Synchronization of two pulse-coupled oscillators

29 June 1998 PHYSICS LETTERS A

ELSEVIER

Physics Letters A 243 (1998) 205-207

Synchronization of two pulse-coupled oscillators George Schmidt Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA Received 1 November 1997; revised manuscript received 23 February 1998; accepted for publication 10 March 1998 Communicated by A.R. Bishop

Abstract

In several Letters Pereira studied phase locking of Andronov clocks. A modified version of Pereira's scheme leads to much stronger coupling between oscillators as well as phase synchronization. Generalizations to other systems, including nonlinear oscillators, are also discussed. © 1998 Elsevier Science B.V.

More than three centuries ago Huygens observed that two pendulum clocks hung side by side on the wall became synchronized [ 1 ]. More recently it was found that a variety of periodic biological, chemical as well as physical systems, weakly coupled, tend to be synchronized. For instance fireflies in South East Asia flash in synchrony [ 1 ], and heart pacemaker cells [ 2 ], and Josephson junction arrays synchronize [3], to mention just a few. Many Letters have recently been published giving a mathematical description of synchronization in coupled nonlinear systems. These descriptions fall into two categories. In the first, limit cycle oscillators, moving on a circular orbit in phase space are given an attracting coupling, and the statistical development is studied [4]. In the second, pulse-coupled oscillators are used, where it is assumed that when an oscillator "fires", the others are given a kick [ 5]. Pereira [6-8] studied pendulum clocks with Coulomb friction as described by Andronov et al. [9] described by the linear equations = y,

) = -x-

fl sgn(y),

with an energy input at each cycle

(1)

y--~ (y2-Fh2)1/2,

whenx=-fl,

y>0.

(2)

When the energy input supplied by the escapement is large enough h > 4fl the system approaches a stable limit cycle. In the x - y phase plane this consists of a circle quadrant in the upper half plane, centered at x = - f l , y = 0, ending at y = 0, x > 0, a semicircle in the lower half plane centered at x = fl, y = 0, and another quadrant in the upper half plane centered at x = - f l , y = 0. The trajectory is closed by a vertical line of length 4fl at x = - f l representing the instantaneous energy input. In the model introduced by Pereira [ 6 ] the coupling between oscillators is provided by a weak signal propagating from a clock at the time of the energy input and reaching the other clock. The short signal is antisymmetric in time and leads to phase locking between the two clocks with a difference of zr/2. It will be shown that a symmetric signal leads to synchronization (phase difference zero), and also much stronger coupling. The following derivation of Eqs. ( 3 ) - ( 9 ) closely follows that of Ref. [7], the main difference being the symmetry of the signal. The signal reaches the second clock at t = to, lasts a short time 7" and has the form

0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 5 - 9 6 0 1 ( 9 8 ) 0 0 2 2 4 - 2

206

G. Schmidt/Physics Letters A 243 (1998) 205-207

p(t)= ~

aksin[ka(t-

to) ]

if to < t < to + r,

k odd

=0

otherwise,

(3)

where a = rr/r produces a symmetric signal around to + r/2. The second equation of Eqs. (1) is now modified to ~9 = - x - / 3 sgn(y) + p ( t ) , which can easily be solved to yield

x( t) = ~

ak/[1 - (ka) 2] sin[ ka( t - t0)]

Zakka/[

-

l - ( k a ) 2] sin(t - to) + x<°~ ( t ) , (4)

where the second term is needed in order to have y(to) = y
6r = [x2(t ') + y Z ( t ' ) ] l / 2 - { [ x ( ° ) ( t ' ) ] 2 h-

[y(O)(tt)]2}l/2,

(5)

Since the signal is weak l/r << 1. A comparison with Pereira's model of antisymmetric perturbation [7] yields two significant differences. In that case 6r and &# go to zero as r ~ 0, and ~q~ is proportional to sin q~, leading to phase locking, ~o = 0 or rr depending on the sign of I. For Eq. (8) the fixed points correspond to 6q~ = 0, yielding ~o = +7r/2, where if l > 0, ~ = +7r/2 is stable giving phase synchronization, and if 1 < 0, ~p = - ~ - / 2 giving antisynchronous locking with a phase difference 7r. In order to study the evolution of two oscillators, first one notes that since the angular velocity is independent of r, 6r plays no role in the synchronization process. Start out with two clocks such that when the second clock receives its energy input (¢'2 = 7r/2) and sends the signal, the first clock has a phase ~on, 0 < ~Pl < ~r/2. The phase difference is ~F = ¢'2 - ~Pl = 7r/2 - go1. This changes the phase of the first clock by

