Dynamics in a system of coupled oscillators

No.1

Qian,

Dynamics

in a System

et al: Dynamics

in a System

of coupled

5

of . . .

Oscillators

1

Min QIAN & Shu ZHU ’ (School of Mathematical Sciences, Peking University, Beijing 100871, China) This article considers the dynamical behavior of a particular system of N coupled oscillators with nearest neighbor coupling, the coupling constant K is assumed to be positive, independent of N. It is demonstrated that when the system does not have equilibrium points, the global attractor of this system is a one-dimensional restricted horizontal curve, therefore any solution will be frequency locked in the long time limit. On the other hand, if there are equilibrium points in the system, any solution is bounded and converges to an equilibrium point. coupled oscillators, horizontal curve, running periodic solution, frequencyKey Words: locking. Abstract:

Introduction In recent years, there has been much interest in the study of dynamical behavior of various systems of coupled oscillators. Such systems were introduced to describe the behavior of real physical phenomena such as the motion of Josephson junction arrays [1,7]; they were also used to model oscillating chemical reactions [Z] and neural network patterns [6,8]. In many applications, the main interest is in whether the ensemble of oscillators will be frequency locked or phase locked, and what is the influence of the input frequencies wj. In this paper, we consider a particular system of ordinary differential equations describing the dynamics of N identical overdamped oscillators with nearest neighbor coupling on a line il. =w-sinh+K(42-41), . . . . .. . . . & =wj-sin&++(+jj-i -2&++j+i), j =2,...,N-1, (1) . . . . .. . . . I in which K > 0 is the coupling coefficient, 4j (1 < j 5 N) is the phase angle of the jth oscillator, and wj E R. This is a special case of the models in one-dimensional parallel arrays of Josephson junctions. While in most references mentioned only numerical or approximate methods could be applied, in our case we will give exact analytic theorems. The method we use was first developed in [4], but in the present case, its power can be demonstrated in a more transparent fashion. In this paper, we analyse the system for N = 1 and N = 2 in some detail, then we state the results for general N 2 3.

1. Dynamics

of a single oscillator

For a single oscillator, its motion is governed by

& =w-sirid,

-

dt

where w 2 0. Because of the periodicity in 4, we may consider the phase space of (2) as a circle. (2) can be solved explicitly. It is easy to see that when w < 1, there are two ‘The 2This

paper was received author is supported

on Dec. by the

23th, 1996 Science Foundation

for Young

Scientists

of Peking

University.

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in Nonlinear

Science & Numerical

Simulation

1997

equilibrium points in the system, one is stable while the another one is unstable. Any solution will tend to the stable equilibrium point in the long time limit. The distance between the two equilibrium points decreases as w increases, and shrinks to zero as w tends to 1, so that the two equilibrium points become one homoclinic point when w = 1. If w > 1, there is no equilibrium point. In this case, any solution spins on the circle with period 27~/4= and rotation number ‘V = dwT/27r.

2. Dynamics

of a-coupled

oscillators

When there are two nonlinear oscillators, equation of motion of the group of coupled oscillators is given by $1 = WI - sin41 + $2 = w2 - sin42 +

(1) Horizontal Let ($l(t),&(t)) R2 + R2, 3iho,~20) b(J = 1+ 2/K. DEFINITION

a horizontal

K(42

- $I),

K(qh

-

(3)

42),

curves be the solution of (3) with initial value (410, ~$20). The map Pt : is called a shifi map of (3) with time t. Let = (h(t),$2(t)),

2.1 A curve in R2, C = ((3, h(s)) 1 s E R}, w h ere h is continuous, is said to be curve of p-type if h satisfies

-$I -

~2)

I h(sd - h(a) 5 b(sl -

~2)

for any s1 > ~2, where p 2 /?o. In addition to the above, if h(s + 27r) = h(s) + 2n, then .! is called a restricted horizontal curve. It is natural to consider (3) in R2. But because the system is invariant under the transformation $i + 4i + 2rk, where k is independent of i, we may also take the cylinder S1 x R as its phase space. A restricted horizontal curve above defined can either be regarded as a continuous curve on R2 or a closed curve winding on the cylinder S1 x R. By analyzing the variation equation of (3), it not difficult to prove the following lemmas which play an important role in the study of the problem. LEMMA 2.1 Let &, 5 /3 < 2,&. If C is a (restricted) horizontal curve of P-type, then for any t 2 0, Pte is also a (rest&ted) horizontal curve of P-type. 2.2 Let 1’ be a horizontal curve of 280-type. a horizontal curve of PO-type for all t 2 To.

LEMMA

Then there is To > 0 such that Pte is

As a consequence of Lemma 2.1, for each shift map P’, there exists a restricted horizontal curve e which is invariant under P’. Furthermore, by Lemma 2.2, any invariant curve is of B0-type.

(2) Rotation

number

Let C be an invariant restricted horizontal horizontal curve can be regarded as a circle, ing map on this circle. Therefore, limn++cx, 2.1 into account, we can show the following

curve of P’ for some 7 E R. Since a restricted P’ can be considered as an orientation preservPnT ($1 , 42) /2rnr exists. Then taking Lemma lemma.

