On the problem of synchronization of identical dynamical systems: The Huygens's clocks
OOn the problem of synchronization of identicaldynamical systems: The Huygens’s clocks
Rui Dil˜ao
Abstract
In 1665, Christiaan Huygens reported the observation of the synchroniza-tion of two pendulum clocks hanged on the wall of his workshop. After synchro-nization, the clocks swung exactly in the same frequency and 180 o out of phase —anti-phase synchronization. Here, we propose and analyze a new interaction mech-anism between oscillators leading to exact anti-phase and in-phase synchronizationof pendulum clocks, and we determine a sufficient condition for the existence of anexact anti-phase synchronization state. We show that exact anti-phase and in-phasesynchronization states can coexist in phase space, and the periods of the synchro-nized states are different from the eigen-periods of the individual oscillators. Weanalyze the robustness of the system when the parameters of the individual pendu-lum clocks are varied, and we show numerically that exact anti-phase and in-phasesynchronization states exist in systems of coupled oscillators with different param-eters. In 26 February 1665, Christiaan Huygens, in a letter to his father, [1], reported theobservation of the synchronization of two pendulum clocks closely hanged on thewall of his workshop. After synchronization, the clocks swung exactly in the samefrequency and 180 o out of phase. For attachment distances, less than 1 meter, theclocks always synchronize with the 180 o phase difference. For larger attachmentdistances ( > . Rui Dil˜aoNonLinear Dynamics Group, Instituto Superior T´ecnico Av. Rovisco Pais, 1049-001 Lisbon, Por-tugal, e-mail: [email protected] a r X i v : . [ n li n . C D ] A p r Rui Dil˜ao
Huygens observations were the first time that synchronization effects have beendescribed scientifically.Huygens justified the observed synchronization phenomena by the “sympathythat cannot be caused by anything other than the imperceptible stirring of the airdue to the motion of the pendulum”.In recent years, there has been a growing interest in the detailed analysis of syn-chronization phenomena, both from the theoretical and the experimental points ofviews. From the experimental point of view, Bennett et al. , [2], built an experimentaldevice consisting of two interacting pendulum clocks hanged on a heavy support,and this support was mounted on a low-friction wheeled cart. This device movesby the action of the tensions due to the swing of the two pendulums, and the inter-action between the two clocks is caused by the mobility of the heavy base of theclocks. With this device, the anti-phase synchronization mode is reached when thedifference between the natural or eigen-frequencies of the two clocks is less than0 . . ∆ ω = . ∆ (cid:96) = √ g (cid:96) / ∆ ω / π , which gives, for (cid:96) = g = . − , ∆ (cid:96) = (cid:96) = .
178 m (the length of the pendulum rods used by Huy-gens, [1]), ∆ (cid:96) = .
02 mm, a precision that Huygens certainly could not achieve.According to Bennett et al. [2, p. 578], Huygens’s results depended on both talentand luck.In the experiment of Bennett et al. , [2], the in-phase synchronization is the natu-ral way of synchronization of the two pendulum clocks. However, due to a detaileddescription of the Huygens findings, we can believe that the in-phase synchroniza-tion was never observed by Huygens.Another experimental model mimicking the Huygens’s clocks system, consistsof two pendulums whose suspension rods are connected by a weak string, and oneof the two pendulums is driven by an external rotor, [3] and [4]. In this system,the in-phase synchronization is approximately achieved with a small phase shift,and the experimental measurements and the model analysis both agree. The nu-merical results of Fradkov and Andrievsky for this device, [4], show simultaneousand approximate in-phase and anti-phase synchronization, tuned by different initialconditions. In another experimental device made of two rotors controlled by ex-ternal torques ([5, 6]), Andrievsky et al. , [5], reported approximate anti-phase andin-phase synchronization of the two oscillators. In this experiment, the synchroniza-tion parameter is the stiffness of a string connecting the two rotors.For demonstration purposes, Pantaleone, [7], reported an experimental deviceconstructed with two metronomes on a freely moving base. In this metronomes ex-periment, the phase difference of the synchronous state is close to 0 o . Increasingthe damping effect on the freely moving base, the author also reported approximatesynchronization with a difference in phase close to 180 o . From the theoretical pointof view, the equations describing this experimental device lead to the Kuramoto syn- ynchronization of Huygens’s clocks 3 chronization model, [8], where the synchronization mechanism is associated with anon-linear effect associated with the phase difference between the oscillators.In the Bennett et al. , [2], and the Pantaleone, [7], experimental systems, the in-teraction