Critical phenomenon of two coupled phase oscillators

The Journal of China Universities of Posts and Telecommunications December 2013, 20(Suppl. 2): 121–127 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Critical phenomenon of two coupled phase oscillators LI Bo (), CHEN Zi-chen, QIU Hai-bo, XI Xiao-qiang School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

Abstract Measure synchronization (MS) is a novel phenomenon which can be found in some coupled Hamiltonian systems (HS). In this paper, we built a new model called two coupled phase oscillators system (CPOS), and found that, under the appropriate initial conditions, it will meet a critical phenomenon in the phase space as the coupling strength increases. We also found the critical state by introducing the Poincaré section technique. We established a new formula for the critical coupling strength by analyzing the HS model, and extended this formula to the CPOS. The numerically analytical results show the reason why the measure synchronization of CPOS does not exist. Keywords

MS, critical phenomenon, Poincaré section, two coupled phase oscillators

1 Introduction

10]. Oloum et al. discussed the relationship between Gaussian curvature and Hamiltonian chaos [11]. While the mathematical analysis of the MS in coupled Hamiltonians system is developing slowly. In this paper, we found that the value of elements in HS model have exchanged when MS is reached. By using this characteristics of HS model, we construct a new formula of the critical coupling strength ( K c ), and

Synchronization is a typical nonlinear phenomenon, which was first reported by Huygens’s observation of two pendulum clocks closely hanged on the wall of his workplace in 1665 [1–2]. Since 1990, with the rapid development of nonlinear science, many different types of synchronization phenomenon have been found. In 1999, a novel synchronization phenomenon, called MS was found in a non-dissipative system of two coupled HS by Hampton and Zanett. MS is characterized by a commonly shared phase space with invariant measure as the coupling strength increases and it attracted more and more interest [3]. Vincent found the partial measure synchronization in duffing model [4]. Chen et al. proposed numerical method used to judge whether MS occurs or not [5]. Tian et al. found the separatrix crossing is the critical dynamical transition behind the MS transition behavior [6–8]. The key technology in chaos communication is chaos controlled. Wang et al. studied the φ 4 model and found

canonical equations of HS are:

the chaos of Hamiltonian can be controlled by MS [9–

θ1 = p1 , p1 =

Received date: 29-10-2013 Corresponding author: LI Bo, E-mail: [email protected] DOI: 10.1016/S1005-8885(13)60208-3

apply a new mathematical analysis to MS in coupled Hamiltonians system to explain why it does not have MS in two coupled phase oscillators model, by comparing HS model and the two coupled phase oscillator model, combined with the Poincaré section. 2 Comparison of two models 2.1 HS model

Based on HS model, Hampton and Zanette proposed how to calculate the critical coupling strength ( K c ). The K ⎫ sin(θ 2 − θ1 ); mod 2π ⎪ ⎪ 2 ⎬ K θ2 = p2 , p 2 = sin(θ1 − θ 2 ); mod 2π ⎪ ⎭⎪ 2

(1)

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Here the initial conditions are taken as (θ1 (0),θ 2 (0), p1 (0), p2 (0)) = (0, 0, 0.1, 0.2) . The calculation formula of K c is [3] : Kc =

ξ(0) 2 2 [1 + cos(ξ (0)) ]

(2)

where ξ (0) = θ1 (0) − θ 2 (0), ξ(0) = p1 (0) − p2 (0) . For the specific

initial

conditions,

we

can

compute

K c = 0.002 5 . The numerical result shows that MS will

be reached around K c = 0.002 5 . As the coupling strength increases, when K < K c , the orbit of the oscillator 1 will expand outwards, while the orbit of the oscillator 2 will expand inward (see Figs. 1(a) and 1(b)). A critical phenomenon will occur when K = K c (see

(c) K=0.002 6 > Kc

Fig. 1(b)). The two oscillators will share the same phase space when K > K c (see Figs. 1(c) and 1(d)). For other coupled HS which exist MS (except the chaotic MS), as the coupling strength reach the critical value, the critical phenomena will also occur.

(d) K=0.002 6 > Kc Fig. 1 Trajectories of coupled phase oscillators of Eq. (1) in the (θ , p) plane.

