Synchronization of clocks and metronomes:
A perturbation analysis based on multiple timescales

Synchronization occurs in diverse physical, biological, and chemical systems Winfree (1980); Pikovsky, Rosenblum, and Kurths (2001); Strogatz (2003); Blekhman (1988). Examples include the synchronous flashing of fireflies, the chorusing of crickets, the rhythmic applause of concert audiences, the coordinated beating of cardiac pacemaker cells, the pathological neural synchrony associated with epileptic seizures, and the coherent voltage oscillations of superconducting Josephson junction arrays.

Historically, the study of synchronization began with Christiaan Huygens’s discovery of an effect he described as “marvelous” Huygens (1893); Strogatz (2003); Blekhman (1988); Yoder (2004); Ramirez and Nijmeijer (2020). While confined to his room in February 1665 with a “slight indisposition,” Huygens noticed that two of the pendulum clocks he had recently built were swinging in perfect time together. Suspecting that they must be interacting somehow, perhaps through vibrations in their common support, Huygens did a series of experiments to test the idea. In one experiment, he attached two clocks to a board suspended on the backs of two chairs (Fig. 1) and noticed, to his amazement, that no matter how he started the clocks, within about thirty minutes their pendulums always settled into antiphase synchrony, repeatedly swinging toward each other and then apart.

In the centuries since then, and especially in the past twenty years, Huygens’s clocks have been revisited by many authors, using a variety of experimental set-ups, simplified models, analytical approximations, and computational techniques Ellicott (1740a, b); Ellis (1873); Korteweg (1906); Bennett et al. (2002); Oud, Nijmeijer, and Pogromsky (2006); Senator (2006); Dilão (2009); Czolczynski et al. (2009); Czołczyński et al. (2011); Czolczynski et al. (2011, 2013); Kapitaniak et al. (2012); Jovanovic and Koshkin (2012); Ramirez, Fey, and Nijmeijer (2013); Ramirez et al. (2014a, b, 2016); Ramirez and Nijmeijer (2016, 2020); Oliveira and Melo (2015); Willms, Kitanov, and Langford (2017); Wiesenfeld (2017). The subject has been reinvigorated by the rise of nonlinear science. New tools have made it possible to simulate and solve the equations of motion for these coupled nonlinear oscillators, and geometric ideas have illuminated many aspects of the system’s dynamics and bifurcations.

As further motivation for studying this class of problems, consider the curious behavior of coupled metronomes Pantaleone (2002); Kuznetsov et al. (2007); Ulrichs, Mann, and Parlitz (2009); Wu et al. (2012). Videos of self-synchronizing metronomes have attracted millions of views on YouTube Bahraminasab (2007); Ikeguchi-Laboratory (2012); MythBusters (2014). In these experiments, following the work of Pantaleone Pantaleone (2002), anywhere from two to 32 metronomes are placed on a light platform that is free to move sideways, typically by rolling on empty soda cans or other light cylinders (Fig. 2). As with Huygens’s clocks, synchronization gradually occurs after several minutes. A striking difference from Huygens’s clocks, however, is that the metronomes in the videos tend to synchronize in phase rather than in antiphase, and one naturally wonders why.

Although a great deal of progress has been made in understanding the synchronization of clocks and metronomes, many questions remain Ramirez and Nijmeijer (2020). One challenge is to model the individual clocks, keeping in mind that real pendulum clocks come in many shapes, sizes, and styles. There are monumental clocks Ramirez et al. (2016) which stand upright like grandfather clocks. There are smaller clocks that can be placed on a mantel or hung on a wall Czolczynski et al. (2011); Oliveira and Melo (2015). And there are stripped-down, simplified clocks designed for laboratory experiments Bennett et al. (2002); Wiesenfeld (2017). The same bewildering diversity is true of metronomes.

A further challenge is to model the escapement mechanism that keeps the clock or metronome running. Depending on the level of realism one seeks, the escapement can be modeled with discontinuous impulses Bennett et al. (2002), step functions Oliveira and Melo (2015), Gaussian-like functions Oud, Nijmeijer, and Pogromsky (2006), piecewise linear functions Ramirez et al. (2016) or a nonlinear damping term of the sort seen in van der Pol oscillators Pantaleone (2002).

