tooth (located near 1 o’clock on theescapement wheel) is contacting the end of the right pallet,thereby applying a force on it perpendicular to the pallet’send (this is where the end shape of matters). Because theforce points in the direction of the blue arrow shown in Fig. 4,the anchor-pendulum system experiences an impulse that in-creases its kinetic energy.Once the green tooth is no longer in contact with the rightpallet, the escapement wheel accelerates clockwise due to thetorque caused by the weight. Then the escapement wheelstops abruptly when the pink tooth (located near 11 o’clockon the wheel) meets the left pallet. Meanwhile, the anchor-pendulum system continues turning counterclockwise. Attime ¯ t in Fig. 4, the pendulum makes its largest angle withthe vertical. While the pink tooth is in contact with the palletface, the tooth applies a force that points toward the pivot of FIG. 3. Components of our model clock.FIG. 4. Snapshots of the deadbeat anchor escapement at differentpoints in its cycle. the anchor-pendulum system (because the pallet face is a cir-cular arc at a constant radial distance from the pivot). Hencethis force does not apply any torque to the anchor-pendulumsystem with respect to the pivot, and so the dynamics of theanchor-pendulum system is not affected when the tooth is incontact with the pallet face.Similar events occur in the next half of the cycle, with times¯ t and ¯ t playing the parts of ¯ t and ¯ t . Energy is pumped intothe pendulum at time ¯ t , and only then.The self-sustained oscillations of the pendulum continueuntil the cord that holds the weight is no longer wound aroundthe axle of the escapement wheel. The periodic input of en-ergy that the anchor-pendulum system receives from the es-capement wheel-weight system makes up for the energy lostdue to damping. III. MODEL OF TWO COUPLED CLOCKS
In the cartoon shown in Fig. 2, both ¯ θ and ¯ θ are functionsof time ¯ t . We use primes to denote derivatives with respect to¯ t . The position of the center of mass of the platform is denoted¯ x e , where e is the constant unit dimensionless vector parallelto the platform in the counterclockwise direction when belowthe pivots.To model the action of the escapement on the pendulum i (1 ≤ i ≤ J anda critical angle ¯ θ c such that pendulum i receives a positiveimpulse ¯ J whenever ¯ θ i = ¯ θ c and ¯ θ ′ i >
0. Such an impulseoccurs at time ¯ t in Fig. 4. Similarly, a negative impulse − ¯ J is received whenever ¯ θ i = − ¯ θ c and ¯ θ ′ i < t inFig. 4). Let { ¯ T ir } be the set of times when pendulum i receivesa positive impulse, and let { ¯ T i ℓ } be the set of times when itreceives a negative impulse. We define¯ f ( ¯ t ) = ∑ ¯ t ⋆ ∈ ¯ T r ¯ J δ ( ¯ t − ¯ t ⋆ ) − ∑ ¯ t ⋆ ∈ ¯ T ℓ ¯ J δ ( ¯ t − ¯ t ⋆ ) and ¯ f ( ¯ t ) = ∑ ¯ t ⋆ ∈ ¯ T r ¯ J δ ( ¯ t − ¯ t ⋆ ) − ∑ ¯ t ⋆ ∈ ¯ T ℓ ¯ J δ ( ¯ t − ¯ t ⋆ ) , where δ is the delta function.To complete the model, let m be the mass of each pendu-lum; M is the combined mass of both metronomes or clocks,including their pendulums, and the platform; L is the lengthof each pendulum, namely the distance from the pivot to thecenter of mass of the pendulum; ¯ ν is a damping constant; and g is the acceleration due to gravity. For simplicity, we neglectthe mass of the escapement