bifurcation, amplitude death and oscillation patterns in a ... · of oscillators and nonlinear...

Bifurcation, amplitude death and oscillation patterns in a system of threecoupled van der Pol oscillators with diffusively delayed velocity coupling

Yongli Song,1,a) Jian Xu,2 and Tonghua Zhang3

1Department of Mathematics, Tongji University, Shanghai 200092, China2School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China3Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Mail H38, PO Box 218Hawthorn, Victoria 3122, Australia

(Received 8 February 2011; accepted 22 March 2011; published online 20 April 2011)

In this paper, we study a system of three coupled van der Pol oscillators that are coupled through the

damping terms. Hopf bifurcations and amplitude death induced by the coupling time delay are first

investigated by analyzing the related characteristic equation. Then the oscillation patterns of these

bifurcating periodic oscillations are determined and we find that there are two kinds of critical

values of the coupling time delay: one is related to the synchronous periodic oscillations, the other is

related to eight branches of asynchronous periodic solutions bifurcating simultaneously from the

zero solution. The stability of these bifurcating periodic solutions are also explicitly determined by

calculating the normal forms on center manifolds, and the stable synchronous and stable phase-

locked periodic solutions are found. Finally, some numerical simulations are employed to illustrate

and extend our obtained theoretical results and numerical studies also describe the switches of stable

synchronous and phase-locked periodic oscillations. VC 2011 American Institute of Physics.

[doi:10.1063/1.3578046]

Coupled dynamical system arises in a variety of contexts

in the physical, biological, and social sciences and has

broad relevance to many areas of research. Synchroniza-

tion phenomena in the coupled system is ubiquitous and

of interest to researchers in different research fields.

Generally speaking, time delay is inevitable in the

coupled systems since the information flows between the

individual units are never conveyed instantaneously, but

only after some time delay. A nature question is how

delay coupling affects the synchronization patterns. Here,

we consider a delay-coupled system consisting of three

van der Pol oscillators and attempt to analytically investi-

gate how the coupling time delay and the coupling

strength affect the stability and synchronization patterns.

Hopf bifurcations and amplitude death induced by the

coupling time delay are investigated and the correspond-

ing relation between the synchronization patterns of

Hopf bifurcating periodic oscillations and the critical val-

ues of the coupling time delay is explicitly found.

I. INTRODUCTION

Since synchronization phenomena is frequently encoun-

tered in nature and can help to explain many phenomena in

biology, chemistry, physics and has potential applications in

engineering and communication, it has been considered as

one of the fundamental characteristics of collective dynamics

of coupled systems and has permanently remained an object

of intensive research in nonlinear science. System of coupled

self-oscillators is often regarded as a basic model in the theory

of oscillators and nonlinear dynamics dealing with synchroni-

zation phenomenon. A system of coupled nonlinear oscillators

is capable of displaying a rich dynamics and has applications

in various areas of science and technology, such as physical,

chemical, biological, and other systems.1–4 In particular, the

effect of synchronization in systems of coupled oscillators

nowadays provides a unifying framework for different phe-

nomena observed in nature (for more reviews, see Refs. 5–7).

The van der Pol equation has long been studied as a

quintessential example of a self-excited oscillator and has

been used to represent oscillations in a wide variety of appli-

cations.8,9 It evolves in time according to the following sec-

ond order differential equation:

€yðtÞ þ eðy2ðtÞ � 1Þ _yðtÞ þ xðtÞ ¼ 0; (1)

where y is position coordinate and e is a positive real number

indicating the damping strength. It is well known10 that

Eq. (1) has a unique periodic solution which attracts all other

orbits except the origin, which is the unique equilibrium

point of Eq. (1) and unstable. It has been shown in Refs. 11

and 12 that linear feedback can annihilate limit cycles and

stabilize the origin.

Recently, there are increasing interests on the study of

the synchronization behavior in the coupled van der Pol

oscillators (see, e.g., Refs. 13–17 and references therein). In

Ref. 13, the synchronization of four coupled van der Pol

oscillators has been carried out. It has been shown that, for

the coupling constant below a critical value, there exists a

region in which a diversity of phase-shift attractors is pres-

ent, whereas for values above the critical value an in-phase

attractor is predominant. It is also observed that the presence

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected].

1054-1500/2011/21(2)/023111/15/$30.00 VC 2011 American Institute of Physics21, 023111-1

CHAOS 21, 023111 (2011)

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of an antiphase attractor in the subcritical region is associated

with sudden changes in the period of the coupled oscillators.

In Ref. 14, the synchronization in the system of coupled noni-

dentical nonisochronous van der Pol–Duffing oscillators with

inertial and dissipative coupling has been studied. In Ref. 15,

Ookawara and Endo have investigated the effect of element

value deviation on the degenerate modes in a ring of three and

four coupled van der Pol oscillators. By using the averaging

method, they proved that, for a ring of three coupled oscilla-

tors, two frequencies bifurcate from the degenerate mode, syn-

chronize if they are close enough, but lose synchronization

when they are separated to some extent; while for a ring of

four coupled oscillators, the two frequencies generally cannot

be synchronized, even if they are close enough. In Ref. 16,

Woafo and Enjieu Kadji have investigated different states of

synchronization in a ring of mutually van der Pol oscillators.

Using the properties of the variational equations of stability,

they calculated the good coupling parameters leading to com-

plete and partial synchronization or disordered states and

obtained a stability map showing domains of synchronization

to an external excitation locally injected in the ring. In Ref. 17,

the stability of the synchronization manifold in a ring and in an

open-ended chain of nearest neighbor coupled self-sustained

systems, each self-sustained system consisting of multilimit

cycle van der Pol oscillators, has been investigated and it has

been found that synchronization occurs in a system of many

coupled modified van der Pol oscillators, and it is stable even

in the presence of a spread of parameters.

However, the coupling delay is not considered in the lit-

eratures mentioned above. In fact, the coupling is certainly

not instantaneous and might exhibit propagation delays

which are comparable to or larger than the characteristic os-

cillatory time scale of the individual oscillators. It has been

shown that a coupling delay can induce complex behavior in

a network and the nodes organize in different synchroniza-

tion patterns. For example, the coupling delay can induce the

amplitude death even for zero frequency mismatch between

the limit cycle oscillators,18–20 which does not occur in the

case without coupling time delay, can lead to chaos when

two lasers are coupled face to face21 and cause the oscillators

to switch from being in-phase to being out-of-phase in vari-

ous types of delay-coupled oscillators.22–25

In the following, we consider a system of three delay-

coupled van der Pol oscillators where three such oscillators

are diffusively coupled through the damping terms

€yiðtÞ þ eðy2i ðtÞ � 1Þ _yiðtÞ þ yiðtÞ

¼ kð _yiþ1ðt� sÞ þ _yiþ2ðt� sÞ � _yiðtÞÞ; i¼ 1;2;3 ðmod3Þ;(2)

where k � 0 represents the coupling strength and s � 0 is

the coupling time delay. It is well known that the dynamics

of the coupled system depends on the properties of each of

the units and on their interactions. Clearly, in system (2), the

parameters k and s reflect the interactions of three van der

Pol oscillators. Here, we aim at addressing the influence of

the coupling strength and time delay on the stability and

synchronized patterns of system (2). We will show that the

coupling time delay will induce to stable synchronous peri-

odic solutions, stable phase-locked periodic solutions, and

unstable mirror-reflecting and standing waves.