(a) K=0.000 1, the system does not reach the measure synchronization

From Eq. (1), the system takes part in a four-dimensional phase space. And in the absence of dissipation, the energy manifolds of four dimensional phase space are restricted to three dimensional phase space. In order to visualize the dynamics of this system, we can use the Poincaré section technique to reduce the recorded orbit points to a two-dimensional phase plane [12]. By solving canonical Eq. (1) and plotting ( θ1 , p1 ) at each time that θ 2 = 0 and θ2 > 0 , and plotting ( θ 2 , p2 ) at each time that θ1 = 0 and θ1 > 0 . We show the overlapped Poincaré section with several discrete coupling strength around K c . In Fig. 2(a), each curve in ( θ1 , p1 ) phase plane represents a specific

(b) K = K c = 0.002 5

coupling intensity K. Here, we set K=0.000 1, 0.01, 0.02, 0.023, 0.025 000 1, 0.002 55, 0.002 6, 0.002 65. The innermost closed curve corresponds to K=0.000 1. As the coupling intensity increases, the section slice expands in size. The crossing separatrix corresponds to K=0.025 000 1 which means MS transition happen. As

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the coupling intensity keeps increasing, the Poincaré section slice is shaped like crescent moon and the size shrinks as k increases further.

(a) The Poincaré section in ( θ1 , p1 ) phase plane

(b) The Poincaré section in ( θ 2 , p2 ) phase plane

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crossing separatrix appears at K=0.025 000 1 corresponding to MS transition. As the coupling intensity increases further, Poincarl section slice is shaped like crescent moon and the size shrinks. We can also see a critical point when K = K c (see Fig. 2(c)). It shows that MS can be judged by the Poincaré section, and the separatrix crossing is the critical dynamical transition behind the MS transition behavior [3]. By analyzing the Poincare section, we can see that the MS is quasi-periodic. The coordinate of the critical point is [0, 0.15]. The essence of the separatrix crossing is energy crossing. It can be said that the oscillator 1 and 2 will achieve energy exchange when the coupling strength reaches a special value. The oscillator 1 can reach the initial energy of the oscillator 2 and vice visa (it can be achieved at the same time or at the different time). It is easy to find that the maximum value of p1 (t ) and p2 (t ) is equal (see Figs. 3(a) and 3(b)). If p is regarded as the speed, p1 = − p 2 , p1,2 = p1,2 (0) + υ (t ) . The two oscillators always have the same acceleration in amplitude with opposite direction. The two oscillators cannot achieve energy exchange under the critical coupling strength (K= 0.002 5). When K > K c , the two oscillators can exchange energy and the system can reach MS. According to this special properties of the HS model, we create another formula for K c . We first assume that K can drive the two oscillators into MS, the initial conditions is [ p1 (0), p2 (0), θ1 (0), θ 2 (0)] , p1 (t ) and p2 (t ) can reach [ p1 (0) + p2 (0)]/ 2 at the same time. So we get p12 (0) + p22 (0) K cos(θ 2 (0) − θ1 (0)) p12 ( x) + p22 ( x ) 2

−

2 K cos(θ 2 ( x) − θ1 ( x)) 2 p (0) + p2 (0) p( x) = p1 = p2 = 1 =a 2

=

2

−

(3) (4)

From Eq. (3) and (4) we can get (c) The Poincaré section when K = K c Fig. 2 The Poincaré section under different coupling strength K

In Fig. 2(b), The Poincarl section in ( θ 2 , p2 ) phase plane. Before MS transition point, the outermost closed ring-like curve corresponds to k=0.000 1. As the coupling intensity increases, the size of section slice shrinks. The

K=

p12 (0) + p22 (0) − 2a 2 cos(θ 2 (0) − θ1 (0)) − cos(θ 2 ( x) − θ1 ( x))

(5)

We know the change of the parameter K can make the system reach MS, and we assume that the parameter K can increase to make the system reach the MS, so the minimum K is equal to K c . The calculation shows that Eqs. (2) and (5) have the same results.

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2.2 The coupled phase oscillators model

The Hamilton equation of this system is: ⎫ p12 + p22 − K ( p1 − p2 ) sin(θ1 − θ 2 ) ⎪ 2 ⎬ ⎪ H i = K ( p1 − p2 ) sin(θ1 − θ 2 ) ⎭ H=

(6)

The corresponding canonical equation: (a) The trajectories of p1 (t ) when K=0.002 5

θ1 = p1 − k sin(θ1 − θ 2 ); mod 2π p1 = k ( p1 − p2 ) cos(θ1 − θ 2 ); mod 2π θ = p − k sin(θ − θ ); mod 2π 2

2

2

1

p 2 = −k ( p1 − p2 ) cos(θ1 − θ 2 );

(b) The trajectories of p2 (t ) when K=0.002 5

⎫ ⎪ ⎪ ⎬ ⎪ mod 2π ⎪⎭

The system is conservative, so we can use the same method as that of in HS model. We set the initial conditions as (θ1 (0), θ 2 (0), p1 (0), p2 (0)) = (0, 0, 0.1, 0.2) . From Fig. 4, the small circle is (θ1 , p1 ) , the big one is (θ 2 , p2 ) , we can find the orbits of two oscillators in phase plane approach each other as the increase of the coupling strength K. When the coupling strength increases even further, the orbits will reach a critical state. If the coupling strength keeps increasing, it will always maintain this critical state. The evolvement of the two oscillators is different from the orbits of HS model in phase plane.