In addition to modeling the individual clocks, there is also the challenge of modeling the coupling between them. Clocks can interact by transmitting sound pulses through a wall on which they are both hanging Oliveira and Melo (2015), or by shaking their common support or the beam from which they are suspended Ramirez et al. (2016); Ramirez and Nijmeijer (2020). Metronomes interact by jiggling their shared platform slightly from side to side every time they swing, thereby transmitting phase information to each other Pantaleone (2002).

Along with these variations in experimental conditions, there is a correspondingly large variety of analytical methods that can be used to study such systems. Early researchers used linear techniques such as normal modes Korteweg (1906). Others have applied qualitative geometric methods, such as Poincaré maps Bennett et al. (2002) and equivariant bifurcation theory Willms, Kitanov, and Langford (2017). Still others have used perturbation methods, such as Poincaré’s method Ramirez and Nijmeijer (2016) or the method of multiple scales Pantaleone (2002). When the goal is to make quantitative predictions, many researchers have relied on numerical integration of the governing ordinary differential equations for a simplified three-degree-of-freedom model, or for further realism, Ramirez and colleagues have used finite element models of the coupling bar through which the clocks interact Ramirez et al. (2014a).

Finally, along with all these methodological choices, there is a zoo of phenomena to be analyzed. Besides in-phase and antiphase synchronization, pendulum clocks sometimes exhibit quasiperiodicity as well as other forms of unsynchronized behavior. In some circumstances, one of the clocks may stop ticking altogether. This phenomenon Ellicott (1740a, b); Bennett et al. (2002), now called “beating death,” occurs if one of the pendulums swings at such low amplitude that it fails to engage its escapement mechanism.

In this paper we focus on one of these many questions. Our goal is to clarify what causes clocks or metronomes to synchronize in phase or in antiphase. Our strategy is to recast the governing equations for such systems into a form amenable to perturbation theory, by rescaling them so that they become a weakly perturbed pair of harmonic oscillators. Then the method of multiple scales yields the evolution equations for the oscillators’ slowly-varying amplitudes and phases. These evolution equations determine whether the system will ultimately synchronize, and if so, in which mode. The perturbation methods we use are well known, but they have rarely been applied in this setting.

Our work is closest in spirit to that of Pantaleone Pantaleone (2002). He used the method of multiple scales in a similar fashion to analyze the dynamics of coupled metronomes. But whereas he assumed a van der Pol approximation for the escapement mechanism, we were curious to see what qualitative differences might arise from a more realistic impulsive model of the escapement. We have also assumed a more general model of the coupling between the oscillators. Pantaleone Pantaleone (2002) assumed that the metronomes rest on a platform that is neither damped nor subject to any restoring forces. We allow for both. By doing so, we find results consistent with some transition scenarios reported experimentally Wu et al. (2012). In particular, we find a sequence of transitions as the damping of the platform is increased. At low enough damping, and assuming the pendulums’ amplitudes are small but not too small, the system tends to synchronize in phase (as seen in many experiments on coupled metronomes). At intermediate damping, both in-phase and antiphase synchronization become stable, each with its own basin of attraction. And at sufficiently high damping, only antiphase synchronization is stable. These results agree with those found in the metronome experiments of Wu et al. Wu et al. (2012)

Other notable features of our approach are: (1) the attention given to modeling the escapement mechanism, (2) a small-angle approximation expanded past the linear term, and (3) the scaling of the model equations that disentangles different physical effects through a two- and a three-timescale asymptotic analysis. While each of these ingredients can be found in the literature, this is the first time, to the best of our knowledge, that all three have been considered simultaneously.

This article is organized as follows. In Section II, we discuss the mechanics of pendulum clocks and metronomes, with particular emphasis on the modeling of the escapement mechanism. Section III develops the rest of the model. In Section IV we identify the timescales over which the various physical effects take place. These considerations guide our choices of the relevant dimensionless parameters and variables. We call the resulting dimensionless system of equations the scaled system. In Section V we carry out a two-timescale asymptotic analysis on the scaled system to obtain the slow flow (often referred to in the perturbation theory literature as the averaged system). This slow flow governs the long-term dynamics. It has one fixed point that corresponds to in-phase synchronization, and another that corresponds to antiphase synchronization. In Section VI, we explore how the damping of the platform affects the stability of these synchronized states. The results are presented in a set of bifurcation diagrams, with explicit formulas given for the bifurcation curves appearing in the diagrams. To simplify the analysis further, in Section VII we assume that the influence of the clocks or metronomes on the motion of the platform is extremely weak. This assumption produces an extra separation of timescales, and with it, some extra insight into the underlying physics. The paper concludes in Section VIII with a brief discussion.