We would like to mention that there are several works on

delayed coupling van der Pol oscillators similar to system (2).

In Ref. 26, Wirkus and Rand have studied the dynamics of a

pair of van der Pol oscillators where the coupling is chosen to

be through the damping terms but not of diffusive type. They

used the method of averaging to reduce the problem to the

study of a slow-flow in three dimensions and investigated the

steady state solutions of this slow-flow, with special attention

given to the bifurcations accompanying their change in num-

ber and stability. Sen an Rand,27 entirely by numerical meth-

ods, investigated the dynamics of a pair of relaxation

oscillators of the van der Pol type with delay position coupling

and have found the various phase-locked motions including

the in-phase and out-of-phase modes. Li et al.28 investigated

the dynamics of a pair of van der Pol oscillators with delayed

position and velocity coupling. Using the method of averaging

together with truncation of Taylor expansions, they have found

that the dynamics of 1:1 internal resonance is more complex

than that of non-1:1 internal resonance. More recently, the sys-

tem of a pair of coupled van der Pol oscillators has been stud-

ied by Atay29 and Song.24 In Ref. 29, Atay has investigated

the synchronization and amplitude death using averaging

theory. The parameter ranges for the coupling strength and

delay are calculated such that the oscillators exhibit amplitude

death or stable in-phase or antiphase synchronized oscillations

with identical amplitudes. In Ref. 24, Song has investigated

the stability and oscillation patterns induced by the coupling

time delay. It was shown that for appropriate damping and

coupling strengths there are stability switches and the

synchronized dynamics changes from in-phase to out-of-phase

oscillations (or vice versa) as the coupling time delay is

increased and the critical values of coupling time delay lead-

ing to different oscillation patterns were explicitly calculated.

This paper is organized as follows. Local stability and

delay-induced Hopf bifurcations are presented in Sec. II.

Spatio-temporal patterns of bifurcating periodic solutions are

investigated in Sec. III. In Sec. IV, we calculate the normal

forms on center manifolds and then determine stability of

bifurcating periodic orbits. Numerical simulations are

employed to support and extend the theoretical analysis in

Sec. V. A summary of the results is presented in Sec. VI.

II. LOCAL STABILITY AND DELAY-INDUCED HOPFBIFURCATIONS

Letting YiðtÞ ¼ ðyi; _yiÞ 2 R2; i ¼ 1; 2; system (2) can

be written as

_YiðtÞ ¼ MYiðtÞ þ N Yiþ1ðt� sÞ þ Yiþ2ðt� sÞð Þ þ gðYiðtÞÞ;i ¼ 1; 2; 3 ðmod3Þ; (3)

where

M ¼0 1

�1 e� k

� �; N ¼

0 0

0 k

� �;

ga

b

� �¼

0

�ea2b

� �; ða; bÞT 2 R2:

(4)

023111-2 Song, Xu, and Zhang Chaos 21, 023111 (2011)


The linearization of (3) at the zero solution leads to

_YiðtÞ ¼ MYiðtÞ þ N Yiþ1ðt� sÞ þ Yiþ2ðt� sÞð Þi ¼ 1; 2; 3 ðmod3Þ: (5)

The characteristic matrix of linearized system (5) is given by

Msð0; kÞ ¼kI2 �M �Ne�ks �Ne�ks

�Ne�ks kI2 �M �Ne�ks

�Ne�ks �Ne�ks kI2 �M

0B@

1CA;

where I2 is a 2� 2 identity matrix. Assuming that gj 2 R2

are eigenvectors, respectively, of kI2 �M � v1j Ne�ks

� v2j Ne�ks; j ¼ 0; 1; 2, then it is easy to verify that

Msð0; kÞvj ¼ diag T2; T2; T2f gvj; j ¼ 0; 1; 2; (6)

where T2 ¼ kI2 �M � vjNe�ks � v2j Ne�ks, vj ¼ ðgj; vjgj;

v2j gjÞT , and vj ¼ e2pj=3i: So, we have

detMsð0; kÞ ¼Y2

j¼0

detðkI2 �M � vjNe�ks � v2j Ne�ksÞ;

and then the characteristic equation of the linearization of (3)

at the zero solution is

Dðs; kÞ ¼ detMsð0; kÞ ¼ D1D22 ¼ 0; (7)

where

D1 ¼ ðk2 þ ðk � eÞkþ 1� 2kke�ksÞ;D2 ¼ ðk2 þ ðk � eÞkþ 1þ kke�ksÞ:

Note the fact that the root k of the characteristic equation 7

corresponds to the zero eigenvalue of the matrix

kI2 �M � vjNe�ks � v2j Ne�ks; j ¼ 0; 1; 2, which will be an

important information to calculate the eigenvector of system

(5) for the purely imaginary roots of the characteristic equa-

tion 7 in the following two sections.

In the remaining of this section, we investigate the distri-

bution of roots of Eq. (7), which determines the local stability

of the zero solution of (2). It is convenient to start with a more

general second order transcendental polynomial equation

k2 þ pkþ r þ ske�ks ¼ 0; p; r; s 2 R and s 6¼ 0; (8)

which has been extensively studied (see, e.g., Refs. 30 and

31). From Refs. 30 and 31, we first have the following lemma.

Lemma 2.1. Assuming that r > 0, then we have thefollowing.

(i) If jsj < jpj, then Eq. (8) has no purely imaginaryroot.

(ii) If jsj > jpj, then Eq. (8) has a pair of purely imagi-nary roots 6ixþ (6ix�, respectively) at s ¼ ~sþj(s ¼ ~s�k , respectively) such that ~sþj 6¼ ~s�k for anynon-negative integer numbers j, k, while Eq. (8) hastwo pairs of purely simple imaginary roots 6ixþand 6ix� at s ¼ ~sþj (or s ¼ ~s�k ) for some non-nega-tive integer numbers j and k such that ~sþj ¼ ~s�k .

(iii) If jsj ¼ jpj, then Eq. (8) has a pair of purely imagi-nary roots 6i

ffiffirp

at s ¼ ~sþj .(iv) Letting

kðsÞ ¼ gðsÞ þ ixðsÞ

is a solution of Eq. (8) satisfying

gð~s6j Þ ¼ 0; xð~s6

j Þ ¼ x6;

then we have

dReðkð~sþj ÞÞds

> 0;dReðkð~s�j ÞÞ

ds< 0; for jsj > jpj (9)

and

dReðkð~sþj ÞÞds

¼ 0; for jsj ¼ jpj:Here

x6 ¼ffiffiffi2p

2s2 þ 2r � p26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs2 þ 2r � p2Þ2 � 4r2

q� �1=2

and

~sþj ¼ 1xþ

2jpþ p� arccos ps

� �� ;

~s�j ¼ 1x�

2jpþ pþ arccos ps

� �� ;

for s > jpj;

~sþj ¼ 1xþ

2jpþ pþ arccos ps

� �� ;

s�j ¼ 1x�

2jpþ p� arccos ps

� �� ;

for s < �jpj;

~sþj ¼ 1ffiffirp 2jpþ pð Þ; for s ¼ p;

~sþj ¼ 1ffiffirp 2ðjþ 1Þpð Þ; for s ¼ �p;

8>>>>>>>>>>><>>>>>>>>>>>:

with j 2 f0; 1; 2;…g.From Lemma 2.1, we have the following two lemmas

on the distribution of roots of the equations D1 ¼ 0 and

D2 ¼ 0.