mechanism between oscillators is obtained by a moving base, an idea ad-vanced by Kortweg in 1906, [9].In the experimental systems described above, there is no clear evidence of whatmechanisms are in the origin of the anti-phase synchronization, as described byHuygens. In some experiments, it appears that if the pendulums have slightly differ-ent periods, the two oscillators may not synchronize, [2, 4].However, as this special type of collective rhythmicity occurs in biological sys-tems and several other natural phenomena, [10], where individual periods are differ-ent, it is important to derive and to understand the interaction mechanisms leadingto exact synchrony. Besides all the attention of the scientific community for thissynchronization phenomenon, there is no clear evidence of a mechanism, leading toanti-phase synchronization.In this paper, we propose and analyze a new interaction mechanism between os-cillators leading to exact anti-phase and in-phase synchronization. The synchroniza-tion mechanism is obtained with a damped elastic string, and the oscillators underanalysis can be simple harmonic oscillators, pendulums, and any type of non-linearoscillators with a limit cycle in phase space, as is the case of pendulum clocks, [11].The main result of this paper is to show that exact anti-phase synchronization canalways be achieved for systems of coupled oscillators.This paper is organized as follows. In section 2, we introduce the synchroniza-tion model in its full generality, discussing its physical assumptions, and we derivethe equations of motion of the interacting oscillators. Then, we make the approxi-mation of small oscillations. In section 3, we introduce a simplified pendulum clockmodel, and we prove that the ordinary differential equation describing the dynamicsof the clock has a limit cycle in phase space. Based on the linear model derived insection 2, in section 4, we discuss the concept of anti-phase synchronization, andwe find a sufficient condition for the existence of an exact anti-phase synchronizedstate for the two pendulum clock system. This result is stated as Theorem 1, the mainresult of this paper. After exploring numerically the phase space structure of the so-lutions of the model equations as a function of a control parameter, we show theexistence of exact anti-phase and in-phase synchronization states for the Huygens’stwo pendulum clocks system. The tuning between the two types of synchronizationregimes can be controlled through a damping constant associated with the (elastic)interaction mechanism and by the choice of initial conditions. These results justifysome numerical results published previously, [12]. In section 5, we analyze numer-ically the persistence of in-phase and anti-phase synchronization phenomena whenwe vary the parameters of the individual pendulum clocks. We show that the anti-phase and the in-phase synchronization regimes persist even if the two oscillatorshave different eigen-periods and parameters. Finally, in section 6, we resume theconclusions of the paper. Rui Dil˜ao
In the Huygens two pendulum clocks system, the pendulums are hanged in a com-mon support, and the only possible interaction is due to the tension forces generatedby the oscillatory motion of the two pendulums. These tension forces propagatethrough the common support, that we consider to be elastic. The role of the tensionforces in the synchronization mechanism is corroborated by the Huygens’s findingthat when the planes of oscillation of the two hanged pendulums are mutually per-pendicular, no synchronization is observed. In fact, the components of the tensionforces generated by the motion of the pendulums are in the plane of motion of thependulums, and force the motion of the two attachment points. To be more specific,we consider the geometric arrangement of Fig. 1, where the two pendulums havemasses m and m , and lengths (cid:96) and (cid:96) , respectively. Fig. 1
Model to analyze the synchronization of the Huygens’s two-pendulum clocks system. Thetwo pendulums are a representation of two nonlinear oscillators. The interaction between the pen-dulums is done by the tension forces at the attachment points, and they actuate through an elasticand resistive media. Each attachment points is considered to have mass M . The pendulums are considered connected by a massless string with stiffness con-stant k . The perturbations that propagate along the string are damped, and the damp-ing force is proportional to the velocity of the attachment points of the string, withdamping constant ρ . The damped string simulates the interaction effects that propa-gate through the elastic and resistive media. We consider that the attachment pointsof the pendulums, located at the horizontal coordinates x = x and x = x , have equalmasses, and we denote this mass by M . As we shall see, the introduction of this massis necessary to obtain explicitly the equations of motion.The system of Fig. 1, considered without the damping forces, is described by thefour degrees of freedom Lagrangian, L = m ( (cid:96) ˙ θ + ˙ x + (cid:96) ˙ x ˙ θ cos θ ) + m g (cid:96) cos θ + m ( (cid:96) ˙ θ + ˙ x + (cid:96) ˙ x ˙ θ cos θ ) + m g (cid:96) cos θ + M ( ˙ x + ˙ x ) − k ( x − x ) , (1) ynchronization of Huygens’s clocks 5 where θ and θ are the angular coordinates of the two pendulums, g is the acceler-ation due to the gravity force, and the last two terms describe the interaction mech-anism between the two pendulums. From (1), the Lagrange equations of motion ofthe two interacting pendulums are, m (cid:96) ¨ θ + m g sin θ = − m ¨ x cos θ m (cid:96) ¨ θ + m g sin θ = − m ¨ x cos θ ( M + m ) ¨ x + m (cid:96) ¨ θ cos θ = m (cid:96) ˙ θ sin θ + k ( x − x )( M + m ) ¨ x + m (cid:96) ¨ θ cos θ = m (cid:96) ˙ θ sin θ − k ( x − x ) . (2)Introducing the damping effects into system of equations (2), we obtain, m (cid:96) ¨ θ + f ( θ , ˙ θ ) + m g sin θ = − m ¨ x cos θ m (cid:96) ¨ θ + f ( θ , ˙ θ ) + m g sin θ = − m ¨ x cos θ ( M + m ) ¨ x + ρ ˙ x + m (cid:96) ¨ θ cos θ = m (cid:96) ˙ θ sin θ + k ( x − x )( M + m ) ¨ x + ρ ˙ x + m (cid:96) ¨ θ cos θ = m (cid:96) ˙ θ sin θ − k ( x − x ) , (3)where ρ is the damping constant of the attachment points of the pendulums, and thefunctions f ( θ , ˙ θ ) and f ( θ , ˙ θ ) describe the escaping mechanism of the pendu-lum clocks ( § M > (cid:96) > m > (cid:96) > m >
0, the system (3) can be solvedalgebraically in order to the higher derivatives, and we obtain, m (cid:96) ¨ θ + f ( θ , ˙ θ ) + m g sin θ = − m cos θ F m (cid:96) ¨ θ + f ( θ , ˙ θ ) + m g sin θ = − m cos θ F ¨ x = F ¨ x = F , (4)where, F = f ( θ , ˙ θ ) cos θ + m g sin θ cos θ − ρ ˙ x + m (cid:96) ˙ θ sin θ + k ( x − x ) M + m sin θ F = f ( θ , ˙ θ ) cos θ + m g sin θ cos θ − ρ ˙ x + m (cid:96) ˙ θ sin θ − k ( x − x ) M + m sin θ . (5)The system of ordinary differential equations (5) are a synchronization model forthe Huygens’s two pendulum clocks system, [12]. Here, we will consider only thecase M >
0. The case M = θ and θ , the system of equations (4)-(5)simplify, and we obtain, Rui Dil˜ao m (cid:96) ¨ θ + f ( θ , ˙ θ ) + m g θ = − m M (cid:0) f ( θ , ˙ θ ) + m g θ − ρ ˙ x + k ( x − x ) (cid:1) m (cid:96) ¨ θ + f ( θ , ˙ θ ) + m g θ = − m M (cid:0) f ( θ , ˙ θ ) + m g θ − ρ ˙ x − k ( x − x ) (cid:1) ¨ x = M (cid:0) f ( θ , ˙ θ ) + m g θ − ρ ˙ x + k ( x − x ) (cid:1) ¨ x = M (cid:0) f ( θ , ˙ θ ) + m g θ − ρ ˙ x − k ( x − x ) (cid:1) . (6)In this paper, our goal is to analyze the synchronization properties of the solutionsof the system of ordinary differential equations (6).To model the Huygens’s two pendulum clocks experiment, we have to choose aspecific form for the functions f ( θ , ˙ θ ) and f ( θ , ˙ θ ) , describing the oscillatorybehavior of pendulum clocks. To restore the energy lost by a pendulum clock during one period, the sustained os-cillations can be maintained by an impulsive force acting on the pendulum rod, orby the force originated by the smooth unwinding of a circular string attached to thependulum balance wheel, [11, pp. 169-200]. In any case, the dynamics of a pendu-lum clock is modeled by a two-dimensional dynamical system with a limit cycle inphase space. This same qualitative behavior can be obtained with an oscillator actu-ated by a non-linear damping force, piecewise proportional to the angular velocityof the pendulum. For small amplitude of oscillations, the proportionality constantis positive, and, for large amplitudes of oscillations, the proportionality constant isnegative.To simplify our analysis, we take, as a qualitative model for a pendulum clock,the following second order differential equation, m (cid:96) ¨ θ + f ( θ ; λ , ˜ θ ) ˙ θ + mg θ = , (7)where, f ( θ ; λ , ˜ θ ) = (cid:26) − λ if | θ | < ˜ θ λ if | θ | ≥ ˜ θ . (8)The function − f ( θ ; λ , ˜ θ ) ˙ θ is the damping force of the pendulum clock, and λ and˜ θ are positive constants. Proposition 1. If λ > , ˜ θ > , m > , (cid:96) > , and g > , the second order differentialequation (7), with the damping function (8), has a unique and stable limit cycle inphase space.Proof. Assume that λ >
0, ˜ θ > m > (cid:96) >
0, and g >
0. Define the function, ynchronization of Huygens’s clocks 7 F ( θ ; η , ˜ θ ) = m (cid:96) (cid:90) θ f ( s ; λ , ˜ θ ) ds = (cid:26) − ηθ if | θ | < ˜ θ ηθ − η ˜ θ if | θ | ≥ ˜ θ , where, η = λ / ( m (cid:96) ) , and ω = g /(cid:96) . With the new coordinate, x = ˙ θ + F ( θ ; η , ˜ θ ) ,the differential equation (7), with damping function (8), has the Li´enard form, (cid:26) ˙ θ = x − F ( θ ; η , ˜ θ ) ˙ x = − ω θ . (9)where F ( θ ; η , ˜ θ ) is a continuous and odd function of θ . The existence and stabilityof a limit cycle solution for equation (9) follows from the Li´enard theorem, [13, pp.179-181]. (cid:117)(cid:116) In the conditions of Proposition 1, the differential equation (7), with dampingfunction (8), has a unique limit cycle in phase space. In Fig. 2, we show the limitcycle in phase space of this non-linear oscillator, for two values of the parameter ω = g /(cid:96) . Fig. 2
Limit cycles solutions of the differential equation (7), with the damping function (8). In a)we have chosen ω = .