We begin by describing the mechanics of the escapement mechanism Kapitaniak et al. (2012); Lepschy, Mian, and Viaro (1992); Moon and Stiefel (2006); Roup et al. (2003); Rowlings (1944). This is the mechanism that provides the source of energy for pendulum clocks and metronomes.

The left panel of Fig. 3 shows the components of a so-called deadbeat anchor escapement in clocks (the escapements for metronomes work similarly, except their energy source is a spring that unwinds instead of a weight that descends). The main components are the axle, the escapement wheel, and the weight. The escapement wheel is a gear with teeth. The axle extends in the direction perpendicular to the page and goes through the center of the escapement wheel. The escapement wheel and the axle rotate together. The weight provides energy to the system; it hangs from a cord wound around the axle and as it descends, it applies a torque to the axle to turn the axle-escapement wheel system in the clockwise direction.

The right panel of Fig. 3 shows a pendulum rigidly attached to an anchor. The sides of the anchor are known as pallets. The pendulum, anchor, and pallets all oscillate together about their common pivot, as shown in Fig. 4, which in turn causes the teeth of the wheel to interact with the pallets. Whenever a tooth strikes a pallet, it does so without recoil; this is where the “dead” in “deadbeat” comes from. Moreover, a tooth in contact with a pallet slides along the pallet face without applying torque to the system. Torque is applied only when a tooth reaches the end of a pallet. Note that the right and left ends of the pallets are differently shaped, as shown in the right panel of Fig. 3; this shape difference is crucial to obtain the desired clock dynamics (but, for visual clarity, those shape differences are suppressed in Fig. 4).

To see how energy is transferred from the escapement to the pendulum, consider four key moments in a swing cycle (Fig. 4). At time $\bar{t}_{1}$, the pendulum is swinging counterclockwise, and the green tooth (located near 1 o’clock on the escapement wheel) is contacting the end of the right pallet, thereby applying a force on it perpendicular to the pallet’s end (this is where the end shape of the pallet matters). Because the force points in the direction of the blue arrow shown in Fig. 4, the anchor-pendulum system experiences an impulse that increases its kinetic energy.

Once the green tooth is no longer in contact with the right pallet, the escapement wheel accelerates clockwise due to the torque caused by the weight. Then the escapement wheel stops abruptly when the pink tooth (located near 11 o’clock on the wheel) meets the left pallet. Meanwhile, the anchor-pendulum system continues turning counterclockwise. At time $\bar{t}_{2}$ in Fig. 4, the pendulum makes its largest angle with the vertical. While the pink tooth is in contact with the pallet face, the tooth applies a force that points toward the pivot of the anchor-pendulum system (because the pallet face is a circular arc at a constant radial distance from the pivot). Hence this force does not apply any torque to the anchor-pendulum system with respect to the pivot, and so the dynamics of the anchor-pendulum system is not affected when the tooth is in contact with the pallet face.

Similar events occur in the next half of the cycle, with times $\bar{t}_{3}$ and $\bar{t}_{4}$ playing the parts of $\bar{t}_{1}$ and $\bar{t}_{2}$. Energy is pumped into the pendulum at time $\bar{t}_{3}$, and only then.

The self-sustained oscillations of the pendulum continue until the cord that holds the weight is no longer wound around the axle of the escapement wheel. The periodic input of energy that the anchor-pendulum system receives from the escapement wheel-weight system makes up for the energy lost due to damping.

A cartoon of our model is shown in Fig. 5. Both of the pendulum angles $\bar{\theta}_{1}$ and $\bar{\theta}_{2}$ are functions of time $\bar{t}$. We use primes to denote derivatives with respect to $\bar{t}$. (The overbars denote dimensional quantities; they will disappear soon, after we nondimensionalize the system.)

To keep track of the motion of the platform or support, we select an arbitrary point on it. The position of this point is denoted by $\bar{x}\,{\bf e}$, where ${\bf e}$ is the constant unit dimensionless vector that points to the right, as illustrated in Fig. 5. Note that for simplicity we are regarding the platform as having only one degree of freedom.

