Phase locking of two limit cycle oscillators with delay coupling S. A. Usacheva and N. M. Ryskin Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 023123 (2014); doi: 10.1063/1.4881837 View online: http://dx.doi.org/10.1063/1.4881837 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Design of coupling for synchronization in time-delayed systems Chaos 22, 033111 (2012); 10.1063/1.4731797 Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise Chaos 20, 033126 (2010); 10.1063/1.3488977 Universal occurrence of the phase-flip bifurcation in time-delay coupled systems Chaos 18, 023111 (2008); 10.1063/1.2905146 Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators Chaos 16, 023111 (2006); 10.1063/1.2197168 Synchronization in large directed networks of coupled phase oscillators Chaos 16, 015107 (2006); 10.1063/1.2148388

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CHAOS 24, 023123 (2014)

Phase locking of two limit cycle oscillators with delay coupling S. A. Usacheva and N. M. Ryskin Saratov State University, 83 Astrakhanskaya st., Saratov 410012, Russia

(Received 3 February 2014; accepted 26 May 2014; published online 4 June 2014) Mutual phase locking in the system of two limit cycle oscillators with delay coupling is studied. Conditions of phase locking are derived as a result of analysis of a generalized Adler equation. The analytical results are compared with numerical simulation. Depending on the phase shift of the coupling signal propagating between the two oscillators, either in-phase or anti-phase mode of synchronization may arise. The number of possible modes of synchronization increases with the C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4881837] delay time. V

Mutual synchronization is a fundamental phenomenon observed in ensembles of coupled oscillators. Delay of the coupling signal significantly affects synchronization in ensembles of interacting microwave oscillators because distance between the elements of the ensemble is large in comparison with the wavelength. Time-delayed dynamical systems are infinite dimensional and thus may show a very complex behavior. In this paper, we present the results of analysis and numerical simulation of mutual phase locking of two limit cycle oscillators with radiative coupling, when the oscillators are coupled via partial penetration of output radiation of one oscillator into the circuit of its counterpart and vice versa. We demonstrate that the picture of phase locking develops as a result of counterplay between dissipative and reactive types of coupling. Which one dominates, depends on the phase shift of the coupling signal propagating between the two counterparts. Moreover, when the delay time is large enough, higher-order modes of synchronization appear, which is inherent in infinitedimensional time-delayed dynamical systems.

I. INTRODUCTION

The problem of synchronization in coupled oscillator arrays is important for many applications in electronic engineering, laser physics, biophysics, etc.1–4 Using of arrays of coupled microwave oscillators is a promising technique for power combining.5–7 General theory of synchronization of two coupled limit cycle oscillators is well developed.1–4 Two basic mechanisms of synchronization of periodic oscillations can be defined (see, e.g., Ref. 4). When natural frequencies of two interacting oscillators are close to each other, a resonance limit cycle appears on the torus as a result of saddlenode bifurcation. With increase of coupling, frequencies of the interacting oscillators gradually approach each other and finally merge into a single peak. This mechanism is named phase (or frequency) locking.4 The other mechanism named suppression of natural dynamics occurs when frequency mismatch and coupling strength are large enough. In that case, torus birth occurs via Andronov–Hopf bifurcation. Phase locking of high-power vacuum tube oscillators, such as magnetrons or vircators has been studied theoretically and experimentally in many works.8–15 However, at 1054-1500/2014/24(2)/023123/9/$30.00