Lemma 2.2. For the equation D1 ¼ 0, we have the fol-lowing results:

(i) When s ¼ 0; the equation D1 ¼ 0 has only two rootsand they both have negative real parts for k < �e,positive real parts for k > �e, and zero real partsfor k ¼ �e.

(ii) If �e < k < e=3, then the equation D1 ¼ 0 has nopurely imaginary root.

(iii) If either k < �e or k > e=3, then the equationD1 ¼ 0 has a pair of purely simple imaginary roots6ixþ1 (6ix�1 , respectively) at s ¼ sþ1j (s ¼ s�1k,respectively) such that sþ1j 6¼ s�1k for any non-nega-tive integer numbers j, k, while the equation D1 ¼ 0

has two pairs of purely simple imaginary roots6ixþ1 and 6ix�1 at s ¼ sþ1j (or s ¼ s�1k) for somenon-negative integer numbers such that sþ1j ¼ s�1k.

(iv) If either k ¼ �e or k ¼ e=3, then Eq. (8) has a pairof purely imaginary roots 6i at s ¼ sþ1j.

Here,

x61 ¼

ffiffiffi2p

2

hð3k � eÞðk þ eÞ

þ 26ffiffið

p3k � eÞðk þ eÞ ð3k � eÞðk þ eÞ þ 4½ �

i1=2

(10)

023111-3 Amplitude death and oscillation patterns Chaos 21, 023111 (2011)


and

sþ1j ¼ 1xþ

1

2jpþ pþ arccos e�k2k

� �� ;

s�1j ¼ 1x�

1

2jpþ p� arccos e�k2k

� �� ;

for k > e=3;

sþ1j ¼ 1xþ

1

2jpþ p� arccos e�k2k

� �� ;

s�1j ¼ 1x�

1

2jpþ pþ arccos e�k2k

� �� ;

for k < �e;

sþ1j ¼ 2jpþ p; for k ¼ e=3;

sþ1j ¼ 2ðjþ 1Þp; for k ¼ �e:

8>>>>>>>>>>><>>>>>>>>>>>:

(11)

Lemma 2.3. For the equation D2 ¼ 0, we have the fol-lowing results:

(i) When s ¼ 0; the equation D2 ¼ 0 has only two rootsand they both have negative real parts for k > e=2,positive real parts for k < e=2, and zero real partsfor k ¼ e=2.

(ii) If k < e=2, then the equation D2 ¼ 0 has no purelyimaginary root.

(iii) If k > e=2, then the equation D2 ¼ 0 has a pair ofpurely imaginary roots 6ixþ2 (6ix�2 , respectively)at s ¼ sþ2j (s ¼ s�2k, respectively) such that sþ2j 6¼ s�2k

for any non-negative integer numbers j, k, while theequation D2 ¼ 0 has two pairs of purely simpleimaginary roots 6ixþ2 and 6ix�2 at s ¼ sþ2j (ors ¼ s�2k) for any non-negative integer numbers j andk such that sþ2j ¼ s�2k.

(iv) If k ¼ e=2, then the equation D2 ¼ 0 has a pair ofpurely imaginary roots 6i at s ¼ sþ2j.

Here,

x62 ¼

ffiffiffi2p

2eð2k � eÞ þ 26

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieð2k � eÞ eð2k � eÞ þ 4½ �

ph i1=2

(12)

and

sþ2j ¼ 1xþ

2

2jpþ p� arccos k�ek

� �� ;

s�2j ¼ 1x�

2

2jpþ pþ arccos k�ek

� �� ;

for k > e=2;

sþ2j ¼ 2ðjþ 1Þp; for k ¼ e=2:

8>><>>: (13)

From (11), it is easy to conclude that for k > e3; s�10 < sþ10 if

and only if the following assumption (A1) holds

ðA1Þ arccose� k

2k

� �>

xþ1 � x�1xþ1 þ x�1

p;

and from (13), we have sþ20 < s�20 for k > e=2. In addition,

combining (11) and (13), we obtain that for k > e=2;s�10 < sþ20 if and only if the following assumption (A2) holds

ðA2Þ xþ2 arccose� k

2k

� ��x�1 arccos

k� ek

� �> p xþ2 �x�1� �

:

Rearranging the critical values of the coupling time delay

and setting

sþj 2 fsþ1m; sþ2m;m ¼ 0; 1; 2;…g;

s�j 2 fs�1m; s�2m;m ¼ 0; 1; 2;…g

(14)

such that sþj < sþjþ1; s�j < s�jþ1; j ¼ 0; 1; 2;…. Note the fact

that (A2) is not satisfied provided that k < e since for k < e,

xþ2 arccose� k

2k

� �� x�1 arccos

k � ek

� �<

p2

xþ2 � x�1� �

:

Thus, by Lemmas 2.2 and 2.3, we have the following results

on the characteristic equation 7.

Lemma 2.4. Assuming that s6j are defined by (14), we

have the following:

(i) If either k < e, then Eq. (7) has at least a pair of rootswith positive real parts for all s � 0.

(ii) If k > e and the assumptions (A1) and (A2) hold, thenthere exists a positive integer n such thats�n�1 < sþn�1 < sþn < s�n and all roots of Eq. (7) havenegative real parts for s 2 ðs�0 ; sþ0 Þ [ ðs�1 ; sþ1 Þ [ðs�n�1; s

þn�1Þ and at least a pair of roots with positive

real parts for s 2 ½0; s�0 Þ [ ðsþn�1;þ1Þ.

It is well known from the theory of functional differen-

tial equations FDEs; Ref. 32 that the characteristic equation

determines local stability of the steady state. If all the eigen-

values have negative real parts, the steady state is stable and

if the characteristic equation has at least one root with posi-

tive real part, the steady state is unstable. And if the charac-

teristic equation has a pair of pure imaginary roots at the

critical values and the transversality condition holds, Hopf

bifurcations occur when the parameter crosses its critical val-

ues. Therefore, by Lemmas 2.2–2.4 and the transversality

condition (9), we have the following results on the stability

and bifurcation phenomena of the zero steady state of the

delayed coupling van der Pol system (2).

Theorem 2.1: Assuming that s61j , s6

2j , and s6j are defined,

respectively, by (11), (13), and (14), we have the following:

(i) If either k < e, then the zero equilibrium of system(2) is unstable for all s � 0.

(ii) If k > e and the assumptions (A1) and (A2) hold,then there exists a positive integer n such thats�n�1 < sþn�1 < sþn < s�n and the zero equilibrium ofsystem (2) is asymptotically stable for s 2 ðs�0 ; sþ0 Þ[ðs�1 ; sþ1 Þ [ ðs�n�1; s

þn�1Þ and unstable for s 2 ½0; s�0 Þ [

ðsþn�1;þ1Þ.(iii) Near the zero equilibrium, system (2) undergoes

Hopf bifurcations at the critical values s61j and

undergoes equivalent Hopf bifurcations at the criti-cal values s6

2j .