8, and in b), ω = . ω = g /(cid:96) , m = (cid:96) =
1. The parametervalues of the damping function are, λ = . θ = .
1. The two vertical lines for θ = ± ˜ θ showthe discontinuity of the tangential derivative along the limit cycles. The non-linear oscillator model defined by the differential equation (7), withdamping function (8), has all the qualitative properties found in more accurate pen-dulum clock models, [11], and will be used in the next sections to model Huygens’sclocks.
Rui Dil˜ao
By (6), the two interacting pendulum clocks with equal parameters are modeled thesystem of differential equations (6),¨ θ + ( m (cid:96) + M (cid:96) ) f ( θ ; λ , ˜ θ ) ˙ θ + ω ( + mM ) θ − ρ M (cid:96) ˙ x = − kM (cid:96) ( x − x ) ¨ θ + ( m (cid:96) + M (cid:96) ) f ( θ ; λ , ˜ θ ) ˙ θ + ω ( + mM ) θ − ρ M (cid:96) ˙ x = kM (cid:96) ( x − x ) ¨ x − M f ( θ ; λ , ˜ θ ) ˙ θ − mM g θ + ρ M ˙ x = kM ( x − x ) ¨ x − M f ( θ ; λ , ˜ θ ) ˙ θ − mM g θ + ρ M ˙ x = − kM ( x − x ) , (10)where f ( θ ; λ , ˜ θ ) is given by (8), and ω = g /(cid:96) .Our goals here is to show that the solutions of the system of differential equations(10) synchronize in anti-phase. For that, we begin by analyzing the solutions of thesystem of equations (10) for the particular case where the initial conditions in θ and θ are bounded by ˜ θ , that is, | θ ( ) | < ˜ θ and | θ ( ) | < ˜ θ . As the system of equations(10), together with (8), is piecewise linear, and if | θ ( t ) | < ˜ θ , and | θ ( t ) | < ˜ θ , forevery t ∈ [ , t ∗ ] , where t ∗ is a positive constant, then the piecewise linear system ofequations (10) can be written as the linear first order system of differential equations, ˙ θ ˙ v ˙ θ ˙ v ˙ x ˙ w ˙ x ˙ w = A B kM (cid:96) ρ M (cid:96) − kM (cid:96)
00 0 0 1 0 0 0 00 0
A B − kM (cid:96) kM (cid:96) ρ M (cid:96) mgM − λ M − kM − ρ M kM
00 0 0 0 0 0 0 10 0 mgM − λ M kM − kM − ρ M θ v θ v x w x w , (11)where we have introduced the new variables, v = ˙ θ , v = ˙ θ , w = ˙ x , w = ˙ x ,and the new constants, A = − ω ( + m / M ) , and B = λ ( / ( M (cid:96) ) + / ( m (cid:96) )) . As, | θ ( t ) | < ˜ θ and | θ ( t ) | < ˜ θ , for every t ∈ [ , t ∗ ] , then, for the same initial conditions,the small amplitude solutions of (10) and (11) coincide in the interval [ , t ∗ ] .Denoting by Q the matrix in (11), we have, Det Q =
0. A simple inspection of Q , shows that Q has only one zero eigenvalue, corresponding to the eigendirectiondefined by the equation x = x . We investigate now the stability of the line of fixedpoints x = x of both equations (10) and (11). Proposition 2. If λ > , M > , (cid:96) > , m > , and ρ > is sufficiently small, thenthe systems differential equations (10) and (11) have a line of fixed points with The vector x T = ( , , , , , , , ) is such that Mx = coordinates, θ = , v = , θ = , v = , w = , w = , x = x , and this lineof fixed points is Lyapunov unstable.Proof. We assume that, λ > M > (cid:96) > m >
0. The existence of the line offixed points follows by solving the equation Qy =
0, where, y T = ( θ , v , θ , v , x , w , x , w ) . For ρ =
0, the characteristic polynomial of the matrix Q in (11) is, q ρ = ( x ) = x ( A + ( B − x ) x ) (cid:0) gkm + M ( A + ( B − x ) x ) (cid:0) Mx + k (cid:1) (cid:96) − kx λ (cid:1) , where A = − ω ( + m / M ) and B = λ ( / ( M (cid:96) ) + / ( m (cid:96) )) . As the polynomial q ρ = ( x ) has the two roots, λ = (cid:16) B ± (cid:112) B + A (cid:17) , with positive real parts, B >
0, then, by continuity of Q ( ρ ) , for sufficiently small ρ >
0, the characteristic polynomial q ρ > ( x ) of the matrix Q , has also eigenvalueswith positive real parts. Therefore, for sufficiently small ρ > λ > M > (cid:96) > m >