microwave frequencies, time delay of a signal travelling between the two counterparts should be taken into account, because the distance between two oscillators can essentially exceed the wavelength.8–12 From the standpoint of the theory of oscillations, systems with time delay have infinitedimensional phase space, and thus it is reasonable to expect that the picture of synchronization in such systems will have many differences from systems with low number of degrees of freedom. In particular, a number of special features have been observed in Ref. 16 where the problem of forced synchronization of a delayed-feedback oscillator driven by an external harmonic signal has been addressed. The effect of time delay in ensembles of coupled oscillator has been studied in several works, especially with applications to biological systems (recent review is presented in Ref. 17). The interaction of two active nonlinear lumped circuits coupled through a common resistive load has been studied.18–20 However, in Refs. 18–20, the main attention has been paid to the effect of amplitude death which is typical for resistively coupled oscillators. On the contrary, in microwave oscillator arrays, coupling is usually provided by partial penetration of output radiation of one oscillator into the resonator of its counterpart (so-called radiative coupling). In this paper, we investigate phase locking of two radiatively coupled oscillators with delay coupling. A model of the coupled oscillators described by two delay-differential equations (DDEs) for the slowly varying amplitudes is derived in Sec. II. In Sec. III, we present the results of theoretical analysis. We derive the generalized Adler equation for the phase difference between the two coupled oscillators and analyze the phase locking in the small delay limit. In Sec. IV, we compare these results with the numerical integration of the original DDEs. II. PHENOMENOLOGICAL MODEL OF MICROWAVE OSCILLATORS WITH DELAY COUPLING

An injection locked self-excited microwave oscillator is usually represented as an equivalent RLC lumped circuit which is described by the equation5–12

24, 023123-1

  d2 V x0 Ye dV x0 dVinj ðtÞ ; þ x20 V ¼  1 Q YL Q dt dt2 dt

(1)

C 2014 AIP Publishing LLC V

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where V is the equivalent circuit voltage, Vinj ðtÞ is the injection voltage, YL is the admittance of the circuit together with the load, Ye is the electronic admittance which accounts pffiffiffiffiffiffiffiffiffifor ð Þ C=L is electron beam interaction withprf-field, Q ¼ 1=Y L ffiffiffiffiffiffi the loaded Q-factor, x0 ¼ 1= LC is the circuit resonance frequency, L and C are the equivalent inductance and capacitance of the circuit, respectively. If one approximates the electronic admittance as Ye  g0  g2 V 2 , that is a typical assumption in the case of weak nonlinearity, (1) becomes the forced van der Pol equation.1–4 Note that in (1) we neglect reactive nonlinearity, i.e., assume the oscillator to be isochronous. In the system of two mutually coupled oscillators the signal delivered to the first oscillator is the part of the output signal of the second one and vice versa. Thus, instead of (1) we can write down the following system of two DDEs (the subscripts j ¼ 1; 2, k ¼ 2; 1 indicate individual oscillators):  xj qjk dVk ðt  l=cÞ d2 Vj xj  2 dVj þ x2j Vj ¼ : (2)  k  b V j j j 2 dt Qj dt Qj dt Here, kj ¼ g0;j =YL:j  1 are the parameters of excitation, bj ¼ g2;j =YL;j , qjk are the parameters of coupling, l is the distance between the oscillators. The coupling parameters characterize the ratio of incoming and outgoing power,8–10 qjk ¼ ðPjin =Pkout Þ1=2 . In (2), delay caused by finite transit times of the coupling signals is taken into account. Assume that all the parameters for both oscillators are identical, i.e., k1;2 ¼ k, Q1;2 ¼ Q, b1 ¼ b2 ¼ b, q12 ¼ q21 ¼ q, except a small mismatch of the individual free-running frequencies, x1;2 ¼ x0 6Dx=2;

Dx  x0 :

(3)

Introducing slowly varying amplitudes Vj ðtÞ ¼ b1=2 Aj ðtÞ expðix0 tÞ þ c:c: (c.c. denotes complex conjugate) and applying the well-known method of averaging21 to (2), we obtain the system of “slow-flow” equations dA1 iD  A1  ðk  jA1 j2 ÞA1 ¼ qeiw A2 ðt0  sÞ; dt0 2   dA2 iD þ A2  k  jA2 j2 A2 ¼ qeiw A1 ðt0  sÞ; 0 dt 2

(4)

where t0 ¼ x0 t=2Q is the slow time scale, D ¼ 2QDx=x0 is the normalized frequency mismatch, s ¼ x0 l=ð2QcÞ is the normalized delay time, and w ¼ x0 l=c. III. THEORETICAL ANALYSIS A. Small delay limit: Generalized Adler equation and its analysis