According to Theorem 2.1, the stability switches region

in the plane of damping and coupling strengths is plotted in

Fig. 1. In the shaded region of Fig. 1, there exist stability

switches for the zero equilibrium of system (2) with increas-

ing the coupling time delay s, and in other regions, the zero

equilibrium of system (2) is always unstable for all s � 0:For fixed damping strength, say e ¼ 0:03, the islands of am-

plitude death in the plane of the coupling time delay and

strength, where all oscillations are quenched, is qualitatively

same as Fig. 2. Throughout the remainder of the present pa-

per, the damping and coupling strengths are always restricted

to the shaded region of Fig. 1.



III. SPATIO-TEMPORAL PATTERNS OF BIFURCATINGPERIODIC SOLUTIONS

It follows from Sec. II that due to the symmetry of sys-

tem (2), the purely imaginary eigenvalues ix62 are always

multiple at the critical values s62j . Hence, the standard Bopf

bifurcation theory of functional differential equations (see,

e.g., Hale32 and Hassard et al.)33 cannot be applied. In this

section, we first explore the symmetry of system (3) and then

apply the symmetric Hopf bifurcation theorem for delay dif-

ferential equations established in Ref. 34 to describe the spa-

tio-temporal patterns of bifurcating periodic solutions as the

coupling time delay crosses the critical values s62j . We would

also like to mention that this theory has been successfully

employed to study the spatio-temporal patterns in the delay-

coupled neural networks (see Refs. 25, 35–38 and references

therein).

Let G : C! Rn and C be a compact Lie group. By a

compact Lie group C, we mean a closed subgroup of

GLðRnÞ, the group of all invertible linear transformations of

the vector space Rn into itself. Note that the space of n� nmatrices may be identified with Rn2 , which contains GLðRnÞas an open subset. It follows from Ref. 39 that the system

_uðtÞ ¼ GðutÞ is said to be C -equivariant if GðcutÞ ¼ cGðutÞfor all c 2 C: Let C ¼D3 be the dihedral group of order 6,

which is generated by the cyclic group Z3 order 3 together

with the flip of order 2 (see Ref. 39, for more details). Denote

by q, the generator of the cyclic subgroup Z3 and j the flip.

Define the action of D3 on R6 by

q

Y1

Y2

Y3

0B@

1CA ¼

Y2

Y3

Y1

0B@

1CA; j

Y1

Y2

Y3

0B@

1CA ¼

Y2

Y1

Y3

0B@

1CA

for all Yi 2 R2; i ðmod 3Þ;

(15)

and then it is easy to verify the following lemma.

Lemma 3.1. System (3) is D3 equivariant.Assume that the infinitesimal generator AðsÞ of the C0-

semigroup generated by the linear system (5) has a pair of

purely imaginary eigenvalues 6ix62 at the critical value s6

2j .

From Sec. II, we also get that for the critical value s62j , there

are two independent eigenvectors v1 and v2, where vj

¼ ðgj; vjgj; v2j gjÞT ; j ¼ 1; 2; with vj ¼ eð2pj=3Þi and

g1 ¼ g2 ¼ 1; ixþ2 ;� �

for sþ2 ; and g1 ¼ g2 ¼ 1; ix�2� �

for s�2 :

Denoting the action of C ¼D3 on R2 by

qu1

u2

� �¼

� 12�ffiffi3p

2ffiffi3p

2� 1

2

!u1

u2

� �;

ju1

u2

� �¼

� 12�ffiffi3p

2

�ffiffi3p

212

!u1

u2

� �;

we have the following lemma.

Lemma 3.2. R2 is an absolutely irreducible representa-tion of C, and KerDðs6

2j ; ix62jÞ is isomorphic to R2 �R2.

Here a representation R2 of C is absolutely irreducible ifthe only linear mapping that commutes with the action of Cis a scalar multiple of the identity.

Proof: The proof for R2 to be two-dimensional abso-

lutely irreducible representation of C is straightforward and

can be found in, for example, Ref. 39. Clearly,

KerDðs62j ; ix

62 Þ

¼ a1 þ b1ið Þv1 þ a2 þ b2ið Þv2; a1; a2; b1; b2 2 Rf g:

Define a mapping J from KerDðs6j ; ix

62 Þ to R4 by

J a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þ¼ a1 þ a2; b1 � b2; b1 þ b2; a2 � a1ð ÞT

Clearly, J : KerDðs6j ; ib

62jÞ ffi R4 is a linear isomorphism.

Note that

q a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þ¼ a1 þ b1ið Þq v1ð Þ þ a2 þ b2ið Þq v2ð Þ¼ a1 þ b1ið Þeið2p=3Þv1 þ a2 þ b2ið Þe�ið2p=3Þv2

¼ � 12

a1 �ffiffi3p

2b1

þ i � 1

2b1 þ

ffiffi3p

2a1

v1

þ � 12

a2 þffiffi3p

2b2

þ i � 1

2b2 �

ffiffi3p

2a2

v2

FIG. 1. (Color online) Possible region of amplitude death for the coupling

time delay in the e� k plane. There exist stability switches for the zero equi-

librium of system (2) by increasing the coupling time delay s in the shaded

region and the zero equilibrium of system (2) is always unstable for any

s � 0 otherwise. l1 and l2 are curves determined by the equations

arc cosðe� k=2kÞ � ðxþ1 � x�1 Þ=ðxþ1 þ x�1 Þp ¼ 0 and xþ2 arccosðe� k=2kÞ� x�1 arccosðk � e=kÞ � p

2ðxþ2 �x�1 Þ ¼ 0; respectively.

FIG. 2. (Color online) Islands of amplitude death in the plane of the cou-

pling time delay s and the coupling strength k for fixed e ¼ 003.



and

j a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þ¼ a1 þ b1ið Þeið2p=3Þv2 þ a2 þ b2ið Þe�ið2p=3Þv1

¼ � 12

a1 �ffiffi3p

2b1

þ i � 1

2b1 þ

ffiffi3p

2a1

v2

þ � 12

a2 þffiffi3p

2b2

þ i � 1

2b2 �

ffiffi3p

2a2

v1:

Therefore,

J q a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þð Þ

¼ � 12

a1 þ a2ð Þ �ffiffi3p

2b1 � b2ð Þ;� 1

2b1 � b2ð Þ

þ

ffiffi3p

2a1 þ a2ð Þ;� 1

2b1 þ b2ð Þ �

ffiffi3p

2a2 � a1ð Þ;

� 12

a2 � a1ð Þ þffiffi3p

2b1 þ b2ð Þ

¼ q J a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þð Þ

and

J j a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þð Þ

¼ � 12

a1 þ a2ð Þ �ffiffi3p

2b1 � b2ð Þ; 1

2b1 � b2ð Þ

�

ffiffi3p

2a1 þ a2ð Þ;� 1

2b1 þ b2ð Þ �

ffiffi3p

2a2 � a1ð Þ;

� 12

a2 � a1ð Þ �ffiffi3p

2b1 þ b2ð ÞÞ

¼ j J a1 þ b1ið Þv1 þ a2 þ b2ið Þv2ð Þð Þ:

This completes the proof. h

Let x ¼ 2p=x6i ; i ¼ 1; 2; and denote by Px the Banach

space of all continuous x�periodic mappings x : R! R6.