To model the action of the escapement on pendulum $i$, where $i=1,2$, we assume there is a constant impulse $\bar{J}$ and a critical angle $\bar{\theta}_{c}$ such that pendulum $i$ receives a positive impulse $\bar{J}$ whenever it reaches its critical angle while swinging to the right, i.e., whenever $\bar{\theta}_{i}=\bar{\theta}_{c}$ and $\bar{\theta}_{i}^{{}^{\prime}}>0$. Such an impulse occurs at time $\bar{t}_{1}$ in Fig. 4. Similarly, a negative impulse $-\bar{J}$ is received whenever $\bar{\theta}_{i}=-\bar{\theta}_{c}$ and $\bar{\theta}_{i}^{{}^{\prime}}<0$ (as at time $\bar{t}_{3}$ in Fig. 4). Let $\{\bar{T}_{ir}\}$ be the set of times when pendulum $i$ receives a positive impulse, and let $\{\bar{T}_{i\ell}\}$ be the set of times when it receives a negative impulse. We define

and

where $\delta$ is the delta function.

This model of the escapement mechanism is idealized. It is the simplest model that, in our opinion, contains the main physics relevant to our studies. More realistic models of the escapement mechanism assume that the input of energy to the pendulum is not applied instantaneously as an impulse, but rather as a force applied over a short but nonzero period of time. Such forces have been modeled with Gaussian-like Oud, Nijmeijer, and Pogromsky (2006) or piecewise linear Ramirez et al. (2016) functions. Although they differ in detail, these alternative escapement models are not qualitatively different from ours. As long as the period of time over which the force acts is much shorter than the time of an oscillation, these short-acting forces can be approximated by delta functions.

Having modeled the escapement forces, we turn now to the forces on the coupling platform. We assume that it is subjected to a linear restoring force ${\bar{\bf F}}_{r}$ that follows Hooke’s law, ${\bar{\bf F}}_{r}=-\bar{\kappa}\,\bar{x}\,{\bf e}$. Here, the origin of $\bar{x}$ has been implicitly chosen such that the restoring force is equal to ${\bf 0}$ when $\bar{x}=0$. The platform is also assumed to be subjected to a linear damping force ${\bar{\bf F}}_{d}=-\bar{\mu}\bar{x}^{{}^{\prime}}{\bf e}$.

The remaining parameters in the equations of motion are as follows: $m$ is the mass of each pendulum; $M$ is the combined mass of both metronomes or clocks, including their pendulums, and the platform; $L$ is the length of each pendulum, namely the distance from the pivot to the center of mass of the pendulum; $\bar{\nu}$ is a damping constant (due to the pendulum motion); and $g$ is the acceleration due to gravity. For simplicity, we neglect the mass of the escapement wheels. Then Newton’s second law yields the following system, which we refer to as the governing equations:

The first and second equations are obtained by taking torques about the pivots of the corresponding pendulum and dividing by $L$. The third equation expresses horizontal force balance for the platform.

The governing equations apply to both clocks and metronomes. For example, in the original Huygens set-up shown schematically in Fig. 1, the restoring force on the support is zero but the damping force on it is not. Thus, if the damping force is assumed to be proportional to the support velocity, the dynamics of the system in Fig. 1 is mimicked by the cartoon in Fig. 5 and by the equations above with $\bar{\kappa}=0$ and $\bar{\mu}\neq 0$. Similarly, Fig. 2 is a simplified model of the metronome experiments. If the rollers underneath the coupling platform are assumed to be massless and do not slip, the dynamics of the system in Fig. 2 are given by the equations above with $\bar{\mu}=\bar{\kappa}=0$. In what follows, we will analyze the governing equations for general values of $\bar{\mu}$ and $\bar{\kappa}$, but with special attention to cases where one or both of these are zero.

Since synchronization takes place after hundreds of swings, the relevant physics occurs at two different timescales. We introduce a small parameter $\epsilon\ll 1$ that encodes the separation of these timescales. We will choose the rest of the dimensionless parameters and variables so that the different physical effects take place on either the timescale of a single swing of a pendulum, or a much longer timescale given by $1/\epsilon$ times the pendulum’s period.