Consider the situation when the normalized delay time is small, i.e., s  1. In that case, we can neglect the delay in the right-hand sides of (4)   iD A_ 1 þ A1 ¼ k  jA1 j2 A1 þ qeiw A2 ; 2   iD _ A 2  A2 ¼ k  jA2 j2 A2 þ qeiw A1 : 2

Hereinafter, dots denote the time derivatives. Introducing in   (5) real amplitudes and phases, Aj ¼ Rj exp iuj , gives   R_ 1 ¼ k  R21 R1 þ qR2 cosðw þ uÞ;   (6) R_ 2 ¼ k  R2 R2 þ qR1 cosðw  uÞ; 2

D R2 ¼ q sinðw þ uÞ; R1 2 D R1 u_ 2  ¼ q sinðw  uÞ; R2 2 u_ 1 þ

where u ¼ u1  u2 is the phase difference. Subtracting Eqs. (7), we obtain

R1 R2 u_ þ D ¼ q sinðw  uÞ  sinðw þ uÞ : R2 R1

(7)

(8)

Thus, the system of DDEs (4) reduce to the third-order system of ordinary differential equations (ODEs) (6), (8). Nevertheless, this system still takes into account the effect of delay owing to the phase shift w. In the case of zero frequency mismatch, D ¼ 0, this system has been studied in detail.22 In the mode of phase locking both oscillators have the same frequency, i.e., uj ¼ xt þ uj0 , and the amplitudes are constant R1;2 ¼ const. Eqs. (6), (8) give R2 cosðw þ uÞ; R1 R1 R22 ¼ k þ q cosðw  uÞ; R2

R1 R2 sinðw  uÞ  sinðw þ uÞ : D¼q R2 R1 R21 ¼ k þ q

(9)

(10)

Also, from (7) we obtain the equation for the locked frequency

q R2 R1 x¼ sinðw þ uÞ þ sinðw  uÞ : (11) R2 2 R1 Consider the limit of weak coupling (q  k) when the amplitudes of the oscillations remain close pffiffiffi to their freerunning values. Setting in (8) R1 ¼ R2 ¼ k, we obtain the following equation for the phase difference:5–10 u_ þ D ¼ 2q cos w sin u;

(12)

which has the form of the classic Adler equation.23 However, in the case of delayed coupling this equation does not give the complete picture of phase locking. Evidently, (12) is not valid when w  p=2. More delicate analysis is required. Following Refs. 24 and 25, express the amplitudes R1;2 from (9) as R21  k þ q cosðw þ uÞ; R22  k þ q cosðw  uÞ;

(13)

(5) and calculate the ratio of the amplitudes in the first-order approximation

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pffiffiffi k þ 2pq ffiffik cosðw þ uÞ R1 q  pffiffiffi  1  sin w sin u: q ffiffi R2 k p cosðw  uÞ kþ

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(14)

2 k

(1) When w ¼ 2pn the coupling is purely dissipative and (16) reduces to the standard Adler equation u_ þ D ¼ 2q sin u:

(19)

Similarly, R2 q  1 þ sin w sin u: R1 k

In the mode of phase locking u_ ¼ 0 and (19) reads (15) sin u ¼ 

Substituting (14) and (15) into (8), after some calculations we obtain the generalized Adler equation u_ þ D ¼ 2q cos w sin u 

q2 2 sin w sin 2u: k

(16)

The two terms in the right-hand side of (16) represent dissipative (diffusive) and reactive (inertial) coupling, respectively.25 Equations (10) and (11) which define the locked state now read

D : 2q

(20)

The locking band is given by the well-known formula1–3 D2  ð2qÞ2 :

(21)

Because cos u > 0 inside the locking band, this mode of phase locking is called the in-phase mode.22 For D ¼ 0 the phases of two oscillators are exactly the same, u ¼ 0. (2) When w ¼ 2pn þ p Eqs. (19) and (20) become u_ þ D ¼ 2q sin u:

(22)

2

q sin2 w sin 2u; k

(17)

q2 2 sin u sin 2w: 2k

(18)

D ¼ 2q cos w sin u  x ¼ q sin w cos u 

A special attention should be paid to three particular cases.

sin u ¼

D : 2q

(23)

For this case the locking band is still given by (21), but cos u < 0 for w ¼ 2pn þ p. This is the out-of-phase mode.22 For D ¼ 0 the oscillators are locked strictly in anti-phase, u ¼ p.