Then D3 � S1 acts on Px, where S1 is a circle group, by

ðc; hÞxðtÞ ¼ cxðtþ hÞ; ðc; hÞ 2 D3 � S1:

Denote by SPx the set of all x�periodic solutions of (5)

with s ¼ s62j . Then, for s6

2j ,

SPx ¼ 11�621 þ 12�

622 þ 13�

623 þ 14�

624; 11; 12; 13; 14 2 R

� �;

where

�621 ¼ Re eix62

hv1

� �¼ cos x6

2 h� �

Re v1ð Þ � sin x62 h

� �Im v1ð Þ;

e622 ¼ Im eix6

2hv1

� �¼ sin x6

2 h� �

Re v1ð Þ þ cos x62 h

� �Im v1ð Þ;

�623 ¼ Re eix62

hv2

� �¼ cos x6

2 h� �

Re v2ð Þ � sin x62 h

� �Im v2ð Þ;

�624 ¼ Im eix62

hv2

� �¼ sin x6

2 h� �

Re v2ð Þ þ cos x62 h

� �Im v2ð Þ:

(16)

For SPx, we have the following property.

Lemma 3.3. SPx is an D3�invariant subspace of Px

and

j �621

� �¼ � 1

2�623 �

ffiffiffi3p

2�624; j �622

� �¼

ffiffiffi3p

2�623 �

1

2�624;

j �623

� �¼ � 1

2�621 þ

ffiffiffi3p

2�622; j �624

� �¼ �

ffiffiffi3p

2�621 �

1

2�622;

q �621

� �¼ � 1

2�621 �

ffiffiffi3p

2�622; q �622

� �¼

ffiffiffi3p

2�621 �

1

2�622;

q �623

� �¼ � 1

2�623 þ

ffiffiffi3p

2�624; q �624

� �¼ �

ffiffiffi3p

2�623 �

1

2�624:

Proof: Note that g1 ¼ g2 ¼ ð1; ix62 Þ

T : It is easy to verify

that

j Re v1ð Þð Þ ¼ � 12

Re v2ð Þ �ffiffi3p

2Re v2ð Þ;

j Im v1ð Þð Þ ¼ffiffi3p

2Re v2ð Þ � 1

2Im v2ð Þ;

j Re v2ð Þð Þ ¼ � 12

Re v1ð Þ þffiffi3p

2Re v1ð Þ;

j Im v2ð Þð Þ ¼ �ffiffi3p

2Re v1ð Þ � 1

2Im v1ð Þ;

and

q Re v1ð Þð Þ ¼ � 12

Re v1ð Þ �ffiffi3p

2Re v1ð Þ;

q Im v1ð Þð Þ ¼ffiffi3p

2Re v1ð Þ � 1

2Im v1ð Þ;

q Re v2ð Þð Þ ¼ � 12

Re v2ð Þ þffiffi3p

2Re v2ð Þ;

q Im v2ð Þð Þ ¼ �ffiffi3p

2Re v2ð Þ � 1

2Im v2ð Þ:

By these formulae and (16), the lemma follows from a

straightforward calculation. h

Let Rh ¼ fðc; eið2p=xÞhÞ; h 2 ð0;xÞg and FixðRh;SPxÞbe the fixed point set of Rh under the action D3 � S

1. We

consider

FixðRh;SPxÞ¼fxðtÞ2SPx; ðc;eið2p=xÞhÞxðtÞ¼cxðtþhÞ¼xðtÞfor all ðc;eið2p=xÞhÞ2Rhg:

Setting

R6ð2;3Þ ¼ j;61ð Þh i;R6

q ¼ hðq; e6ið2p=3ÞÞi;

we can prove the following lemma.

Lemma 3.4. FixðR62;3; SPxÞ and FixðR6

q ; SPxÞ are allsubspaces of SPx of dimension 2.

Proof: (1) First, x 2 FixðRþ2;3; SPxÞ if and only if

jðxÞ ¼ x: For x ¼P4

i¼1 1i�62i ; we have

jx¼X4

i¼1

1ij �62i

� �¼� 1

213þ

ffiffi3p

214

�621þ

ffiffi3p

213� 1

214

�622

þ � 1211þ

ffiffi3p

212

�623�

ffiffi3p

211þ 1

212

�624

It follows that x 2 FixðRþ2;3; SPxÞ if and only if

� 12�ffiffi3p

2ffiffi3p

2� 1

2

!13

14

� �¼

11

12

� �:

This implies that FixðRþ2;3; SPxÞ is spanned by ��1 and ��2,

where

��1 ¼ �1

2�621 þ

ffiffiffi3p

2�622 þ �623; ��2 ¼ �

ffiffiffi3p

2�621 �

1

2�622 þ �624:

(2) Second, if and only if It is easy to see that �62iðtþ x2ÞÞ

¼ ��62i and then xðtþ x2Þ ¼ �

P4i¼1 1i�

62i : This implies that

x 2 FixðR�2;3; SPxÞ if and only if



� 12�ffiffi3p

2ffiffi3p

2� 1

2

!13

14

� �¼ �

11

12

� �:

Therefore, Fix ðR�2;3; SPxÞ is spanned by ��3 and ��4, where

��3 ¼1

2�621 �

ffiffiffi3p

2�622 þ �623; ��4 ¼

ffiffiffi3p

2�621 þ

1

2�622 þ �624:

(3) Third, for x ¼P4

i¼1 1i�62i ; we have

qðxÞ ¼ � 1211 þ

ffiffi3p

212

�621 �

ffiffi3p

211 þ 1

212

�622

� 1213 þ

ffiffi3p

214

�623 þ

ffiffi3p

213 � 1

214

�624

In addition,

cos x62 t� x

3

� �� ¼ cos x6

2 t� 2p3

� �¼ � 1

2cos x6

2 t� �

þffiffi3p

2sin x6

2 t� �

;

sin x62 t� x

3

� �� ¼ sin x6

2 t� 2p3

� �¼ �

ffiffi3p

2cos x6

2 t� �

� 12

sin x62 t

� �:

This, together with the expression of each �62j and

x ¼P4

j¼1 1j�62j ; leads to

x t� x3

� �¼ � 1

211 þ

ffiffi3p

212

�621 þ

ffiffi3p

211 � 1

212

�622

� 1213 þ

ffiffi3p

214

�623 þ

ffiffi3p

213 � 1

214

�624

So, x 2 FixðRþq ; SPxÞ, i.e., qðxðtÞÞ ¼ x t� x3

� �, if and only if

� 1211 þ

ffiffi3p

212 ¼ � 1

211 �

ffiffi3p

212;

�ffiffi3p

211 � 1

212 ¼

ffiffi3p

211 � 1

212;

� 1213 �

ffiffi3p

214 ¼ � 1

213 �

ffiffi3p

214;ffiffi

3p

213 � 1

214 ¼

ffiffi3p

213 � 1

214:

8>>>>><>>>>>:

That is, x 2 FixðRþq ; SPxÞ if and only if 11 ¼ 12 ¼ 0; and

x 2 FixðR�q ; SPxÞ if and only if 13 ¼ 14 ¼ 0: Therefore,

FixðRþq ; SPxÞ is spanned by �623 and �624, and FixðR�q ; SPxÞ is

spanned by �621 and �622. h

Note that, if Y(t) is a periodic solution of (3), then so is

ðc; eið2p=xÞhÞYðtÞ for every ðc; eið2p=xÞhÞ 2Dn � S1: It fol-

lows from the symmetric bifurcation theory of delay differ-

ential equations due to Wu34 that, under usual nonresonance

and transversality conditions, for every subgroup R Dn

�S1 such that the R�fixed-point subspace of SPx is of

dimension 2, symmetric delay differential equations has a

bifurcation of periodic solutions whose spatial–temporal

symmetries can be completely characterized by R. By

Lemma 3.4, the subgroups R6ð2;3Þ and R6

q can be employed to

describe the symmetry of periodic solutions of (3) which ex-

hibit certain spatial–temporal patterns (see Ref. 39 and 34,

for more details). Thus, Lemmas 3.1–3.4 allow us to apply

the general symmetric Hopf bifurcation theorem due to

Wu34 to describe the spatio-temporal patterns of local Hopf

bifurcating periodic oscillations.