wheels. Then, Newton’s secondlaw yields mL ¯ θ ′′ = − mg sin ¯ θ − ¯ ν L ¯ θ ′ − m ¯ x ′′ cos ¯ θ + ¯ f mL ¯ θ ′′ = − mg sin ¯ θ − ¯ ν L ¯ θ ′ − m ¯ x ′′ cos ¯ θ + ¯ f MmL ¯ x ′′ = − ¯ θ ′′ cos ¯ θ + ¯ θ ′ sin ¯ θ − ¯ θ ′′ cos ¯ θ + ¯ θ ′ sin ¯ θ . The first and second equations are obtained by taking torquesabout the pivots of the corresponding pendulum and divid-ing by L . The third equation reflects the assumption that thewheels have no mass, which implies that the center of mass ofthe whole system experiences no acceleration in the directionparallel to the platform. IV. CHARACTERISTIC SCALES ANDNONDIMENSIONALIZATION
Since synchronization takes place after hundreds of swings,the relevant physics occurs at two different time scales. We in-troduce a small parameter ε ≪ / ε timesthe pendulum’s period.Specifically, we scale the variables and parameters as fol-lows. The natural choice for the dimensionless time is t = ¯ t p g / L so that the periods of the pendulums are O ( ) in t . An O ( ) phase adjustment of the pendulums, due to inertial forcingfrom the motion of the platform, occurs in long times of O ( M / m ) in t ; thus, we want M / m = O ( / ε ) , or equivalently,the mass ratio m / M = O ( ε ) . This choice leads us to introduce a dimensionless parameter b = m / ( M ε ) , assumed to be O ( ) . Physically, b quantifies how stronglythe pendulums’ motion affects the platform’s motion, and viceversa. Indirectly, b also controls how much one pendulumcouples to the other.To scale the angle variables, note first that the ¯ θ i are ofthe order ¯ θ c . To make the nonlinear equations of motion astractable as possible, we want to use a small-angle approxima-tion, but we also want to retain the leading effects of nonlinearterms. With these ideas in mind, note that sin ¯ θ i ≈ ¯ θ i + O ( ¯ θ i ) ,so the leading nonlinear effects take place in times of O ( / ¯ θ i ) in t . Thus, we want 1 / ¯ θ c = O ( / ε ) , which motivates the fol-lowing scaling: θ c = ¯ θ c / √ ε r , θ i = ¯ θ i / √ ε r , where the dimensionless parameters r and θ c are O ( ) .To scale the remaining quantities in the model, we esti-mate that the position of the center of mass ¯ x satisfies ¯ x = O ( L ¯ θ i m / M ) . Since ¯ θ i = O ( √ ε r ) and m / M = O ( ε ) , we intro-duce x = ¯ x / ( L ε √ ε r ) , so that x is O ( ) . The damping due to friction takes place intimes of O (cid:16) ( m / ¯ ν ) p g / L (cid:17) in t . Since we want this quantityto be O ( / ε ) , we introduce the O ( ) dimensionless parameter ν = ( ¯ ν / m ε ) p L / g . The impulse ¯ J causes an increase in the amplitude of oscil-lations of O (cid:0) ¯ J / ( m √ gLr ε ) (cid:1) in the dimensionless variables θ i .We want this quantity to be O ( ε ) so the cumulative effects ofthe impulses take place in times of O ( / ε ) in t . Therefore wedefine J = ε − / ¯ J / ( m p gLr ) , and assume it to be O ( ) .Next, we nondimensionalize the governing equations. Let { T ir } and { T il } be the set of dimensionless times t when pen-dulum i receives a positive or negative impulse, respectively.We define f ( t ) = ∑ t ⋆ ∈ T r δ ( t − t ⋆ ) − ∑ t ⋆ ∈ T ℓ δ ( t − t ⋆ ) (1)and f ( t ) = ∑ t ⋆ ∈ T r δ ( t − t ⋆ ) − ∑ t ⋆ ∈ T ℓ δ ( t − t ⋆ ) . (2)Then, neglecting terms of O ( ε ) in the first two equations