0, the line of fixed points of the linear system (11) is Lyapunov unstable. Asboth systems of equations (10) and (11) have the same phase space orbits near thecommon line of fixed points, the local properties of the flows are the same, andthis is sufficient to prove the local instability of the flow defined by the system ofequations (10). (cid:117)(cid:116)
Proposition 2 gives the conditions of nonconvergence to the quiescent state ofthe solutions of the system of linear equation (11). This quiescent state is the line offixed points x = x in the eight-dimensional phase space. Under the conditions ofthe Proposition 2, the two pendulum clocks have sustained oscillations, in the sensethat, lim t → ∞ θ ( t ) (cid:54) =
0, and lim t → ∞ θ ( t ) (cid:54) = t n = t + nh , where n = , , · · · , and t is the initial time, we musthave, lim t n → ∞ θ ( t n ) = − lim t n → ∞ θ ( t n ) lim t n → ∞ x ( t n ) = − lim t n → ∞ x ( t n ) lim t n → ∞ v ( t n ) = − lim t n → ∞ v ( t n ) lim t n → ∞ w ( t n ) = − lim t n → ∞ w ( t n ) . (12)for every h > . In this context, we say that the two pendulum clocks synchronize in anti-phase if, for every h > t n → ∞ θ ( t n ) = − lim t n → ∞ θ ( t n ) . As we shall see in §
5, this definition can only be used in thecase of identical pendulum clocks.0 Rui Dil˜ao
We now define the new variables, θ = θ + θ , v = v + v , x = x + x and w = w + w . Then, adding the corresponding variables in the system of equations(11), we obtain, ˙ θ ˙ v ˙ x ˙ w = − ω ( + mM ) λ ( M (cid:96) + m (cid:96) ) ρ M (cid:96) mgM − λ M − ρ M θ vxw . (13)In the following, we call the system of equations (13) the reduced system of equa-tions associated to system (11). The reduced system of equations (13) has a line offixed points with coordinates, θ = v = w = x = constant. If the line offixed points of the reduced linear system (13) is asymptotically stable, then we have,lim t → ∞ θ ( t ) = lim t → ∞ θ ( t ) + lim t → ∞ θ ( t ) = t → ∞ v ( t ) = lim t → ∞ v ( t ) + lim t → ∞ v ( t ) = t → ∞ w ( t ) = lim t → ∞ w ( t ) + lim t → ∞ w ( t ) = . (14)If, for every h >
0, the first condition in (12) is verified, the first condition in (14)is also verified, and synchronous solutions of equation (10) are also stable solutionsof equations (13).As, by Proposition 2, lim t n → ∞ θ ( t n ) (cid:54) =
0, and lim t n → ∞ θ ( t n ) (cid:54) =
0, for any initialcondition away from the line of fixed points, any solution of the differential equation(10) that anti-phase synchronizes is also an asymptotically stable solution of thereduced system of equations (13).Hence, if the line of fixed points of the reduced system (13) is asymptoticallystable and the line of fixed points of the linear system of equations (11) is unstable(Proposition 2), then the solutions θ ( t ) and θ ( t ) of the system of equations (10)anti-phase synchronize. The asymptotic stability condition for the reduced the linearsystem (13), together with the instability condition of Proposition 2, both give asufficient condition for the existence of exact anti-phase synchronization of the twoidentical non-linear pendulum clocks.To analyze the stability properties of the line of fixed points of the system ofequations (13), we calculate the characteristic polynomial of the matrix P in (13), p ( y ) = y ( mM (cid:96) y + y ( m (cid:96) ρ − m λ − M λ )+ y ( m (cid:96) ω + mM (cid:96) ω − λ ρ ) + mg ρ )= yp ( y ) . (15)To the eigenvalue x = e x . This eigendirection is theline of fixed points of the linear system (13). If all the eigenvalues of the polynomial p ( y ) in (15) have negative real parts, any initial condition away from the line offixed points x = constant, evolve in time to this line of fixed points. Therefore, wehave: ynchronization of Huygens’s clocks 11 Theorem 1.