Theorem 3.1: System (3) has eight branches of asyn-

chronous periodic solutions of period ps near 2p=x62 , bifur-

cated simultaneously from the zero solution at the critical

values s ¼ s62j , and these are

(i) two phase-locked oscillations: YiðtÞ ¼ Yiþ1 t6ps=3ð Þ,for i(mod 3) and t 2 R;

(ii) three mirror-reflecting waves: YiðtÞ ¼ YjðtÞ 6¼ YkðtÞ,for t 2 R and for some distinct i; j; k 2 f1; 2; 3g; and

(iii) three standing waves: YiðtÞ ¼ Yj tþ ðps=2Þð Þ, for

t 2 R and some pair of distinct elements i; j 2f1; 2; 3g.

IV. THE CALCULATION OF NORMAL FORMS ONCENTER MANIFOLDS AND STABILITY OFBIFURCATING PERIODIC ORBITS

In this section, we employ the algorithm and notations

of Faria and Magalhaes40,41 to derive the normal forms of

system (3) on center manifolds and then we can determine

the direction and stability of bifurcating periodic orbits.

Rescaling the time by t 7! t=s to normalize the delay so

that system (3) can be written as a FDE in the phase space

C ¼ Cð½�1; 0�;R6Þ; and then separating the linear terms

from the nonlinear terms, (3) becomes

_uðtÞ ¼ LsðutÞ þ Fðut; sÞ (17)

where ut 2 C; utðhÞ ¼ uðtþ hÞ;�1 h 0; and for u¼ ðu1;u2;u3;u4;u5;u6ÞT 2 C, we have

LsðuÞ ¼ sM1u2� 2ð0Þ þ sM2uð�1Þ (18)

and

Fðu; sÞ ¼ sð0; �eu21ð0Þu2ð0Þ; 0; �eu2

3ð0Þu4ð0Þ; 0;

� eu25ð0Þu6ð0ÞÞT ; (19)

where u ¼ ðu1;u2;u3;u4;u5;u6ÞT 2 C and M1, M2 are as

follows:

M1 ¼M O2 O2

O2 M O2

O2 O2 M

0B@

1CA; M2 ¼

O2 N N

N O2 N

N N O2

0B@

1CA;

where M, N are defined by (4) and O2 is a 2� 2 zero matrix.

There exists a function of bounded variation such that

the linear map Ls may be expressed in the following integral

form:

LsðuÞ ¼ð0

�1

½dgsðhÞ�uðhÞ

Letting C� ¼ Cð½0; 1�;R6�Þ with R6� being the six-dimen-

sional space of row vectors, define the adjoint bilinear form

on C*�C as follows:

hwðsÞ;/ðhÞi ¼ wð0Þ/ð0Þ �ð0

�1

ðh

0

wðn� hÞdgs� ðhÞ/ðnÞdn;

w 2 C�;/ 2 C: (20)

It follows from Sec. II that the characteristic equation of (18)

has purely imaginary roots 6ix�s� at the critical values



s ¼ s� of the coupling time delay, where x� ¼ x61 for

s� ¼ s61j and x� ¼ x6

2 for s� ¼ s62j . Setting K0 ¼ �ix�s�;f

ix�s�g and using the formal adjoint theory for FDEs in Ref.

32, the phase space C can be decomposed by K0 as

C ¼ P� Q; where P is the center space for _uðtÞ ¼ Ls�ut; i.e.,

P is the generalized eigenspace associated with K0: Let Uand W be bases for P and for the space P* associated with

the eigenvalues 6ix�s� of the formal adjoint equation,

respectively, and normalized so that , where Ip is the p� pidentity matrix and p¼ dim P. Assume that B is a p� p real

matrix with the point spectrum rðBÞ ¼ K and satisfies simul-

taneously satisfy _U ¼ UB;� _W ¼ BW:Introducing the new parameter l ¼ s� s� such that

l ¼ 0 is a Hopf bifurcation value of (17), Eq. (17) becomes

_uðtÞ ¼ Ls�ut þ ~Fðut; lÞ;

where

~Fðut; lÞ ¼ Llut þ Fðu; s� þ lÞ:

Enlarging the phase space C by considering the Banach

space BC of functions from [–1, 0] into R4 which are uni-

formly continuous on [–1, 0] and with a jump discontinuity

at 0. Using the decomposition ut ¼ UxðtÞ þ y; with

xðtÞ 2 Cp and y 2 Q1 Q \ C1 (C1 is the subset of C con-

sisting of continuously differentiable functions). Then we

can decompose (17) as

_x ¼ BxþWð0Þ ~FðUxþ y; lÞ;_y ¼ AQ1 yþ ðI � pÞX0

~FðUxþ y; lÞ;

where AQ1 : Q1 ! ker p is such that AQ1/ ¼ _/þ X0½Ls� ð/Þ� _/ð0Þ�. We write the Taylor formula

Wð0Þ ~FðUxþ y; lÞ ¼ 12! f

12 ðx; y; lÞ þ 1

3! f13 ðx; y; lÞ þ � � � ;

ðI � pÞX0~FðUxþ y; lÞ ¼ 1

2! f22 ðx; y; lÞ þ 1

3! f23 ðx; y; lÞ þ � � � ;

(21)

where f 1j ðx; y; lÞ and f 2

j ðx; y; lÞ are homogeneous polyno-

mials in ðx; y; lÞ of degree j with coefficients in Cp, ker p,

respectively, and hot stands for higher order terms. There-

fore, differential equation EDE can be written as

_x ¼ BxþPj�2

1j! f

1j ðx; y; lÞ;

_y ¼ AQ1 yþPj�2

1j! f

2j ðx; y; lÞ:

(22)

From the normal form theory of FDEs due to Faria and Mag-

alhaes,40,41 the normal form of (17) on the center manifold

of the origin at l ¼ 0 is given by

_x ¼ Bxþ 1

2!g1

2ðx; 0; lÞ þ1

3!g1

3ðx; 0; lÞ þ h:o:t:; (23)

where g12ðx; 0; lÞ; g1

3ðx; 0; lÞ are the second and third order

terms in ðx; lÞ; respectively, and hot stands for higher order

terms. In the following part of this section, we will calculate

the normal form (23) at the critical values s61j and s6

2j .

A. For the critical values s61j

From Sec. II, at s ¼ s61j the characteristic equation of

(18) has purely imaginary roots 6ix61 s6

1j which are simple.