&pl = 1/ r cos ~pl = 1/ r sin q<

(10)

For I > 0 this results in the reduction of the phase difference ~qt = -6q~1 = - 1 / r sin ~ . Next the first clock reaches the angle ~o'z = 7r/2 with the second clock behind at q~ = 7r/2 + ~ + 6 ~ , resulting in the change of this angle 6q~ = 1/ r cos qd2 = - 1 / r sin ( Ts + ~qs ) . For one cycle this leads to the map

~q~ = -- sin -1 [y<°)(t')x(t') -- x
q~ = ~F + ~( ~ + ~3q~) = IF - l/r sin( qS - I/r sin q~ ), (ll)

(6) with t r = to + r. Expanding in r gives

x ( g ) = x <°) + o ( r 2 ) , y ( t ' ) = 2 ~f-~ akr/k~r + y(O) + o ( r 2 ) ,

(7)

which yields ~r =/sin~o,

~3~o=l/rcosq~,

(8)

where the strength of the signal is characterized by the integral to+'r

I= / to

p ( t ) dt = 2 ~ k odd

ak/ka.

(9)

representing the change of the relative angle per period. This map converges asymptotically to qs _+ 0. The principal assumption in the procedure was that during the short time period r the unperturbed solution is represented by a circular segment on the x - y plane. Since this is true for curves representing smooth nonsingular solutions in general, the method can be used to study a variety of other problems. An obvious candidate is the replacement of the artificial Coulomb damping by the more usual form

.t=y,

y=-yy-x,

(12)

with the energy input at x = 0, y > 0, and y << 1. The energy input can be more general than the one given in Eq. (2), e.g. a constant velocity change. When the

G. Schmidt/Physics Letters A 243 (1998) 205-207

external signal arrives at t = to, the unperturbed system is situated at Xo, Yo, and one may expand to get = Y,

Y = -TYo - x,

(13)

an equation like Eq. (1) where fl sgn(y) is replaced by TYo. In the close vicinity of xo, Y0 the trajectory can be well approximated by a circular segment centered at -TYo. Eq. (8) again follows trivially and the interacting clocks phase synchronize for l > 0. A further generalization can be carried out for nonlinear oscillators, described by the equation )~ + 7Jc + f ( x ) = 0,

(14)

with f ( 0 ) = 0 . Expanding about x0, Yo gives + YYo + f ( x o ) + f ( x o ) (x - xo) = 0. Introducing ( f ' ( xo ) ) l /2t --. t one arrives at .t=y,

5' = - x - A ,

(15)

where A = 7 y o / f ( x o ) + f ( x o ) / f ( x o ) - xo, the same form as Eqs. ( I ) and (13). The change in r and ¢p described by Eq. (8) immediately follows. Now, however, since the frequency of nonlinear oscillations is amplitude dependent the change in phase during an oscillation period also depends on 8r. Consequently, = 0r/2 where 8~p = 0, are no longer fixed points, and exact synchronization is not possible. Since, however, at ~o= 0, ~-, 8r vanishes there are still a pair of fixed

207

points present giving rise to phase locking. For weak nonlinearity near synchronization can be achieved. A different generalization of Pereira's model was given by Abraham [ 10]. The author wishes to thank S. Strogatz for calling his attention to the work of Pereira. References [1] S.H. Strogatz, I. Stuart, Scientific American, December (1993) p. 102. [2] A.T. Winfree, The Geometry of Biological Time (Springer, Berlin, 1980). [3] A.A. Chemikov, G. Schmidt, Phys. Rev. E 52 (1995) 3415. [4] S.H. Strogatz, R.E. Miroilo, P.C. Matthews, Phys. Rev. Lett. 48 (1992) 2730; S.H. Strogatz, R.E. Mirollo, Physica D 31 (1988) 143. [5] C.S. Peskin, Mathematical Aspect of Heart Physiology, Courant Institute of Mathematical Sciences, New York University (1975), p. 368; R.E. Mirollo, S. Strogatz, SIAM J. Appl. Math. 50 (1990) 1645. [6] J.V. Pereira, in: Dynamical Systems and Microphysics, eds. A. Avez, A. Blaquiere, A. Marzolio (Academic Press, London, 1981 ). 17] A.M. Nunes, J.V. Pereira, Phys. Lett. A 107 (1985) 362. 18] J.V. Pereira, Int. J. Non-Linear Mechanics 24 (1989) 177. [9] A.A. Andronov, A.A. Vitt, S.E. Khaikin, Theory of Oscillators (Pergamon, London, 1961 ). [10] R.H. Abraham, in: A Chaotic Hierarchy, eds. G. Baler, M. Klein (1991) p. 49.