Transition phenomena in two interacting van der Pol oscillators

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PHYSICS LETTERS A

20 November 1989

TRANSITION PHENOMENA IN TWO INTERACHNG VAN DER POL OSCILLATORS Y. MORIMOTO Physics Department, Meiji College of Oriental Medicine, Honoda, Hiyoshi, Funai, Kyoto 629-03, Japan Received 16 May 1989; revised manuscript received 29 August 1989; accepted for publication 14 September 1989 Communicated by A.R. Bishop

Two interacting van der P01 oscillators attain synchronous states as the coupling increases, when transitions are induced between asynchronous and synchronous states, and between synchronous states. The slowing down of characteristic time and the appearance of metastable mode are observed near the transition points.

Transformation ofoscillation modes is induced by external periodic force in self-sustained oscillator systems [1]. Recently the author succeeded in finding anomalous phenomena near the transition point between two oscillation modes by numerical analysis in the forced van der Pol oscillator [2—4].It is the presence of metastable oscillation modes with finite lifetime existing in the external force region where another mode should be stable. After the lifetime the metastable mode is destabilized, and transforms into an intrinsically stable mode. Of course such mode has not been (and cannot be) predicted by the traditional stationary solution analysis [11. Thus the numerical results show that the stationary solution assumption is not appropriate near the transition points. In such a region, two modes compete with each other with comparable order, and the fluctuations seem to play an important role just as near the critical point in the second order phase transitions of equilibrium systems [5]. This may be the reason why the stationary solution analysis breaks down near the transition points. A similar example has been found in the forced self-sustained hard mode oscillation equation [61, which is the presence of an amplitude modulated mode near the transition point between the quasi-periodic and fundamental oscillation modes. The above results indicate that the numerical analyses have the possibility to develop new fields in the nonlinear oscillation theory, especially in the transition phenomena among various oscillation

In this Letter we take up as an example showing transition phenomena two interacting van der Pol oscillators with slightly different frequencies through linear coupling. As is well known, two oscillators synchronize when the coupling becomes strong. Concerning this problem, detailed studies have been done classically [7]. However, the analyses were limited to the stationary states for their practical importance. So we reinvestigate this problem on the stand point of transition phenomena between synchronous and asynchronous states, or between synchronous states, and clarify the transition phenomena appearing in these transitions. Of course the synchronization of plural self-sustained oscillators is a practically important problem, and many extensive studies are now in progress. Especially Aronson et a!. [8] have reported comprehensive analytical and numerical studies on linearly coupled oscillators. The studies are very general, and treat a very wide region of bifurcation problems including period-doubling bifurcation, saddle-node bifurcation from periodic point and chaotic motion. The generality however obscures the specific pictures of the transition phenomena in this system, and so only the transition phenomena are directly treated numerically, and the attention is focused on them. The linearly coupled two van der Pol oscillators with slightly different frequencies of the form dx d2x —p~(1—x2) +w~x=E 1y, -~-~

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d2y

—,11 2(l—Y

)

PHYSICS LETTERS A

+w~y=E2x

(1)

are analyzed numerically. The coupling is taken to be diffusive such as E1 (~ —x) orE2 (x—~) in ref. [81, but there it is taken as E1y or E2x for simplicity. The gain and coupling terms are set to be symmetric, that is, ~ =u2=u and E1 =E2=E for the sake of simplicity. In the traditional theory, the synchronous state is analyzed by assuming x( I) = a cos (Qt —0) and y(t)=bcos(Qt—Ø), that is, by assuming two oscillations with common frequency Q. Substituting these solutions into eq. (1) and equating the terms of cos( Qt —0) or sin (Qt 0) gives the basic relations concerning a, b, 0 and 0. They are 2 _2)2 + (b2 _2)2 = 8, (a b2=a2(Q2—w~)/(Q2—w~), (2) a2Q2[u2Q2(l—~a2)2+(Q2—w~)2]=b2E2, (3) —

tan(O—O)=uQ(l—ka2)/(Q2—co~) (4) Eq. (2) shows that Qw2 when w2>w1. Determining a and b from eq. (2) and substituting them into eq. (3), Q is determined as follows: 6—(w~+w~)Q4+w?w~Q2—E2_—0. (5) Q Since Q—w 11, IQ—w21<w2, = cv~+ A two cases are present, Q 1 and Q 2 (0 0) without loss of generality. Fixing Acv, and increasing E causes a transition from an asynchronous state (A-state hereafter) having no common frequency to the synchronous state with common frequency. As mentioned above, two kinds of synchronous states are 2/ present, oneoscillates has common frequency of about 1 —E is 4L~cv,and nearly in-phase. This state .