andterms of O ( ε ) in the third equation, we find that the equationsof motion become¨ θ + θ = ε r θ − εν ˙ θ + ε J f − ε ¨ x ¨ θ + θ = ε r θ − εν ˙ θ + ε J f − ε ¨ x (3)¨ x = − b (cid:0) ¨ θ + ¨ θ (cid:1) , where dots denote derivatives with respect to t .The upshot is that our choice of scaling has converted thefirst two equations into undamped linear oscillators perturbedby various forces of size O ( ε ) . Although these forces aresmall, their effects accumulate on a long time scale of O ( / ε ) .The cubic terms multiplied by ε r turn out to be especially im-portant. As we will see in subsequent sections, the size of r determines whether the coupled system will ultimately syn-chronize in phase or in antiphase, or whether both of thosestates are locally stable. From a physical standpoint, varying r corresponds to varying the critical angle ¯ θ c and the impulse¯ J in the same proportion while keeping all the other dimen-sional parameters constant. V. NUMERICAL SIMULATIONS
We simulated the system of equations (3) for several valuesof the parameters and initial conditions. In all our simulations,the pendulums synchronized either in antiphase or in phase.As a first example, consider Fig. 5. In this simulation westarted the two pendulums nearly in phase (top panel) and theycontinued to swing in that mode for a while. But after verylong times, on the order of hundreds of swing cycles, the pen-dulums settled into a state of antiphase synchronization (bot-tom panel), much like what Huygens observed in his experi-ments on coupled pendulum clocks. The parameters used inthis simulation were not selected accidentally. We were led tothem by the asymptotic analysis given in the next section.As a second example, consider Fig. 6. Now the pendu-lums were started near the antiphase state, but in this casethey slowly drifted away from it and ultimately converged tothe in-phase state, much like what one sees in experiments oncoupled metronomes. Note that the same parameter valueswere used as in Fig. 5, except for the value of r . As we willsee in the next section, this parameter r plays a decisive role indetermining whether the long-term mode of synchronizationis in phase or antiphase. VI. ASYMPTOTIC ANALYSIS
As discussed in previous sections, we are assuming themass ratio m / M is of the same order as a small parameter ε ≪
1. With suitable scaling of the other physical parameters,
FIG. 5. Convergence to antiphase synchronization. Curves show theevolution of the pendulum angles θ (dashed line) and θ (solid line),when the parameters are ε = . b = . r = . θ c = p / ν = J = π , and the initial conditions are θ ( ) =
0, ˙ θ ( ) = θ ( ) = θ ( ) = .
1. Although the system was started nearthe in-phase state (top panel), it evolves toward the antiphase state(bottom panel) after hundreds of swing cycles.FIG. 6. Convergence to in-phase synchronization. Curves show theevolution of θ (dashed line) and θ (solid line), when the parametersare ε = . b = . r = . θ c = p / ν = J = π , and theinitial conditions are θ ( ) =
0, ˙ θ ( ) = θ ( ) = θ ( ) = − .