FIG. 1. Domains of stability of the in-phase (I) and anti-phase (A) mode of phase locking on the D; q plane for k ¼ 0:5, w ¼ 0 (a), 0.25p (b), 0.4p (c), 0.5p (d), 0.6p (e), and 0.75p (f).

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(3) When w ¼ 2pn6p=2 the coupling is purely reactive.25 In that case Eq. (16) reads u_ þ D ¼ 

q2 sin 2u; k

(24)

and the locking band is D2  q4 =k2 :

(25)

Unlikely the case of dissipative coupling, when the locking band is proportional to q, in the case of inertial coupling it is proportional to q2 . Note that the two modes of phase locking are degenerate, i.e., inside the locking region either in-phase or out-of-phase locked state arises, depending on the initial conditions. The theory based on the classic Adler Eq. (12) fails to describe this situation. In a general situation, both dissipative and inertial coupling are essential. The edge of the locking region can be found from the condition

FIG. 2. An example of graphical solution of Eq. (30) for w ¼ 0:2p, s ¼ 1:0, q ¼ 0:2 (1), 0.5 (2), and 0.8 (3). Stable and unstable solutions are shown with black and white circles, respectively.

To analyze the phased-locked state in the case of finite delay, one should replace sinw ! sinðw þ xsÞ, cos w ! cosðw þ xsÞ in Eqs. (17), (18), and (26). This gives D ¼ 2q cosðw þ xsÞsin u 

dD 2q2 2 sin w cos 2u ¼ 0; ¼ 2q cos w cos u  k du

q2 2 sin ðw þ xsÞsin 2u; (27) k

q2 2 sin u sin½2ðw þ xsÞ; (28) 2k k cosðw þ xsÞcos u : (29) q¼ 2 sin ðw þ xsÞcos 2u

x ¼ q sinðw þ xsÞcos u 

which gives q¼

k cos w cos u : sin2 w cos 2u

(26)

Equations (17) and (26) are the parametric representation of the domain of stability of the phase-locked states in D; q parameter plane. In Fig. 1, these domains are shown for k ¼ 0:5 and different w. When w ¼ 0, only the in-phase mode exists (Fig. 1(a)), and its stability boundary has the classic shape of a synchronization tongue. When w ¼ 6 0, the boundary distorts and the locking band reduces. Also the domain of the out-of-phase locking appears (Fig. 1(b)). Unlikely the in-phase mode, it does not touch the q ¼ 0 axes, i.e., there exist the threshold of synchronization. The minimal value of coupling required for the out-of-phase locking can be found assuming u ¼ p in (26) that gives q ¼ kcosw=sin2 w. Because q < 1 by definition, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi out-of-phase mode appears at cos w < 1 þ ðk=2Þ2  k=2. With the increase of w the boundary of the out-of-phase mode shifts downwards (Fig. 1(c)), and, at w ¼ p=2 merges with the boundary of the in-phase mode (Fig. 1(d)). At p=2 < w < 3p=2 the out-of-phase mode dominates, and the stability boundary of the in-phase mode shifts upward qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Fig. 1(e)). At cos w ¼ k=2  1 þ ðk=2Þ2 the in-phase domain vanishes. B. Phase locking in the case of finite delay time

Studying the case when the delay is not negligibly small is much more complicated task because DDEs, in general, have infinite number of eigenmodes. As a result, with the increase of delay there appear a multitude of synchronized states.17,26

FIG. 3. Domains of stability of the in-phase (I) and anti-phase (A) mode of synchronization on the D; q plane for k ¼ 0:5, s ¼ 1:0, w ¼ 0:1p (a) and w ¼ 0:48p (b). The results of theoretical analysis and numerical simulation are shown by solid lines and circles, respectively. BT denote Bogdanov–Takens points.