Specifically, we scale the variables and parameters as follows. The natural choice for the dimensionless time $t$ is

so that the periods of the pendulums are $O(1)$ in $t$. An $O(1)$ phase adjustment of the pendulums, due to inertial forcing from the motion of the platform, occurs in long times of $O(M/m)$ in $t$; thus, we want $M/m=O(1/\epsilon),$ or equivalently, the mass ratio $m/M=O(\epsilon).$ This choice leads us to introduce a dimensionless parameter

where $b$ is assumed to be $O(1)$. Physically, $b$ quantifies how strongly the pendulums’ motion affects the platform’s motion. Consequently, $b$ also controls how much one pendulum couples to the other, mediated by the driving they each impart to the platform. (All of this will become clearer after we nondimensionalize the governing equations; see Eq. (3) below.)

To scale the angle variables, note first that the $\bar{\theta}_{i}$ are of the order $\bar{\theta}_{c}$, the critical angle at which the escapement engages. To make the nonlinear equations of motion as tractable as possible, we want to use a small-angle approximation, but we also want to retain the leading effects of nonlinear terms. With these ideas in mind, note that $\sin\bar{\theta}_{i}\approx\bar{\theta}_{i}+O(\bar{\theta}_{i}^{3})$, so the leading nonlinear effects take place in times of $O(1/\bar{\theta}_{i}^{2})$ in $t$. Thus, we want $1/\bar{\theta}_{c}^{2}=O(1/\epsilon)$, which motivates the following scaling:

where the dimensionless parameters $r$ and $\theta_{c}$ are $O(1)$.

To scale the remaining quantities in the model, we estimate that $\bar{x}=O(L\bar{\theta}_{i}m/M)$. Since $\bar{\theta}_{i}=O(\sqrt{\epsilon r})$ and $m/M=O(\epsilon)$, we introduce

so that $x$ is $O(1)$. The damping due to friction in the pendulums takes place in times of $O\left((m/\bar{\nu})\sqrt{g/L}\right)$ in $t$. Since we want this quantity to be $O(1/\epsilon)$, we introduce the $O(1)$ dimensionless parameter

The impulse $\bar{J}$ causes an increase in the amplitude of oscillations of $O\left(\bar{J}/(m\sqrt{gLr\epsilon})\right)$ in the dimensionless variables $\theta_{i}$. We want this quantity to be $O(\epsilon)$ so the cumulative effects of the impulses take place in times of $O(1/\epsilon)$ in $t$. Therefore we define

and assume it to be $O(1)$.

The restoring force and damping force on the platform should affect the dynamics of the platform in times of $O(1)$. This leads to the following dimensionless stiffness and damping parameters for the platform:

Next, we nondimensionalize the governing equations. Let $\{T_{ir}\}$ and $\{T_{il}\}$ be the set of dimensionless times $t$ when pendulum $i$ receives a positive or negative impulse, respectively. We define

and

Then, neglecting terms of $O(\epsilon^{2})$ in the first two governing equations and terms of $O(\epsilon)$ in the third equation, we find that the governing equations become

where dots denote derivatives with respect to $t$. We refer to Eq. (3) as the scaled system.

Notice that our choice of scaling has converted the first two governing equations into undamped linear oscillators perturbed by various forces of size $O(\epsilon)$. Although these forces are small, their effects accumulate on a long timescale of $O(1/\epsilon)$. The energy source that drives this slow evolution is the train of small impulses coming from the escapements (the $\epsilon Jf$ terms). Meanwhile, the parts of the system interact through two-way coupling: the pendulums are inertially forced by the platform’s accelerated motion (the $\epsilon\ddot{x}$ terms), while the pendulums act back on the platform through the $b\ddot{\theta}$ terms.

In deriving this scaled system, we kept only the dominant terms, as is customary in perturbation theory. The terms of $O(\epsilon^{2})$ neglected in the first two equations can be proven not to affect the existence or stability of the in-phase or antiphase synchronized states, or of any other periodic solutions, as long as those periodic solutions satisfy certain genericity conditions; this fact follows from a basic theorem of averaging theory Guckenheimer and Holmes (1983). Admittedly, the $O(\epsilon^{2})$ terms can make tiny quantitative changes to the solutions, but these are asymptotically negligible for $\epsilon\ll 1$. For the same reason, it suffices to consider only the $O(1)$ terms in the third equation.