1. The system was started near the antiphase state (top panel) butgradually became synchronized in phase (bottom panel). the dynamics then take place on two time scales, one of O ( ) and the other of O ( / ε ) in t . Specifically, the pendulum an-gles θ , θ and the dimensionless location x of the system’scenter of mass are oscillatory variables with periods of O ( ) in t , but their amplitude and phase differences change by O ( ) on time scales of O ( / ε ) in t . Thus, we make the followingansatz: θ i ( t ) ∼ θ i ( t , τ ) + εθ i ( t , τ ) + · · · , i = , x ( t ) ∼ x ( t , τ ) + ε x ( t , τ ) + · · · (4)where ∼ means asymptotic approximation in the parameterregime ε ≪
1, and τ = ε t is a slow time variable. Each θ i j and x i are functions of t and τ , and these functions are periodic intheir first argument t .We carry out a standard two-time scale analysis. Namely,we plug the ansatz (4) into the system of equations (3), thenreplace ddt by ∂∂ t + ε ∂∂τ (5)in that system (this is because of the form of the ansatz (4)),and finally collect terms having like powers of ε .From the terms that contain the power ε in the expansionof (3), we obtain ∂ θ ∂ t ( t , τ ) + θ ( t , τ ) = ∂ θ ∂ t ( t , τ ) + θ ( t , τ ) = ∂ x ∂ t ( t , τ ) + b (cid:18) ∂ θ ∂ t ( t , τ ) + ∂ θ ∂ t ( t , τ ) (cid:19) = . The general solution of these equations (recalling that x isperiodic in t ) is θ ( t , τ ) = A ( τ ) sin ( t + ϕ ( τ )) (6) θ ( t , τ ) = A ( τ ) sin ( t + ϕ ( τ )) (7)and x = − b ( θ + θ ) . As usual, differential equations for the evolution of the slowvariables A , A , ϕ , ϕ will be obtained at the next order of ε .But before we proceed to that order, we need to deal with anunusual feature of our model system (3): it contains delta-function forcing terms due to the repeated impulses providedby the escapement mechanism. Now that we have an asymp-totic approximation for the fast oscillations of the pendulums,we can find the times when the escapement acts; by solvingfor these times and inserting them into the delta functions, weget the following asymptotic approximations for the impulsiveforcing terms f and f in Eqs. (1) and (2): f ( t ) ∼ f ( t , τ ) and f ( t ) ∼ f ( t , τ ) , (8)where f ( t , τ ) = ∑ n ∈ Z δ (cid:18) t − arcsin (cid:18) θ c A ( τ ) (cid:19) + ϕ ( τ ) + n π (cid:19) − ∑ n ∈ Z δ (cid:18) t − arcsin (cid:18) θ c A ( τ ) (cid:19) + ϕ ( τ ) + ( n + ) π (cid:19) and f ( t , τ ) = ∑ n ∈ Z δ (cid:18) t − arcsin (cid:18) θ c A ( τ ) (cid:19) + ϕ ( τ ) + n π (cid:19) − ∑ n ∈ Z δ (cid:18) t − arcsin (cid:18) θ c A ( τ ) (cid:19) + ϕ ( τ ) + ( n + ) π (cid:19) . Now proceeding to the O ( ε ) terms in the expansion of thesystem (3), we find that its first two equations give ∂ θ ∂ t + θ = r θ − ν∂θ ∂ t + J f − ∂ x ∂ t − ∂ θ ∂ t ∂τ∂ θ ∂ t + θ = r θ − ν∂θ ∂ t + J f − ∂ x ∂ t − ∂ θ ∂ t ∂τ , (9)where we have not explicitly displayed that the arguments ofeach variable are ( t , τ ) .Next, to derive the slow time equations for A , A , ϕ , ϕ ,recall an elementary fact from the solvability theory of differ-ential equations: Let h ( t ) be a π -periodic function of t. Let ϕ be any fixed real number. The equation ¨ θ + θ = h has a π -periodic solution θ if and only if R π h ( t ) sin ( t + ϕ ) dt = and R π h ( t ) cos ( t + ϕ ) dt = . This fact is usually stated with ϕ =