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The phase locking conditions can be obtained from solution of these equations. Note that by neglecting the second-order  terms O q2 in (27), (28) we arrive to the formulas of Ref. 26. However, similar to Sec. III A that approximation overlooks the effect of reactive coupling and appearance of the out-of-phase mode. For simplicity, let us examine the case of zero frequency mismatch, D ¼ 0. Evidently, from (27) it follows that sin u ¼ 0. Thus, there exist two kinds of solutions, the inphase solutions for which cos u ¼ 1, and anti-phase ones for which cos u ¼ 1. The Eq. (28) becomes x ¼ 7q sinðw þ xsÞ:

of solutions increases. This is confirmed by Fig. 2 where an example of graphical solution of (30) for different q is presented. The eigenfrequencies are determined by intersections of sinusoid y ¼ 6sinðw þ xsÞ and straight lines y ¼ x=q. In Ref. 27, we proved that the solutions with cosðw þ xsÞ < 0 are unstable while the solutions with cosðw þ xsÞ > 0 are stable. Thus, with the increase of delay new modes of phase locking should appear. For better understanding of this picture, let us simulate Eqs. (4) numerically. IV. NUMERICAL RESULTS

(30)

A similar equation has been studied in Ref. 27 where an oscillator with time-delayed reflection from a remote load has been studied. Analysis of (30) reveals27 that with the increase of either coupling strength q or delay s the number

For the numerical integration we use the 4th-order Runge–Kutta method adapted for DDEs.28 In Fig. 3(a), an example of the domain of phase locking on the (D; q)-plane in the case of finite delay time s ¼ 1:0 and w ¼ 0:1p is presented. Solution of the Eqs. (27)–(29) is shown by solid

FIG. 4. Waveforms and spectra of the oscillations illustrating the phase-locking mechanism of synchronization: D ¼ 0:1, k ¼ 0:5, w ¼ 0:1p, s ¼ 1:0, q ¼ 0:03 (a), 0.05 (b), 0.052 (c), 0.053 (d).

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lines, while the solutions of the original DDEs (4) are shown by circles. For small enough D and q, the results of simulation are in good agreement with the results of theoretical analysis. With the increase of q the validity of the Eqs. (27)–(29) breaks and the results of simulation begin to differ from the

Chaos 24, 023123 (2014)

theory. The simulation shows that there exist two different mechanisms of synchronization. Mutual phase locking of the two oscillators is observed only if the frequency mismatch D is small enough. Typical waveforms and spectra illustrating this mechanism are presented in Fig. 4. When the coupling is small we observe the beating regime when the amplitudes of

FIG. 5. Waveforms and spectra illustrating the suppression mechanism of synchronization: D ¼ 1:0, k ¼ 0:5, w ¼ 0:1p, s ¼ 1:0, q ¼ 0:3(a), 0.4 (b), 0.45 (c), 0.47 (d), 0.48 (e).

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FIG. 6. Waveforms and spectra illustrating the hard transition from the out-of-phase mode to the in-phase mode at w ¼ 0:48p, s ¼ 1:0, D ¼ 0:1, q ¼ 0:19 (a) and 0.18 (b).

oscillations R1;2 oscillate in time (Fig. 4(a)). With the increase of coupling, fundamental frequencies of the oscillators get closer and the beating period increases (Figs. 4(b) and 4(c)). Finally, the beating period tends to infinity and transition to the phase-locked mode happens (Fig. 4(d)). Now the frequencies of the two oscillators become identical and the amplitudes are constant. At larger values of D there exists another mechanism known as synchronization by suppression.4 This mechanism is illustrated by Fig. 5. With the increase of coupling gradual suppression of the spectral component corresponding to the natural frequency of the first oscillator x  0:4 is observed. The main frequencies remain almost constant, but the beating amplitude decreases (Figs. 5(b)–5(d)). Finally, synchronous oscillation on the second oscillator frequency (x  0:2) establishes (Fig. 5(e)). Accordingly, the torus in the phase space gradually shrinks to become a stable limit cycle. On the contrary, at D < 0, the first oscillator suppresses the second one. The transition from the locking scenario to the suppression scenario occurs at D  60:63 in so-called Bogdanov–Takens (BT) points. As was shown in Sec. III, when the phase shift w is close to p=2 inertial coupling dominates and bistability appears, i.e., different modes of synchronization may establish, depending on the initial conditions. In Fig. 3(b), domains of synchronization for w ¼ 0:48p are shown. The results of simulation are in good qualitative agreement with the theory, despite with the increase of D and q the difference becomes evident. To plot the boundaries of synchronization domains for the out-of-phase mode we set the initial conditions as   A1;2 ðtÞ ¼ R0 exp iðxt þ u1;2 Þ for t 2 ð0; s, where R0 and x are the amplitude and frequency on the corresponding mode determined from (30). The phases should be chosen as u1 ¼ u2 þ p because at D ¼ 0 this mode is purely antiphase.