(c) The trajectories of p1 (t ) when K=0.002 6

(a) K=0.01 (d) The trajectories of p2 (t ) when K=0.002 6

(e) min | p2 − p1 | as the parameter K increases Fig. 3 The evolution of p(t) under the different values of K

(7)

(b) K=0.1

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(c) K=0.2

(d) K=10 Fig. 4 The orbits in phase plane of two coupled phase oscillators model with different coupling strength K

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(b) The Poincaré section in ( θ 2 , p2 ) phase plane

(c) Correspond to the amplification of (a) respectively

We also apply the Poincaré section to the two coupled phase oscillators model. We plotted the Poincaré section (θ1 , p1 ) [(θ 2 , p2 )] with θ 2 = 0 ( θ1 = 0 ) and θ2 > 0 ( θ1 > 0 ) for different value of the parameter K. In Figs. 5(a) and 5(c), we can see the area of the section become bigger with the increase of K.

(d) Correspond to the amplification of (b) respectively Fig. 5 The Poincaré section under different coupling strength K

(a) The Poincaré section in ( θ1 , p1 ) phase plane

In the Figs. 5(b) and 5(d), they have the same situation. As the value of K increase, the system seems to keep critical state. From Fig. 5, we cannot see the separatrix crossing behavior which means no MS exists in the system. From Eq. (7) one can find that

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p1 (t ) = p1 (0) + λ (t ) ⎫ ⎬ p2 (t ) = p2 (0) − λ (t ) ⎭

(8)

p1 = p2 = 0.15 , H i = 0 , the system energy is not

conserved, so

p1

We use the same method as we find Eq. (5) and get K=

It can be seen that the system have the same characteristics as that of the HS model, the two oscillators can maintain the same rate of change. For the initial conditions of Fig. 4, as the K increases, p1 and p2 close to 0.15 at the same time. From Eq. (6), when and

p2

are closing to an

intermediate value, but they cannot reach the value. So the system cannot have the energy exchange, the Poincaré section of the system does not have separatrix crossing behavior, which MS cannot be reached.

2013

p1 (0)2 + p2 (0)2 − p1 ( x) 2 − p2 (0) 2 2( p2 ( x) − p1 ( x))sin(θ1 ( x) − θ 2 ( x))

(9)

[ p1 ( x) − p 2 ( x )] → 0 , so K c = lim K = ∞ . For two

coupled phase oscillators model, this method has a certain limitation. This system cannot reach MS, but we can assume the K can make the system reach MS. In some initial conditions, Eq. (9) cannot be used, such as [θ1 (0),θ 2 (0), p1 (0), p2 (0)] = (0.1, 0.2, 0.1, 0.1) , this initial conditions is fixed points of the system [12–13]. Because the system does not have MS, so we can use min[ p2 (t ) − p1 (t )] to calculate the distance between the two orbits in phase plane. The distance decreases with the increase of parameter K. The Fig. 3(c) can be approximated as (1 + 0.15 x) / x , x ∈ [0, +∞] . 3 Conclusions In this paper, we apply a new mathematical analysis to MS, and find out the formula of the critical coupling strength ( K c ). The MS in coupled HS actually is the

(a) The evolution of p1 (t ) when K=0.2

energy exchange behavior. This exchange can occur as the increase of the parameter K. However, in our two coupled phase oscillators model, this exchange cannot occur because the structure of the system. It will be said that the system does not have MS, i.e. the critical coupling strength is infinite. The previous studies are focused on the Hamiltonian system which MS can be reached. It is difficult to the method of using the mathematical derivation to establish the formula about K c for the complex model with chaos. These works will be given a further investigation in the future.

(b) The evolution of p2 (t ) when K=0.2

(c) The evolution of min (p2 (t ) − p1 (t )) with the increase of K Fig. 6 The evolution of P(t) under the different values of K

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