0, but in our analysis it will be convenient to use ϕ = ϕ and ϕ = ϕ .We can now go back to Equations (9), recall that θ i j are2 π -periodic in t , and use the fact stated in the last paragraphto conclude that Z π (cid:18) r θ − ν∂θ ∂ t + J f − ∂ x ∂ t − ∂ θ ∂ t ∂τ (cid:19) × sin ( t + ϕ ) dt = , Z π (cid:18) r θ − ν∂θ ∂ t + J f − ∂ x ∂ t − ∂ θ ∂ t ∂τ (cid:19) × cos ( t + ϕ ) dt = , Z π (cid:18) r θ − ν∂θ ∂ t + J f − ∂ x ∂ t − ∂ θ ∂ t ∂τ (cid:19) × sin ( t + ϕ ) dt = Z π (cid:18) r θ − ν∂θ ∂ t + J f − ∂ x ∂ t − ∂ θ ∂ t ∂τ (cid:19) × cos ( t + ϕ ) dt = . By computing these four integrals (and omitting the alge-braic details, which are long but straightforward), we obtainthe following slow time equations: dA d τ = − ν A + s − θ c A J π + b ( ϕ − ϕ ) A (10) A d ϕ d τ = b A − θ c A J π + b ( ϕ − ϕ ) A − r A (11) dA d τ = − ν A + s − θ c A J π + b ( ϕ − ϕ ) A (12) A d ϕ d τ = b A − θ c A J π + b ( ϕ − ϕ ) A − r A . (13)This system holds for A ( τ ) > θ c and A ( τ ) > θ c , meaningthat the pendulums’ swings are large enough to engage theescapement mechanism at all times.Since we our goal is to identify whether the system evolvesto antiphase or in-phase synchronization or no synchroniza-tion at all, the variable of interest to us is the phase difference ψ = ϕ − ϕ . Dividing Eq. (11) by A , dividing Eq. (13) by A , and sub-tracting the results, we obtain d ψ d τ = θ c J π (cid:0) A − − A − (cid:1) + r (cid:0) A − A (cid:1) + (14) + b (cid:18) A A − A A (cid:19) cos ψ . We rewrite Equations (10) and (12) as dA d τ = − ν A + s − θ c A J π + b A sin ψ (15) dA d τ = − ν A + s − θ c A J π − b A sin ψ . (16)From now on we will focus on the analysis of the system(14), (15) and (16). VII. STABILITY ANALYSIS OF IN-PHASE ANDANTIPHASE SYNCHRONIZATION
The system (14), (15) and (16) has four fixed points. Moreprecisely, for σ = σ = −
1, we define α = (cid:18) πθ c ν J (cid:19) , A ( σ ) c = √ J πν q + σ √ − α . One fixed point is A = A = A ( − ) c and ψ =
0. A secondfixed point is A = A = A ( − ) c and ψ = π . Both of thesefixed points turn out to be unstable, so we ignore them in whatfollows. To ease the notation, let A c = A ( ) c . A third fixed pointis A = A = A c and ψ =
0, which corresponds to in-phasesynchronization . The fourth fixed point is A = A = A c and ψ = π . It represents antiphase synchronization .The eigenvalues of the Jacobian matrix associated with thesystem (14), (15), and (16) can be found explicitly at thesesynchronized states, and thereby provide information abouttheir stability. The calculations involved are tedious but stan-dard, so we omit the details and summarize the results. Let r = νβ θ c √ α − β br = νβ θ c √ α + β b where β = α (cid:0) + √ − α (cid:1) θ c . We find that in-phase synchronization is unstable for r < r and stable for r > r , whereas antiphase synchronization isunstable for r > r and stable for r < r . FIG. 7. Bifurcation diagram in the parameters b and r , when theother parameters are fixed at the values θ c = p / ν =
1, and J = π . The straight lines are r and r as indicated in the text. Theother curves were generated with the numerical bifurcation programM ATCONT . Figure 7 shows a bifurcation diagram in the parameters r and b . The values of the other parameters are: θ c = p / ν =
1, and J = π .. The straight lines correspond to the sta-bility boundaries r = r and r = r . The other curves in thediagram were generated with the numerical bifurcation pro-gram M ATCONT .Fix b to be small, say b = .