LEMMA

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Qian, et al: Dynamics in a System of . . .

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2.3

For any solution (41 (t) ,& (t)) of (A’), the limits

are convergent, identical, and independent of the system.

of its initial value. Therefore, it is an invariant

We call this number the rotation number of system (3), and denote it by V. Lemma 2.3 implies that the rotation number of (3) is the same as the one of P’ on L. Since it is easy to verify that (3) has no periodic solutions, then we have the following lemma. 2.4 The following statements are equivalent. (I) The rotation number V = 0; (II) system (3) has equilibrium points; (III) any solution of (3) is bounded.

LEMMA

If V # 0, let T = l/V and e be an invariant horizontal curve of PT. Then there exists a point (&o,&o) E e such that PT(&o,qbo) = (qbo,qbo) + (27r, 2~) [3]. Therefore, (3) has at least one solution with the following property, ($1 (t + T), $2 (t + T)) = (h(t),

$2 (4 + P,

2~)

for all t E R. We call this kind of solution a running periodic solution. The orbit of such a solution is called a running periodic orbit. LEMMA

2.5 The rotation

number V # 0 if and only if the system has a running periodic

solution.

(3) Uniqueness

of the invariant

manifold

In this subsection, we suppose that V # 0 or K > l/2. With some geometrical considerations, from Lemmas in the above subsection, using the method given in [4], we can show the following lemma. 2.6 Let e be a restricted horizontal Pte = C for all t pi R.

LEMMA

curve such that P’k’ = e for some r > 0. Then

From Lemma 2.6 together with Lemma 2.2, there follows the uniqueness of the invariant restricted horizontal curve. LEMMA 2.7 If e and e’ are two restricted horizontal curves satisfying Pte = L and Pte’ = e’ for all t E R, then C = f?.

(4) Global

stability

of the invariant

manifold

If V # 0, we introduce a map Q : R2 + R2 by Q(&, $2) = PT(&, 42) - (27r, 2w), where T = l/V. It is able to show, from Lemma 2.2, that for any (41, $2) E R2 the sequence converges to a point (47, q5f) E e, where L is the unique invariant restricted {Q’Yh> 42H:S horizontal curve. Therefore, Dist{(&(t),&(t)),e} --+ 0 as t + +oc for any solution of (3). This shows that e attracts any orbit of (3), so that e is the global attractor. When V = 0, from Lemma 2.4 any solution of (3) is bounded. Furthermore, it is easy to construct a Liapunov function of (3) to show that any solution will tend to an equilibrium point. Notice that if K > l/2, there is an invariant restricted horizontal curve e through each equilibrium point. By the uniqueness of the invariant restricted horizontal curve, C contains all equilibrium points of (3). In fact, e consists of equilibrium points and their unstable manifolds, therefore it is the global attractor.

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Communications

3. Conclusions

in Nonlinear

Science & Numerical

Simulation

1997

and final remark

Using similar methods, we can similarly define the concepts of a (restricted) horizontal curve and a running periodic solution and prove all results in the above section for general N 2 3. We state the results in the following theorems. THEOREM

3.1 For any solution (&(t),

. . . , $N(t))

of (I), the limits

are convergent, identical and independent of initial values. Therefore, it is ara invariant the system. The nmnber is called the rotation number of (l), denoted by V.

of

Let Ko = 1/4sin2(n/2N). 3.2 (I:) If V # 0, the global attractor of (1) is th,e unique invariant restricted horizontal curve, which is composed of a mnning periodic orbit. (II) If V = 0 and K > Ko, the global attractor of (1) is the invariant restricted horizontal curve, which consists of all equilibrium points and their unstable manifolds.

THEOREM

Remark 1 When V = 0, one can not expect that the global attractor is one-dimensional in general since the dimensions of the unstable manifolds of some equilibrium points may be greater than one. Remark 2 If V # 0, according to Theorem 3.2(I), any solution will tend to the running periodic orbit. In other words, all oscillators will be frequency locked in the long time limit.

References

PI Hadley,

P., Beasley, M. R. and Wiesenfeld, K., Phase locking of Josephson junction series arrays, Phys. Rev. B, 38 (1988), pp. 8712-8719. PI Kuramoto, Y., Chemical Oscillations, Waves and Turbulence, Springer-Verlag, New York, 1984. PI Nitecki, Z., Differentiable Dynamics, M.I.T. Press, Cambridge, MA, 1971. PI Qian Min, Shen Wenxian and Zhang Jinyan, Global behavior in the dynamical equation of J-J type, J. Differential Equations, 71 (1988), pp. 315-333. [51 Qian Min, Shen Wenxian and Zhang Jinyan, Dynamical behavior in coupled systems of J-J type, J. Differential Equations, 88 (1990), pp. 1755212. PI Schuster, H. G., Nonlinear Dynamics and Neuronal Networks, VCH, Weinheim, 1991. VI Wiesenfeld, K. and Hadley, P., Attractor crowding in oscillator arrays, Phys. Rev. Lett., 62 (1988), pp. 1335-1338. PI Winfree, A. ‘T., The Geometry of Biological Times, Springer-Verlag, New York, 1980.