We consider the system of differential equations (10), with dampingfunction (8). If λ > , ˜ θ > , m > , (cid:96) > , M > , k > , g > , and ρ > issufficiently small, and if the reduced linear differential equation (13) has only non-positive eigenvalues, then, the solutions of equation (10), with damping function (8),synchronize in anti-phase, in the sense that, for every h > , lim t n → ∞ θ ( t n ) = − lim t n → ∞ θ ( t n ) , where, t n = t + nh, lim t n → ∞ θ ( t n ) (cid:54) = , and lim t n → ∞ θ ( t n ) (cid:54) = . Moreover, if thepolynomial, q ( ρ ) = m (cid:96) λ ρ − ( m λ + M λ + gm (cid:96) ) ρ + gm λ + gm M λ + gmM λ , has two real roots ρ and ρ , and ρ obeys to the inequalities, ρ < ρ < ρ ρ > ρ = λ (cid:96) ( + Mm ) , then, for any initial condition away from the line of fixed points of the system ofequations (10), the solutions of equation (10), with damping function (8), anti-phasesynchronize.Proof. The sufficient condition for the existence of anti-phase synchronization ofthe two pendulum clocks has been derived before the statement of the theorem. Theinstability of the line of fixed points of the system of equations (10) has been provenin Proposition 2. To prove the condition of non-positivity of the eigenvalues of thematrix in the system of equations (13), we use the Routh-Hurwitz criterion, [14]. By(15), as p ( y ) = a y + a y + a y + a , and as, by hypothesis, a > a >
0, bythe Routh-Hurwitz criterion, if a > ( a a − a a ) >
0, then the polynomial p ( y ) has only roots with negative real parts. As, ( a a − a a ) = − q ( ρ ) = − m (cid:96) λ ρ + ( m λ + M λ + gm (cid:96) ) ρ − gm λ − gm M λ − gmM λ , the polynomial q ( ρ ) has a global minimum for positive values of ρ and can havetwo real roots. This proves the first inequality of the theorem. The second inequalityfollows from the Routh-Hurwitz condition, a = m (cid:96) ρ − m λ − M λ > (cid:117)(cid:116) Theorem 1 gives a sufficient condition for the existence of exact anti-phase syn-chronization in the Huygens’s two pendulum clocks system. To test numerically theresults of Theorem 1, we take for the parameters of the nonlinear oscillator (7)-(8)the values, g = . m = (cid:96) = λ = .
1, and ˜ θ = .
1. For these parameter values,the period of the solutions on the limit cycle of the equation (7) is, T = . k = M = .
1, and ρ is a free parameter.To test the conditions of Theorem 1, in Fig. 3, we have plotted the eigenvalueswith the largest real part of the matrices Q and P , of the linear systems (11) and (13), respectively, as a function of the damping parameter ρ . The zero eigenvalue has beenexcluded from the characteristic polynomials of the matrices Q and P . Accordingto Theorem 1, if, 0 . = ρ < ρ < ρ = . ρ > ρ = .
11, and the matrix Q has eigenvalues with positive real parts, the two pendulum clocks synchronizein anti-phase. Numerically, for our reference parameter values, if ρ < ρ = . Q of the system of equations (11) has positive eigenvalues. Therefore,the conditions of Theorem 1 imply that, if ρ ∈ [ ρ , ρ ] , the two pendulum clockssynchronize in anti-phase, Fig. 3. Fig. 3
Eigenvalues with the largest real part of the matrices Q and P of the linear systems (11) and(13), respectively, as a function of the damping parameter ρ . The other parameter have been fixedto the values: g = . m = (cid:96) = λ = .
1, ˜ θ = . k =
10 and M = .
1. If ρ > ρ = .
121 and ρ < ρ = . P have non-positive real parts, and the line offixed points of system (13) is Lyapunov stable. If ρ < ρ = . Q has eigenvalues withpositive real parts, and the line of fixed points of system (10) is Lyapunov unstable. By Theorem 1,if ρ ∈ [ ρ , ρ ] , the two pendulum clocks synchronize in anti-phase. By Theorem 1, the anti-phase synchronization between the two non-linear pen-dulum oscillators exists for ρ ∈ [ ρ = . , ρ = . ] . In Fig. 4, we show thetime evolution of the angular coordinates and attachment points of the two pen-dulum clocks, for ρ = .