1. Then for small values of r ,only antiphase oscillations are stable; for intermediate valuesof r , both forms of synchronization are stable; and for largevalues of r , only in-phase oscillations are stable.To interpret these results physically, recall that r is a dimen-sionless measure of the pendulum’s nonlinearity, which canbecome important when the oscillations are small but not toosmall. Indeed, r arose when we scaled the size of the criticalangle at which the escapement engages and impulses are im-parted. Our analysis shows that the nonlinear effects capturedby r are not negligible perturbations; they completely changethe picture. We would not see a transition from antiphase toin-phase synchronization without them.We have also seen that r reflects the dependence of a pen-dulum’s frequency on its amplitude, an effect that becomesincreasingly important at large amplitudes. In short, as theamplitudes increase, antiphase synchronization loses stabilityin favor of in-phase synchronization. This finding may shedsome light on why metronomes tend to synchronize in phase:they have a larger critical angle and typically swing at muchlarger amplitudes than the pendulums in pendulum clocks.For larger fixed values of the coupling constant b , and forlarger values of r , Fig. 7 shows a more complicated scenario.In particular, for r increasing from small values, the stablefixed points corresponding to antiphase oscillations branchinto two stable equilibria with a phase difference ψ that isnearly, but not exactly, equal to π ; meanwhile, the exactly an-tiphase oscillations become unstable. For slightly larger val-ues of r , the two stable, nearly antiphase oscillations bifurcateinto two limit cycles in a Hopf bifurcation. Finally, at evenlarger values of r , the limit cycles lose their stability and onlyin-phase synchrony is stable. We were able to find the nearlyantiphase equilibria and the stable limit cycles in numericalsimulations of the original nondimensional equations. VIII. THREE-TIME SCALE ANALYSIS
In the parameter regime b ≪
1, the coupling effect from themotion of the platform becomes so weak that it takes place ona super-long time scale of t = O ( ε − b − ) . This extra sepa-ration of time scales allow us to simplify the system of equa-tions (14), (15) and (16) even further as explained next.If we were to set b = A and A would ap-proach A c as the time τ increases. This suggests the ansatz A ( τ ) ∼ A c + ba ( τ , b τ ) + . . . A ( τ ) ∼ A c + ba ( τ , b τ ) + . . . (17)in the parameter regime b ≪
1, where a and a are functionsof two variables that we call τ and s , i.e., a = a ( τ , s ) and a = a ( τ , s ) , where s = b τ .If we were to plug the ansatz of Equations (17) into theequation (14) for the phase difference ψ , we would obtain thatthe right hand side of that equation is of O ( b ) . This observa-tion suggests the following ansatz for ψ : ψ ∼ ψ ( b τ ) + b ψ ( τ , b τ ) + . . . , (18)where ψ is a function of only one variable, ψ = ψ ( s ) but ψ is a function of two variables, ψ = ψ ( τ , s ) .We again carry out a two time scale analysis. More pre-cisely, in Eqs. (14), (15) and (16) wereplace dd τ by ∂∂τ + b ∂∂ s , (19)plug the ansatz (17) and (18), and expand in powers of b . Wefind that at first order in b , Eqs. (15) and (16) reduce to: da d τ = − ν + J θ c π A c r − θ c A c a + A c ψ (20) da d τ = − ν + J θ c π A c r − θ c A c a − A c ψ . (21)Simple but tedious algebra (that we do not present here) re-veals that the quantity in large parentheses above is negative.This result implies that both a and a tend to constants on thetime scale of τ . Since we are interested in much longer timescales (on the super-slow time scale of τ / b ) we arrive at the following result for the corrections terms to the amplitudes: a ∼ − − ν + J θ c π A c r − θ c A c − A c ψ (22) a ∼ − ν + J θ c π A c r − θ c A c − A c ψ . (23)We now plug the ansatz (17) and (18) with a and a given bythe above formula into Eq. (14) to conclude, after simple ar-guments and more algebra not presented here, that ψ evolvesaccording to d ψ ds = γ (cid:18) νβ θ c √ α − r (cid:19) sin ψ , (24)where the constant prefactor γ is given by γ = (cid:0) + √ − α (cid:1) p − α + √ − α να h(cid:0) + √ − α (cid:1) p − α + √ − α − α i . Equation (24) shows that the antiphase mode ψ = π is sta-ble for 0 ≤ r < νβ θ c / ( √ α ) and the in-phase mode ψ = r > νβ θ c / ( √ α ) . This finding agrees withour previous analysis. In fact, νβ θ c / ( √ α ) is the interceptwhere the straight-line stability boundaries meet the r -axis inFig. 7. IX. DISCUSSION