called S0-state. The2other has theoscillates commonnearly frequency / 4i~w, and outof about 1+ ~cv+E phase. We call this state the S,~-state.As E increases 408

20 November 1989

from the state ofwhen mode transforms A—S~--+S0states 0, successively, z~w is relativelyassmall. When ~\w is large, the manner of transition sequence changes slightly, but we do not discuss such a case in this Letter. Thus two transitions appear at E~1 (A—~S,~) and at E~2(S,~—~S0). In the following eq. (1) is analyzed numerically near E~1and E~2,and their transition phenomena are investigated. The numerical analyses are undertaken through a second order Runge—Kutta method. We set 1u1=u2=l, w1=l and w2=l+Aco, as mentioned before. E1 = E2 = E is also adopted. For i~w~S0.07 the above-mentioned transition sequence takes place. When Aco 0.07 a different sequence appears. In this is self-sustained, do not depend on the (1) iniLetter i.~cois fixedthe to results be 0.0247. Since the system tial conditions except for short step initial transients. Here we take x(0)=l, ~(0)=2, y(O)=O.S and j’(0)=l.5. The Lissajous diagrams of two oscillations drawn in the x( t), y( t)) plane are utilized to check whether two oscillators are synchronized or not. The examples are represented in fig. 1, where the initial transients are excluded from the data. For E S0-state is attained, is represented in E~2,the fig. lc. The phase difference is easily recognized to be about 0 from fig. 1 c. Thus the above-mentioned transition sequence is confirmed. To investigate the transition phenomena appearing near the transition points, let us observe the time evolution of the point x( t) (>0) satisfying y( t) = 0, that is, the point passing through the positive x-axis of the phase space. It is clear that x( I) is constant in the S. and S0-states except for the short step initial transients. The result forE < E~1is represented in fig. 2a. Since the frequency ratio of two oscillations is not a simple rational in the A-state, x(t) oscillates over the x-axisnumber with some period determined by Aw and E. The period is indicated as T 1 in fig. 2a. As E approaches E~1under the condition E
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2

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des. With the elongation of T1, the stay time ofx( t) in the flat region increases as is clear from fig. 2a. The flat region clearly corresponds to the S,~-mode for E>.E~1(fig. 2b), which means the increment of the S,~-modecomponent in the A-state as E approaches E~1.Thus this phenomenon shows the change of stability between the A and SE-modes, rather than the usually observed slowing down of characteristic time near the transition points [2,6]. At E=E~1,T1 diverges, by which the system enters the synchronous state, the S,~-state.Ofcourse we cannot confirm T1 to be infinite numerically, but we checked it up to 10~steps. When E approaches E~1 under the condition E>E~1,none of the transition phenomena are observed, which is a usually ob-

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Fig. 2. Time evolutions of the points passing through the possitive x-axis ofthe Lissajous diagram. (a) E=O. 114 (A-state). T 1 shows the period of quasi-periodic oscillation, A-mode. (b) E~O.ll6(S,~-state).(c)E=0.302 (S,~-state).SinceEisnearE~, the metastable S0-mode is seen. The lifetime T2 is indicated. The scales of horizontal and vertical axes are common for all cases.