37, and calculated numerically from the system of equa-tion (10). We have chosen for initial conditions the coordinate values, θ ( ) = . θ ( ) = . x ( ) = x ( ) =
0, ˙ θ ( ) =
0, ˙ θ ( ) =
0, ˙ x ( ) = x ( ) = T = . T = . ynchronization of Huygens’s clocks 13 Fig. 4
Numerical solutions of the system of equations (10), with damping function (8), describingthe coupling of two identical pendulum clocks. The parameter values of the simulation are: m = (cid:96) = g = . k = M = . λ = .
1, ˜ θ = . ρ = .
37. The initial conditions are: θ ( ) = . θ ( ) = . x ( ) = x ( ) =
0, ˙ θ ( ) =
0, ˙ θ ( ) =
0, ˙ x ( ) = x ( ) = T = . Fig. 5
Anti-phase synchronization of the two pendulum clock system, for the same parametervalues of Fig. 4, except for the damping parameter ρ that, in this case, has the value ρ = .
1. Inthis simulation, the period of the exact anti-phase oscillations is T = . Fig. 6
In-phase synchronization of the two pendulum clock system, for the same parameter valuesof Fig. 4, except for the damping parameter ρ that, in this case, has the value ρ = .
01. In thissimulation, the initial conditions are : θ ( ) = . θ ( ) = . x ( ) = x ( ) =
0, ˙ θ ( ) = θ ( ) =
0, ˙ x ( ) = x ( ) =
0. The period of the in-phase oscillations is T = . Fig. 7
Anti-phase synchronization of the two pendulum clock system, for the same parametervalues of Fig. 6, and ρ = .
01. In this simulation, the initial conditions are : θ ( ) = . θ ( ) = − . x ( ) = x ( ) =
0, ˙ θ ( ) =
0, ˙ θ ( ) =
0, ˙ x ( ) = x ( ) =
0. Initially, the twopendulums are approximately in anti-phase. The period of the anti-phase oscillations is T = . Fig. 8
Coexistence of anti-phase and in-phase synchronization and aperiodic regimes for theasymptotic solutions of the system of equations (10). We show the limit cycle solutions of the sys-tem equations (10) for the same parameter values of Fig. 4, and several values of ρ : a) ρ = . ρ = .
07, c) ρ = .
06, and d) ρ = .
02. We have plotted the asymptotic solutions in phasespace for two different initial conditions. In one case we have taken θ ( ) = . θ ( ) = . θ ( ) = . θ ( ) = − .
3. For ρ > . ρ ≤ .
02, we have two limit cycles in phase space, one corresponding to anti-phasesynchronization, and the other to in-phase synchronization. The thin line curve is the limit cyclesolution of the reference equation (7), with damping function (8).
In Fig. 5, we have decreased the parameter ρ to values below the transition values ρ and ρ of Theorem 1. In this case, the two non-linear oscillators also anti-phasesynchronize. The numerical integration for several initial conditions shows that, forthe damping parameter value ρ = .
1, the system of equation (10) has a stable limitcycle in the eight-dimensional phase space. In this case, we are outside the condi-tions of Theorem 1.Decreasing further the parameter ρ , for ρ < .
06, and for the same initial con-ditions as in Fig.4, the two oscillators synchronize with the same phase (in-phase),Fig. 6. However, changing the initial conditions for an approximate anti-phase ini-tial state of the two pendulums, we obtain anti-phase synchronization, Fig. 7. Thissimple fact shows that, for ρ < .
07, there are two stable limit cycles in the eight-dimensional phase space, and these limit cycles have their own basins of attraction.
To analyze the transition from the anti-phase to the in-phase synchronizationasymptotic regimes, we have chosen ρ in the interval [ . , . ] , and we havechanged the initial conditions of the numerical simulations. For ρ in the interval [ . , . ] , the initial conditions that eventually lead to an in-phase synchronizedregime show quasi-periodic behavior in time. If ρ < .