We have modeled the behavior of two coupled pendulumswith deadbeat escapement mechanisms driving their motion.In our analysis of this system, we focused on a parame-ter regime that is both physically realistic and analyticallytractable: a weak-coupling regime in which the ratio of a pen-dulum’s mass to the mass of the entire system is assumed tobe small. In this regime, phase adjustments of the pendulumsdue to inertial forcing from the platform occur over long timesrelative to the period of the pendulums. By scaling other phys-ical parameters appropriately, we were able to use a multipletime scale analysis to study “the sympathy of clocks" in a waythat appears simpler than most in the existing literature. It al-lows us to delineate regions in the mass ratio and escapementimpulse parameter space where only in-phase synchronizationis stable, or where only antiphase synchronization is stable, orwhere both are stable.Let us try to situate our work relative to the enormous bodyof earlier work on Huygens’s clocks and related problemsabout coupled metronomes. One of the unusual features ofour approach is that we model the escapement mechanism byusing discrete impulses in the form of δ -functions. Other ap-proaches have used different discontinuous functions to modelthe escapement or continuous functions such as avan der Pol term or some other continuous functionwhich gives self-excitation in the system . Importantly, ourimpulses provide a “boost” to the pendulum before it reachesthe apex of its swing rather than to push it back in the oppositedirection. We believe that this model more faithfully reflectshow deadbeat escapements actually work.Further, when making the small-angle approximation, weexpand sin θ past the linear term to include the cubic term. Itis much more common to either take only the first order ap-proximation to sine so that the analysis is morestraightforward, or to avoid a small-angle approximation al-together although this choice can cause the analysis tobecome unwieldy. We find that including the cubic term iscrucial to the dynamics of our model; we would not see atransition from antiphase to in-phase without it.The asymptotic analyses that we have presented are similarin some respects to others in the literature . However,while previous analyses have predominantly used the mass ra-tio as the small parameter, we consider both the mass ratio anda new small parameter related to the critical angle at which theescapement mechanism engages. As a result, we can clearlytease out the separate roles of the mass ratio, the size of thecritical angle, and the size of the impulse. The latter two arethe quantities (in least in our model) that select which modeof synchronization is favored: only in-phase, only antiphase,or the bistability of both.Regions of bistability have been found in earlier analyticalstudies . Bistability can also occur in reality; althoughwe are perhaps more accustomed to antiphase synchroniza-tion of clocks and in-phase synchronization of metronomes,experimental studies have demonstrated that both kinds of de-vices can display bistability in certain circumstances .One of our main results is that the slight dependence ofa pendulum’s frequency on its amplitude can play an out-sized role in the long-term dynamics of coupled clocks andmetronomes. Although well known for individual pendulums,this effect has not been emphasized in previous analyses ofthese coupled systems. Indeed, we suspect that the dynam-ics of Huygens’s clocks have resisted a complete analysis formore than 350 years, precisely because small effects like thiscan play such a pivotal role.These cautionary words underscore the necessity of find-ing a model that captures the essential physics but remainstractable. In this spirit, we have focused on one small pieceof the puzzle: the modeling of the escapement mechanism.But much remains to be done, especially in the modeling ofthe platform, where damping and restoring forces may also bequalitatively important. ACKNOWLEDGEMENTS
Research of A.N.N was supported by an NSF MathematicalSciences Postdoctoral Research Fellowship, Award NumberDMS-190288.