However, the leading perturbations of size $O(\epsilon)$ in the first two equations must be retained. As is well known from the theory of weakly nonlinear oscillators Guckenheimer and Holmes (1983); Strogatz (1994); Bender and Orszag (1999); Holmes (1995), these small perturbations play a decisive role in determining the system’s long-term qualitative behavior.

The structure of Eq. (3) also clarifies what the parameter $\epsilon$ represents physically. A clue to its meaning is that the only terms with a pure coefficient of $\epsilon$ and no other prefactors are the $\epsilon\ddot{x}$ terms in the first two equations in (3). Then, by tracing this clue back to the original governing equations, one can check that $\epsilon$ is a dimensionless quotient of two characteristic accelerations: the typical accelerations $\bar{x}^{{}^{\prime\prime}}$ of the platform and the (much larger) typical accelerations $L\bar{\theta}^{{}^{\prime\prime}}$ of the pendulums.

The cubic terms multiplied by $\epsilon r$ in Eq. (3) are not usually considered, but turn out to be important. As we will see in subsequent sections, in the parameter regimes of interest the size of the cubic coefficient $r$ determines whether the pendulums will synchronize in phase or in antiphase, or whether both of those synchronized states are locally stable. From a physical standpoint, increasing $r$ corresponds to increasing the critical angle $\bar{\theta}_{c}$ and the impulse $\bar{J}$ in the same proportion while keeping all the other dimensional parameters constant. Because of this close connection between $r$ and $\bar{J}$, we will sometimes find it helpful to interpret $r$ as a dimensionless measure of the impulsive forcing strength, even though $J$ plays this role more directly.

To make the analysis as clear as possible, we begin with the simplest case: $\mu=0$ and $\kappa=0$. (The procedure for analyzing the more general case when these two parameters are nonzero is similar to that presented below. But the resulting equations are complicated enough to obscure the benefits of the method. The messy asymptotic equations for $\mu\neq 0$ and $\kappa\neq 0$ are relegated to Appendix A.)

As discussed in previous sections, we are assuming the mass ratio $m/M$ is of the same order as a small parameter $\epsilon\ll 1$. With suitable scaling of the other physical parameters, the dynamics then take place on two timescales, one of $O(1)$ and the other of $O(1/\epsilon)$ in $t$. Specifically, the pendulum angles $\theta_{1}$, $\theta_{2}$ and the dimensionless location $x$ of the system’s center of mass are oscillatory variables with periods of $O(1)$ in $t$, but their amplitudes and phases change by $O(1)$ on timescales of $O(1/\epsilon)$ in $t$. Thus, we make the following ansatz:

where $\sim$ means asymptotic approximation in the parameter regime $\epsilon\ll 1$, and

is a slow time variable. Each $\theta_{ij}$ and $x_{i}$ are functions of $t$ and $\tau$, and these functions are periodic in their first argument $t$.

We carry out a standard two-timescale analysisGuckenheimer and Holmes (1983); Strogatz (1994); Holmes (1995); Bender and Orszag (1999). Namely, we plug the ansatz (4) into the system of equations (3), then replace $d/dt$ by

in that system (this substitution follows from the form of the ansatz (4)), and finally collect terms having like powers of $\epsilon$. This perturbative method, often called two-timing, is a special case of the method of multiple scalesStrogatz (1994); Holmes (1995); Bender and Orszag (1999). It can be rigorously justified by averaging theoryGuckenheimer and Holmes (1983).

From the terms that contain the power $\epsilon^{0}$ in the expansion of (3), we obtain

The general solution of these equations (recalling that $x_{0}$ is periodic in $t$) is

and

As usual, differential equations for the evolution of the slow variables $A_{1},A_{2},\varphi_{1},\varphi_{2}$ will be obtained at the next order of $\epsilon$. But before we proceed to that order, we need to deal with an unusual feature of our model system (3): it contains delta-function forcing terms due to the repeated impulses provided by the escapement mechanism. Now that we have an asymptotic approximation for the fast oscillations of the pendulums, we can find the times when the escapement acts; by solving for these times and inserting them into the delta functions, we get the following asymptotic approximations for the impulsive forcing terms $f_{1}$ and $f_{2}$ in Eqs. (1) and (2):

where

and

Now proceeding to the $O(\epsilon^{1}$) terms in the expansion of the system (3), we find that its first two equations give