06, the asymptotic in-phasesynchronized state is a limit cycle in phase space. This suggests the existence of anon-local bifurcation for ρ near the value ρ (cid:39) . ρ = . ρ = . ρ . Further numerical tests show that these results are still true in thesingular limit M → In the previous analysis, we have consider that both oscillators are characterized bythe same parameters. However, in real experiments this is not realistic and we mustconsider the persistence of synchronization when the parameters of the pendulumsare different. Here, the persistence of the anti-phase and the in-phase synchroniza-tion states is analyzed numerically. We take, m = m , m = m ( + ε ) , (cid:96) = (cid:96) and (cid:96) = (cid:96) ( + δ ) , where ε and δ can have positive or negative values. In this case, theequations of motion (6) are rewritten as,¨ θ + ( m (cid:96) + M (cid:96) ) f ( θ ; λ , ˜ θ ) ˙ θ + ω ( + mM ) θ − ρ M (cid:96) ˙ x = − kM (cid:96) ( x − x ) ¨ θ + ( + ε )( + δ ) ( m (cid:96) + M (cid:96) ) f ( θ ; λ , ˜ θ ) ˙ θ + ω ( + δ ) ( + mM ( + ε )) θ − ρ M (cid:96) ( + δ ) ˙ x = kM (cid:96) ( + δ ) ( x − x ) ¨ x − M f ( θ ; λ , ˜ θ ) ˙ θ − mM g θ + ρ M ˙ x = kM ( x − x ) ¨ x − M f ( θ ; λ , ˜ θ ) ˙ θ − mM ( + ε ) g θ + ρ M ˙ x = − kM ( x − x ) , (16) ynchronization of Huygens’s clocks 17 where f ( θ ; λ , ˜ θ ) is given by (8). Fig. 9
Anti-phase synchronization of two pendulum clocks with different lengths and masses. Ina), we have the anti-phase synchronized state as in Fig. 5. In b), we show the anti-phase synchro-nized state for ε = . δ = .
4. In c) and d), we show the limit cycles in phase space (thicklines) of the angular coordinates and of the attachment points of the two pendulums. The thin linesare the limit cycles for the cases ε = δ =
0. In b), the period of oscillation is T = . T = . t n → ∞ θ ( t n ) = − lim t n → ∞ θ ( t n ) , derived in §
4, is no longer verified.
We have integrated numerically the system of equations (16) for the same param-eter values of Fig. 5, and the same initial conditions, but with ε = . δ = . ε and δ .Fixing ε to the value ε = .
4, the anti-phase synchronized state still persists for δ ∈ [ − . , . ] . For δ = − .
2, and the same initial conditions as in Fig. 9, the systemin-phase synchronizes.The comparison between figures Fig. 9a) and Fig. 9b) shows that the definitionof anti-phase synchronization used in the previous section is specific to the case ofnon-linear oscillators with equal parameters. In Fig. 9b), the two pendulums clearlysynchronize in anti-phase, but, lim t n → ∞ θ ( t n ) (cid:54) = − lim t n → ∞ θ ( t n ) .As the model presented here shows anti-phase and in-phase synchronization phe-nomena for oscillators characterized by different parameters, these numerical resultsshow that the equality between the eigen-periods of the two pendulums is not re-quired to obtain synchronization. We have proposed a model of interaction between oscillators leading to exact anti-phase synchronization. This phenomena has been observed by the first time, in 1665,by Christiaan Huygens.The interactions parameters in our model are a damping constant ρ , a stiffnessconstant k of a linear string, and a mass parameter M . This mass parameter is as-sociated with the interaction, and not with the individual oscillators. For moderatedvalues of the (wet) damping constant ρ , k >
0, and M >
0, the asymptotic solutionsof the model equations converge to a stable limit cycle in the eight-dimensionalphase space of the model equations. Numerically, this limit cycle is the only stableattractor of the dynamics of the interacting oscillators.For smaller values of the damping constant ρ , two stable limit cycles in phasespace coexist. One corresponds to the anti-phase synchronized state of the two pen-dulum clocks, and the other corresponds to the in-phase synchronized state. The twolimits cycles are reached by different initial conditions in the model equations.The transition between the one-limit cycle solution and the two-limit cycle solu-tion appears by a global bifurcation tuned by ρ . After the bifurcation, the anti-phasestate coexists with an approximate quasi-periodic in-phase state.Changing the parameters of the individual pendulum clocks, we obtain the samesynchronization properties as in the case of oscillators with identical parameters.This fact shows that the interaction mechanism purposed here is robust to changesin the parameters of the nonlinear oscillators.In all the cases analyzed numerically, the anti-phase and the in-phase synchronyoccurs with periods different from the eigen-periods of the individual oscillators.This shows that the equality between the eigen-periods of the individual oscillatorsis not required to obtain anti-phase or in-phase synchronization.An important new issue introduced in the model is the possibility of existenceof small movements of the attachment points of the pendulums clocks, a situationclearly avoid in the modern experimental devices, and diverging from mechanismof synchrony proposed by Kortweg, [9]. This explains why modern experimentalsetups have not been able to reproduce the original Huygens’s results. Acknowledgements
I would like to thank the support of the Ettore Majorana Center for Sci-entific Culture, and the hospitality of the organizers of the conference ”Variational Analysis andAerospace Engineering”, dedicated to Prof. Angelo Miele for his 85th birthday. This work hasbeen partially supported by a Fundac¸˜ao para a Ciˆencia e a Tecnologia (FCT) pluriannual fundinggrant to the NonLinear Dynamics Group (GDNL).