We consider two pendulum clocks suspended at the same wall. When one clock receives the kick, the impact propagates in the wall slightly perturbing the second clock. The perturbation is assumed instantaneous since the time of travel of sound in the wall between the clocks is assumed very small compared to the period. The interaction was studied geometrically and qualitatively by Abraham10,11. However, that approach does not give estimates on the speed of convergence.

In Vassalo-Pereira13, the theoretical problem of the phase locking is tackled. The author makes the assumptions:

1 Dry friction. 2 The pendulums have the same exact natural frequency ω. 3 The perturbation in the momentum is always in the same vertical direction in the phase space, see also10,11. 4 The perturbation imposes a discontinuity in the momentum but not a discontinuity in the dynamic variable. 5 The interaction between clocks takes the form of a Fourier series12.

Vassalo-Pereira deduced that the two clocks synchronize with zero phase difference. This is the exact opposite of Huygens first remarks1 and our experimental observations, where phase opposition was observed. Therefore, we propose here a modified model accounting for a difference in frequency between the two clocks.

Consider two oscillators indexed by i = 1,2. Each oscillator satisfies the differential equation

when , the kinetic energy of each oscillator is increased by the fixed amount h i as in the Andronov model. The coupling term is the normalized force , where is the interaction function and α i a constant with acceleration dimensions.

We consider that the effect of the interaction function is to produce an increment −α in the velocity of each clock leaving the position invariant when the other is struck by the energy kick, as we will see in equations (9). We could consider that the interaction function is the Dirac delta distribution , giving exactly the same result.

The sectional solutions of the differential equation (4) are obtainable when the clocks do not suffer kicks. To treat the effect of the kicks we construct a discrete dynamical system for the phase difference. The idea is similar to the construction of a Poincaré section. If there exists an attracting fixed point for that dynamical system, the phase locking occurs.

Our assumptions are

1 Dry friction. 2 The pendulums have natural angular frequencies ω 1 and ω 2 near each other with ω 1 = ω + ε and ω 2 = ω − ε, where ε ≥ 0 is a small parameter. 3 The perturbation in the momentum is always in the same vertical direction in the phase space10,11. 4 Since the clocks have the same construction, the energy dissipated at each cycle of the two clocks is the same, h 1 = h 2 = h. The friction coefficient is the same for both clocks, μ 1 = μ 2 = μ. 5 The perturbative interaction is instantaneous. This is a reasonable assumption, since in general the perturbation propagation time between the two clocks is several orders of magnitude lower than the periods. 6 The interaction is symmetric, the coupling has the same constant α when the clock 1 acts on clock 2 and conversely. In our model we assume that α is very small.

All values throughout the paper are in SI units when not explicit.

To prove phase locking we solve sectionally the differential equations (4) with the two small interactions. Then, we construct a discrete dynamical system taking into account the two interactions per cycle seen in Fig. 2 and 3. After that, we compute the phase difference when clock 1 returns to the initial position. The secular repetition of perturbations leads the system to near phase opposition as we can see by the geometrical analysis of Fig. 2 and 3.

Figure 2 Interaction of clock 1 on clock 2 at t = 0+. We see the original limit cycle, before interaction and the new one in solid and the original limit cycle in dashed. Note that the value of α and of h are greatly exaggerated to provide a clear view. The effect of the perturbation is secular and cumulative. Full size image

Figure 3 Second interaction. Interaction of clock 2 on clock 1 when clock 2 reaches its impact position. All the features are similar to the Fig. 2. Full size image

The notation is simplified if we consider the function γ(φ) such that

and the function χ(φ) such that

We make the assumption that the natural frequencies are near. A difference of 28 s per day in the movement of the clocks with natural periods in the order of 1.42 s, which is easy to obtain even with very poor clocks, means that ε is on the order of 10−3rads−1.

This means that, in each cycle of each clock, the other one will give one perturbative kick to the other. Suppose that the clocks are bring to contact at t 0 = 0. Consider that the fastest clock (number 1) is at position

Using γ and χ we have

The perturbation of clock 1 on clock 2 adds the value of −α to the velocity , keeping the position q 2 (0−).Thus, the new initial conditions at t = 0+ for the movement of the second clock are

The new phase of clock 2, which is the phase difference of the two clocks φ′ 0 , at 0+ is now

To simplify the notation we consider the function

therefore

With power expansion in α

The correction of the phase difference at is

With first order term in α

Now, both clocks start their natural movement.

We suppose that the clock 2 arrives at the vertical position without being overtaken by clock 1, if that is the case we begin our study after that situation occurs. Clock 2 takes the time to arrive at this position. The phase of clock 2 is . The phase of clock 1 is now

The next interaction is given by the kick from clock 2 to clock 1. Denoting the phase of clock 1 at this stage by

and , we have the phase difference immediately before the second kick

The next interaction is given by the kick from clock 2 to clock 1. Using a process similar to the previous kick we have after the second kick

Expanding this function in power series we have

The new phase difference is

The correction of the phase difference at is now

with first order term in α

and this expression can be further simplified remembering that

giving in first order in α

To complete the study of the phase difference the clock 1 must return to the vertical position, which happens for the time . The time that this clock takes to return to the vertical position is , the phase of clock 1 is now 2π, the phase of clock 2 gives us the new phase difference φ 1 after one cycle of clock 1, that is

If we call the affine function

and the coefficients

the large expression (31) is the composition of four maps

The discrete dynamical system for n ≥ 0 is given by the map Ω such that

Obviously, Ω is a map from the interval [0,2π] to itself. Despite the apparent complexity of Ω, this map is relatively manageable. Under certain conditions we can prove that Ω has a stable fixed point. In this work we deal only with the first degree approximation, relative to the small parameters α and ε, the value of the fixed point φ f which is near to π. The phase difference is asymptotic to the solution φ f . Knowing this value it is possible to prove the existence and stability of the limit cycle of each clock in interaction and the final asymptotic frequency ω f . Under this model we can say that Huygens sympathy occurs.

In first order of α and ε we have the iterative scheme

We define the map

We get in first order of α and ε the dynamical system for the phase difference

There are two fixed points and of Ξ in the interval [0,2π]

The derivative of Ξ at the fixed point must be |Ξ′(x f )| < 1 to have stability and the condition about the argument of the function arcsin gives

Therefore, the limit of the phase difference is, in first order, which is very near to π when the natural frequencies of both clocks are very near, i.e., small ε. When the system reaches this limit the corrections of phase are null for both clocks.