Next, to derive the slow flow equations for $A_{1},A_{2},\varphi_{1},\varphi_{2}$, recall an elementary fact from the solvability theory of differential equations: Let $h(t)$ be a $2\pi$-periodic function of $t$. Let $\varphi$ be any fixed real number. The equation $\ddot{\theta}+\theta=h$ has a $2\pi$-periodic solution $\theta$ if and only if $\int_{0}^{2\pi}h(t)\sin(t+\varphi)dt=0$ and $\int_{0}^{2\pi}h(t)\cos(t+\varphi)dt=0$. This fact is usually stated with $\varphi=0$, but in our analysis it will be convenient to use $\varphi=\varphi_{1}$ and $\varphi=\varphi_{2}$.

The next step is to go back to Eq. (9), recall that $\theta_{ij}$ are $2\pi$-periodic in $t$, and use the fact stated in the last paragraph to conclude that

and

By computing these four integrals (and omitting the algebraic details, which are long but straightforward), we obtain the following slow flow equations:

This system holds for $A_{1}(\tau)>\theta_{c}$ and $A_{2}(\tau)>\theta_{c}$, meaning that the pendulums’ swings are large enough to engage the escapement mechanism at all times.

Since our goal is to identify whether the system evolves to antiphase or in-phase synchronization or no synchronization at all, the variable of interest to us is the phase difference

Dividing Eq. (11) by $A_{1}$, dividing Eq. (13) by $A_{2}$, and subtracting the results, we obtain

We rewrite Eqs. (10) and (12) in terms of $\psi$ as

Equations (14), (15) and (16) form the slow flow system for the special case $\mu=0,\kappa=0$, in which we neglect the damping and restoring forces on the platform. The more general version, where $\mu$ and $\kappa$ are allowed to be nonzero, is given in Appendix A as Eqs. (35), (36), and (37).

There are many physical parameters we could vary to test the model, but we have chosen to focus on the platform damping $\bar{\mu}$, as it is one of the easiest parameters to adjust experimentally. Before delving into the predictions of our model, let us recall what has been seen in the lab.

We can gain further analytical insight by considering a weak-coupling regime in which $b\ll 1$. Then the indirect coupling from one pendulum onto the other (mediated by the motion of the platform) becomes so weak that the pendulums adjust their phase difference $\psi=\varphi_{1}-\varphi_{2}$ on a super-long timescale of $t=O(\epsilon^{-1}b^{-1})$. Thanks to this extra separation of timescales, it becomes possible to simplify the slow flow system even more, because in this regime the amplitudes relax to their equilibrium values much more rapidly than the phase difference adjusts. Hence the amplitudes can be adiabatically eliminated, which allows us to reduce the system to a single equation for the phase difference $\psi$, as we will now show. We will work with the general case where $\mu\not=0$ and $\kappa\not=0$, for which the dynamics are described by the full slow flow system (35), (36) and (37), as given in Appendix A.

If we were to set $b=0$ in the amplitude equations (35) and (36), we would find that both $A_{1}$ and $A_{2}$ would approach the steady-state amplitude $A_{a}$ (defined in Eq. (19)) as the slow time $\tau$ increases. This suggests the ansatz

in the parameter regime $b\ll 1$, where $a_{1}$ and $a_{2}$ are functions of the slow time $\tau$ and a super-slow time $s$, where

Thus, let $a_{1}=a_{1}(\tau,s)$ and $a_{2}=a_{2}(\tau,s)$. If we were to plug the ansatz (26) into the equation (37) for the phase difference $\psi$, we would find that the right hand side of that equation is of $O(b)$. This observation suggests the following ansatz for $\psi$:

where $\psi_{0}$ is a function of only one variable, $\psi_{0}=\psi_{0}(s)$ but $\psi_{1}$ is a function of two variables, $\psi_{1}=\psi_{1}(\tau,s)$.

We again carry out a two-timescale analysis. In Eqs. (35), (36), and (37) we replace $d/d\tau$ by

plug in the ansatz (26) and (27), and expand in powers of $b$. We find that at first order in $b$, Eqs. (35) and (36) reduce to

and