So, in this subsection, p¼ 4, x� ¼ xþ1 for s� ¼ sþ1j and

x� ¼ x�1 for s� ¼ s�1j and then we have

B ¼ diagðix�s�;�ix�s�Þ

It follows from (6) that the base of the center space P at

s ¼ s61j can be taken as U ¼ /1;/2ð Þ; where

/1ðhÞ ¼ eix�s�hv0; /2ðhÞ ¼ e�ix�s�h�v0; �1 h 0;

where

v0 ¼ ðg0; g0; g0ÞT ; and g0 ¼ 1; ixþ1� �

for sþ1j

and g0 ¼ 1; ix�1� �

for s�1j:(24)

One can easily show that the adjoint basis satisfying

hWðsÞ;UðhÞi ¼ I2 is

WðsÞ ¼w1ðsÞw2ðsÞ

� �¼ 1

3

d�vT0 e�ix�s�s

�dvT0 eix�s�s

!; 0 s 1;

where

d ¼ 1

1þ x2� þ s�x2

�ðk � eÞ � s�x� 1� x2�

� �i: (25)

Let V3j ðC

2Þ be the homogeneous polynomials of degree j in

three variables, x1; x2; l, with coefficients in C2, and let M1

j

denote the operator from V3j ðC

2Þ into itself defined by

ðM1j hÞðx; lÞ ¼ Dxhðx; lÞBx� Bhðx; lÞ; (26)

where h 2 V3j ðC

2Þ. Then, since B is the 2� 2 diagonal ma-

trix, it follows from Ref. 41 that

V3j ðC

2Þ ¼ ImðM1j Þ � KerðM1

j Þ; g1j ðx; 0; lÞ 2 KerðM1

j Þ

and

KerðM1j Þ ¼ Kerfxq1

1 xq2

2 llek : q1k1 þ q2k2 ¼ kk;

k ¼ 1; 2; q1; q2; l 2 N0; q1 þ q2 þ l ¼ jg;

where k1 ¼ ix�s�; k2 ¼ �ix�s� and fe1; e2g is canonical

basis of R2: Hence,

KerðM12Þ ¼ span

x1l

0

� �;

0

x2l

� � �;

KerðM13Þ ¼ span

x21x2

0

� �;

x1l2

0

� �;

0

x1x22

� �;

0

x2l2

� � �:

Since f 00ð0Þ ¼ 0; by (21), we have

1

2!f 12 ðx; 0; lÞ ¼ Wð0ÞLlðUxÞ

¼13

d�vT0

13

�dvT0

!ðLlð/1Þx1 þ Llð/2Þx2Þ;



and the second order terms in ðx; lÞ of the normal form on

the center manifold are given by

1

2g1

2ðx; 0; lÞ ¼1

2ProjKerðM1

2Þf

12 ðx; 0; lÞ

¼ ProjKerðM12Þ

13

d�vT0

13

�dvT0

!ðLlð/1Þx1 þ Llð/2Þx2Þ:

Note that

Llð/1Þ ¼ ilx�v0; Llð/2Þ ¼ �ilx�v0:

So,

1

2g1

2ðx; 0; lÞ ¼A1x1l

A1x2l

� �;

with A1 ¼ 13

dx��vT0 v0i: From (24), we have

A1 ¼ dx�ð1þ x2�Þi: (27)

Next we compute the cubic terms g13ðx; 0; lÞ as described in

(23). We first note that

1

3!g1

3ðx; 0; lÞ ¼ ProjKerðM13Þ

1

3!~f 13 ðx; 0; lÞ

¼ 1

3!ProjS

~f 13 ðx; 0; 0Þ þ Oðjxjl2Þ;

for

S ¼ spanx2

1x2

0

� �;

0

x1x22

� � �;

where 13!

~f 13 ðx; 0; lÞ denotes the third order terms after the com-

putation of the normal form up to the second order terms. It is

sufficient to compute only ProjS~f 13 ðx; 0; 0Þ for the purpose of

determining the generic Hopf bifurcation. Since f 00ð0Þ ¼ 0;implying f 1

2 ðx; y; 0Þ ¼ 0; we can deduce that, after the change

of variables that transformed the quadratic terms f 12 ðx; y; lÞ of

the first equation in (22) into g12ðx; y; lÞ; the coefficients of

third order at y ¼ 0; l ¼ 0 are still given by 13! f

13 ðx; 0; 0Þ, i.e.,

1

3!~f 13 ðx; 0; 0Þ ¼

1

3!f 13 ðx; 0; 0Þ:

Thus, from (19) and (21), we have

1

3!~f 13 ðx; 0; 0Þ ¼

1

3!f 13 ðx; 0; 0Þ

¼ Wð0ÞF3ðUx; s�Þ

¼ s�3

d�vT0

�dvT0

!0

�eðx1v01 þ x2�v01Þ2ðx1v02 þ x2�v02Þ0

�eðx1v03 þ x2�v03Þ2ðx1v04 þ x2�v04Þ0

�eðx1v05 þ x2�v05Þ2ðx1v06 þ x2�v06Þ

0BBBBBBBB@

1CCCCCCCCA;

where v0¼ (v01, v02, v03, v04, v05, v06)T. It follows that

13! g

13ðx; 0; 0Þ ¼ 1

3! ProjS~f 13 ðx; 0; 0Þ

¼A2x2

1x2

A2x1x22

!;

where

A2 ¼ �es�d

3ðv2

01�v202 þ v2

03�v204 þ v2

05�v206 þ 2jv01j2jv02j2

þ 2jv03j2jv04j2 þ 2jv05j2jv06j2Þ:

This, together with (24), yields that

A2 ¼ �es�x2�d: (28)

So, the normal form of (17) on the center manifold reads to

_x ¼ BxþA1x1l

A1x2l

� �þ

A2x21x2

A2x1x22

!þ O jxjl2 þ jx4j

� �;

(29)

where A1 and A2 are defined by (27) and (28), respectively.

Changing (29) to real coordinates by the change of variables

x1¼w1 – iw2, x2¼w1þ iw2 and then using polar coordinates

ðq; nÞ; w1 ¼ q cos n; w2 ¼ q sin n; we obtain

_q ¼ K1lqþ K2q3 þ O l2qþ jðq; lÞj4

;

_n ¼ �x�s� þ Oðjðq; lÞjÞ

with K1 ¼ ReðA1Þ; K2 ¼ ReðA2Þ:It is well known42 that the sign of K1 K2 determines the

direction of the bifurcation (supercritical if K1K2 < 0, sub-

critical if K1K2 > 0), and the sign of K2 determines the sta-

bility of the nontrivial periodic orbits (stable if K2 < 0,

unstable if K2 > 0). From (27) and (28), we have

K1 ¼ ReðA1Þ ¼ �x�ð1þ x2�ÞReðdÞ; (30)

and

K2 ¼ ReðA2Þ ¼ �es�x2ReðdÞ: (31)

From (10) and (25), it is easy to verify that for k > e;

ReðdÞ ¼ 1þ x2� þ s�x2

�ðk � eÞC

> 0;

ImðdÞ ¼s�x� 1� x2

�� C

< 0; for s� ¼ sþ1j;

> 0; for s� ¼ s�1j;

( (32)

where

C ¼ 1þ x2� þ s�x

2�ðk � eÞ

� �2 þ s2�x

2� 1� x2

�� 2

:

By (30), (31), and (32), we have

K1 ¼>0; for s� ¼ sþ1j;

<0; for s� ¼ s�1j;

(K2 < 0;

which immediately leads to the following theorem.