served fact in the transitions of forced self-sustained oscillator systems [2—4}.The state of E>E~1can affect the state of E < E~1when E < E~1,however, the latter cannot have an effect on the former when E>E~1,which says that the former is dominant. This is different from the phase transitions in equilibrium systems [5], in which the critical phenomena are observed when the control parameter (temperature, for example) approaches the critical point from both sides. For E~1
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20 November 1989

can see by referring to fig. 2c that the rhombus shape diagram (x( t) is large) appears first, and then loses its stability at T2, and after that the SE-modes (x( t) is small) with elliptic shape diagram stabilizes. For E~E~2,T2 becomes infinite, and the mode appearing before I’2 for E~E~2 becomes stable instead of the SE-mode. That is, the mode is the S0-mode. The

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SE-mode is intrinsically stable for E < E~2.However, for the condition (E~2—E)~ ~ the S0-mode seems to manifest its existence. The S0-mode, so to speak, can exist metastably even in the region of the SE-state, E~E~2. The behaviour of the S0-mode for E~E~2 clearly indicates that it is the metastable os-

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first observation of a metastable mode in the synchronization problem in two interacting oscillators to the author’s knowledge. When E approaches E~2 under the condition E> E~2no remarkable transition phenomena are observed as in the case of the tran-

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Fig. 4. The Lissajous diagram for E=0. 302. The metastable mode transforms into a stable S,,-mode after a finite life-time ~‘2. The initialtransients are excluded.

in fig. 2b. But when E approaches E~2under the condition E < E~2,another type of oscillation mode is generated after short step initial transients, before the SE-mode is stabilized. It persists long enough (>5 X 1 0~steps), after that it loses its stability, and then the SE-mode appears stably. This mode looks as if it were a metastable oscillation mode with finite lifetime [2—41.The behaviour of x( t) is represented in fig. 2c in which the lifetime of the mode is mdicated as T2. Similarly as T~,T2 shows a power law divergence as T2 1.7 x 10~(E~2 E) —0.49 which is represented in fig. 3 by solid circles. The Lissajous diagram for E= 0.302, the same value for fig. 2c, is represented in fig. 4 to see what the mode is. There the initial transients before l0~steps are excluded. Although the time evolution is not clear in fig. 4, we —

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cillation mode with finite lifetime recently observed in the forced van der Pol oscillator [2—4].This is the

teracting van der Pol oscillators with slightly different frequencies. Although the obtained results are at the present stage very preliminary, two interesting features are found, firstly a power law divergence of the characteristic time describing the transition, and secondly the appearance of a metastable oscillation mode with finite lifetime. In the transition region E~~ECLand E~E~2 remarkable features cannot be observed, that is, the transition phenomena take place for E lower than the transition points, which is to be noticed. The metastable mode has already been found in the forced van der Pol oscillator, which suggests the metastability to be a usual phenomenon in the nonlinear Oscillation problem. Aronson et al. mentioned [81 the stability of synchronized solutions. The in-phase solution (here the S0-mode) is stable for any coupling strength (here

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E) if it is larger than some critical value, while the out-phase solution (here the SE-mode) becomes unstable if E becomes larger than some critical value which depends on the frequency difference (here L~w),and shows a bifurcation to the fixed point or Hopf bifurcation depending on i~co.This points out that the manner of transition sequence differs by Eico. In this Letter the case of SE-+SO transition, that is, the bifurcation to the fixed point, is only investigated when z~cois small. When E&cv is large, the Hopfbifurcation may be observed, and different type transition phenomena will be expected. Its investigation is left for a future problem, and is now in progress.

20 November 1989

References [1] C. Hayashi, Nonlinear oscillations in physical systems (McGraw-Hill, New York, 1964). [2] Y. Monmoto, J. Phys. Soc. Japan 55 (1986) 2937. [3] Y. Monmoto, Trans. IEICE Japan E-69 (1986) 913. [4] Y. Monmoto, Trans. IEICE Japan E-70 (1987) 89. [5J H.E. Stanley, Introduction to phase transitions and critical phenomena (Clarendon, Oxford, 1971). [6] Y. Morimoto, Trans. IEICE Japan E-7 1 (1988) 959. [7] T. Suezaki and S. Mori, Trans. lElCEJapan 48 (1965) 1551. [8] D.G. Aronson, EJ. Doedel and H.G. Othmer, Physica D 25 (1987) 20.

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