DATA AVAILABILITY
The data that support the findings of this study are almostall available within the article. Any data that are not availablecan be found from the corresponding author upon reasonablerequest. A. T. Winfree,
The Geometry of Biological Time (Springer-Verlag, 1980). A. Pikovsky, M. Rosenblum, and J. Kurths,
Synchronization: A UniversalConcept in Nonlinear Sciences (Cambridge University Press, 2001). S. H. Strogatz,
Sync (Hyperion, 2003). I. I. Blekhman,
Synchronization in Science and Technology (American So-ciety of Mechanical Engineers Press, 1988). J. G. Yoder,
Unrolling Time: Christiaan Huygens and the Mathematizationof Nature (Cambridge University Press, 2004). J. Pantaleone, “Synchronization of metronomes,” Am J Phys , 992–1000(2002). M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, “Huygens’sclocks,” Proc R Soc A , 563–579 (2002). M. Kumon, R. Washizaki, J. Sato, R. Mizumoto, and Z. Iwai, “Controlledsynchronization of two 1-DOF coupled oscillators,” in
Proceedings of the15th IFAC World Congress, Barcelona (2002) pp. 3–10. M. Senator, “Synchronization of two coupled escapement-driven pendulumclocks,” J Sound Vib , 566–603 (2006). A. L. Fradkov and B. Andrievsky, “Synchronization and phase relations inthe motion of two-pendulum system,” International Journal of Non-LinearMechanics , 895–901 (2007). N. V. Kuznetsov, G. A. Leonov, H. Nijmeijer, and A. Y. Pogromsky, “Syn-chronization of two metronomes.” in
PSYCO (2007) pp. 49–52. K. Czolczynski, P. Perlikowski, A. Stefanski, and T. Kapitaniak, “Cluster-ing of Huygens’ clocks,” Prog Theo Phys , 1027–1033 (2009). H. Ulrichs, A. Mann, and U. Parlitz, “Synchronization and chaotic dynam-ics of coupled mechanical metronomes,” Chaos , 043120 (2009). R. Dilão, “Antiphase and in-phase synchronization of nonlinear oscillators:The Huygens’s clocks system,” Chaos , 023118 (2009). K. Czołczy´nski, P. Perlikowski, A. Stefa´nski, and T. Kapitaniak, “Why twoclocks synchronize: Energy balance of the synchronized clocks,” Chaos ,023129 (2011). V. Jovanovic and S. Koshkin, “Synchronization of Huygens’ clocks and thePoincaré method,” J Sound Vib , 2887–2900 (2012). Y. Wu, N. Wang, L. Li, and J. Xiao, “Anti-phase synchronization of twocoupled mechanical metronomes,” Chaos , 023146 (2012). K. Czolczynski, P. Perlikowski, A. Stefanski, and T. Kapitaniak, “Synchro-nization of the self-excited pendula suspended on the vertically displacingbeam,” Communications in Nonlinear Science and Numerical Simulation , 386–400 (2013). J. Pena Ramirez, R. H. Fey, and H. Nijmeijer, “Synchronization of weaklynonlinear oscillators with Huygens’ coupling,” Chaos , 033118 (2013). H. M. Oliveira and L. V. Melo, “Huygens synchronization of two clocks,”Scientific reports , 11548 (2015). J. P. Ramirez, L. A. Olvera, H. Nijmeijer, and J. Alvarez, “The sympathyof two pendulum clocks: beyond Huygens’ observations,” Scientific reports , 23580 (2016). A. R. Willms, P. M. Kitanov, and W. F. Langford, “Huygens’ clocks revis-ited,” Royal Society Open Science , 170777 (2017). M. Kapitaniak, K. Czolczynski, P. Perlikowski, A. Stefanski, and T. Kapi-taniak, “Synchronization of clocks,” Physics Reports , 1–69 (2012). A. M. Lepschy, G. Mian, and U. Viaro, “Feedback control in ancient waterand mechanical clocks,” IEEE Transactions on Education , 3–10 (1992). F. C. Moon and P. D. Stiefel, “Coexisting chaotic and periodic dynamics inclock escapements,” Phil Trans R Soc A , 2539–2564 (2006). A. V. Roup, D. S. Bernstein, S. G. Nersesov, W. M. Haddad, andV. Chellaboina, “Limit cycle analysis of the verge and foliot clock escape-ment using impulsive differential equations and poincare maps,” Interna-tional Journal of Control , 1685–1698 (2003). A. Rowlings,
The Science of Clocks and Watches (Caldwell Industries, Lul-ing, TX USA, 1944). A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer, and B. Sautois,“New features of the software matcont for bifurcation analysis of dynamical systems,” Mathematical and Computer Modelling of Dynamical Systems14