First, we take D ¼ 0 and choose a sufficiently high value of q which lies in the domain of stability of the higher-order mode of synchronization. We run the numerical simulation until the synchronization regime establishes. After that, we increase the detuning very smoothly step-by-step and perform the simulation with heritage of the initial conditions. We repeat this procedure until we observe a hard transition (switching) to the in-phase mode. In Fig. 6, waveforms and spectra are presented which demonstrate the hard transition. At q ¼ 0:19, we observe the out-of-phase mode when the first oscillator dominates, consequently, the common oscillation frequency is x  0:158. With a very small decrease of coupling, transition to the inphase mode occurs. Now the second oscillator dominates and x  0:143. Note that for the out-of-phase mode we are unable to determine whether synchronization occurs via phase locking or via suppression because with smooth increase of coupling we always arrive to the in-phase mode. Fig. 7 shows transformation of the synchronization domains on the (D; q)-plane with the increase of delay. New stable single-frequency solutions appear, which correspond to the higher-order modes of synchronization (Figs. 7(b) and 7(c)). The stability boundaries for these solutions were plotted using a similar procedure as described above for the out-of-phase mode shown in Fig. 3(b). When we cross the stability boundary, we observe switching to the stable singlefrequency solution with nearest frequency similar to that shown in Fig. 6(b). The stability boundaries shift downwards with the increase of s. Analysis of Eqs. (27)–(29) predicts a similar behavior. V. CONCLUSION

In this paper, mutual phase locking in the system of two limit cycle oscillators with delay coupling is studied. In the

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a result of phase locking but as a result of suppression of natural oscillation of one oscillator by its counterpart. More rigorous theoretical analysis taking into account both mechanisms will be the subject of further studies. With the increase of the delay time the number of possible modes of synchronization increases and higher-order synchronization domains appear on the plane of parameters (D; q). The boundaries of these domains shift downwards with the increase of s. This may cause frequency hopping between the synchronized states. In this regard, the correct account of time delay of a coupling signal is important for understanding phase locking of coupled oscillators at microwave frequencies. ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (Grant Nos. 12-02-01298a and 12-02-31493).

1

FIG. 7. Synchronization domains on the D; q plane at k ¼ 0:5, w ¼ 0:2p and s ¼ 1:0 (a), 3.0 (b), 5.0 (c).

limit of small delay the generalized Adler equation is derived. Its analysis reveals that the phase-locking conditions depend strongly on the phase shift w of the coupling signal propagating between the two counterparts. In particular, if sinw  0, the coupling is purely dissipative and either the in-phase (at w  2pn) or the anti-phase mode of synchronization (at w  2pn þ p) is stable. On the contrary, when cosw  0, the coupling is reactive and the bistability of phase-locked states exists, i.e., both the in-phase and antiphase modes are stable. Thus, a variation of distance between the two oscillators in about a half wavelength may significantly change the synchronization conditions. The results of analysis of the generalized Adler equation are verified by numerical simulations of the DDEs (4). When the coupling strength and frequency mismatch are small enough, the numerical results are in good agreement with the theory. However, at large D synchronization comes on not as

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Phase locking of two limit cycle oscillators with delay coupling.

Mutual phase locking in the system of two limit cycle oscillators with delay coupling is studied. Conditions of phase locking are derived as a result ...
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