Theorem 4.1: Assuming that k > e and the assumptions

(A1) and (A2) hold, then near sþ1j ðrespectively; s�1jÞ, there

exists a supercritical (respectively, subcritical) bifurcation of

stable synchronous periodic solutions of period p, near

2p=xþ1 ðrespectively; 2p=x�1 Þ, bifurcated from the zero



solution of system (2), and those nontrivial periodic orbits

are all stable on the center manifold.

B. For the critical values s62j

From Sec. II, at s ¼ s62j the characteristic equation of

(18) has purely imaginary roots 6ix62 s6

2j which are double.

So, in this subsection, p¼ 4, x� ¼ xþ2 for s� ¼ sþ2j and

x� ¼ x�2 for s� ¼ s�2j and then we have

B ¼ diagðix�s�;�ix�s�; ix�s�;�ix�s�Þ

By (6), the base of the center space P at s ¼ s� can be taken

as U ¼ /1;/2;/3;/4ð Þ; where

/1ðhÞ ¼ eix�s�hv1; /2ðhÞ ¼ e�ix�s�h�v1;

/3ðhÞ ¼ eix�s�hv2; /4ðhÞ ¼ e�ix�s�h�v2;� 1 h 0: (33)

It is easy to check that a basis for the adjoint space P* is

WðsÞ ¼

w1ðsÞ

w2ðsÞ

w3ðsÞ

w4ðsÞ

0BBBBB@

1CCCCCA ¼

1

3

de�ix�s�s�vT1

�deix�s�svT1

de�ix�s�s�vT2

�deix�s�svT2

0BBBBB@

1CCCCCA; 0 s 1: (34)

with hWðsÞ;UðhÞi ¼ I4 for the adjoint bilinear form on

C� � C defined by (20), where d is defined by (25).

Let V5j ðC

4Þ be the homogeneous polynomials of degree

j in 5 variables, , with coefficients in C4, and let M1

j denote

the operator from V5j ðC

4Þ into itself defined by (26) with

h 2 V5j ðC

4Þ. Thus, we have

M1j ðlxqekÞ ¼ ix�s�lðq1 � q2 þ q3 � q4 þ ð�1ÞkÞxqek;

jqj ¼ j� 1;

where j � 2; 1 k 4; and fe1; e2; e3; e4g is the canonical

basis of R4: In particular, if jqj ¼ 1, then

KerðM12Þ \ span lxqek : jqj¼1;k¼1;2;3;4f g

¼ span lx1e1;lx3e1;lx2e2;lx4e2;lx1e3;lx3e3;lx2e4;lx4e4f g:(35)

It follows from (33) and (34) that

Wð0Þ ¼ 1

3

d�vT1

�dvT1

d�vT2

�dvT2

0BBBBB@

1CCCCCA;

Uð0Þx ¼ v1; �v1; v2; �v2ð Þx ¼ x1v1 þ x2�v1 þ x3v2 þ x4�v2

and

Uð�1Þx¼ x1e�ix�s�v1þ x2eix�s��v1þ x3e�ix�s�v2þ x4eix�s��v2:

By (21), we have

12! f

12 ðx; 0; lÞ ¼ Wð0ÞLlðUxÞ

¼ ilx�Wð0Þ x1v1 � x2�v1 þ x3v2 � x4�v2ð Þ

¼ ilx� 1þ x2�

� � d x1 � x4ð Þ�d x3 � x2ð Þd x3 � x2ð Þ�d x1 � x4ð Þ

0BBB@

1CCCA:

(36)

and then by (35) the second order terms in ðx; lÞ of the nor-

mal form on the center manifold are given by

12

g12ðx; 0; lÞ ¼ 1

2ProjKerðM1

2Þf

12 ðx; 0; lÞ

¼

B1x1l

B1x2l

B1x3l

B1x4l

0BBB@

1CCCA;

(37)

where

B1 ¼ ix� 1þ x2�

� �d: (38)

Next we compute the cubic terms g13ðx; 0; lÞ as described in

(23). We first note that

g13ðx; 0; lÞ ¼ ProjKerðM1

3Þ~f

13 ðx; 0; lÞ

¼ ProjKerðM13Þ~f

13 ðx; 0; 0Þ þ Oðjxjl2Þ;

where 13!

~f 13 ðx; 0; lÞ denotes the third order terms after the

computation of the normal form up to the second order

terms. It is easy to see from (36) and (37) that g12ðx; 0; 0Þ

¼ f 12 ðx; 0; 0Þ ¼ 0 So,

1

3!~f 13 ðx; 0; 0Þ ¼ 1

3! f13 ðx; 0; 0Þ

¼ Wð0ÞF3ðUx; s�Þ

¼ s�Wð0Þ

0

�e Uð0Þxð Þ1� �2

Uð0Þxð Þ20


Uð0Þxð Þ40


Uð0Þxð Þ6

0BBBBBBBB@

1CCCCCCCCA;

where

Uð0Þxð Þj ¼ x1v1j þ x2�v1j þ x3v2j þ x4�v2j; j ¼ 1; 2; 3; 4; 5; 6:

Also note that for q¼ 3,

M13ðxqejÞ ¼ 0; if and only if q1 � q2 þ q3 � q4 þ ð�1Þj ¼ 0;

j ¼ 1;2; 3;4:

So,

KerðM13Þ ¼ spanfx2

1x2e1; x21x4e1; x

23x2e1; x

23x4e1; x1x2x3e1;

x1x3x4e1; x22x1e2; x

22x3e2; x

24x1e2; x

24x3e2;

x1x2x4e2; x2x3x4e2; x21x2e3; x

21x4e3; x

23x2e3;

x23x4e3; x1x2x3e3; x1x3x4e3; x

22x1e4; x

22x3e4;

x24x1e4; x

24x3e4; x1x2x4e4; x2x3x4e4g



and then

1

3!g1

3ðx; 0; 0Þ ¼ 13! ProjKerðM1

3Þ~f

13 ðx; 0; 0Þ

¼

B2 x21x2 þ 2x1x3x4

� ��B2 x2

2x1 þ 2x2x3x4

� �B2 x2

3x4 þ 2x1x2x3

� ��B2 x2

4x3 þ 2x1x2x4

� �

0BBB@

1CCCA;

where

B2 ¼ �es� x�ð Þ2d: (39)

So, the normal form of (17) on the center manifold reads to

_x ¼ x�s�

i 0 0 0

0 �i 0 0

0 0 i 0

0 0 0 �i

0BBB@

1CCCAxþ

B1x1l�B1x2l

B1x3l�B1x4l

0BBB@

1CCCA

þ

B2 x21x2 þ 2x1x3x4

� ��B2 x2

2x1 þ 2x2x3x4

� �B2 x2

3x4 þ 2x1x2x3

� ��B2 x2

4x3 þ 2x1x2x4

� �

0BBB@

1CCCAþ Oðjxjl2 þ jxj4Þ; (40)

where B1 and B2 are defined by (38) and (39), respectively.

Changing (40) to real coordinates by the change of variables

x ¼

1 �i 0 0

1 i 0 0

0 0 1 �i

0 0 1 i

0BBB@

1CCCAw

and letting

q21 ¼ x1x2 ¼ w2

1 þ w22; q2

2 ¼ x3x4 ¼ w23 þ w2

4;

we obtain

_w1

_w2

� �¼ x�s�

w2

�w1

� �þ l

Re B1ð Þw1 þ Re B1ð Þw2

�Im B1ð Þw1 þ Re B1ð Þw2

� �

þ q21 þ 2q2

2

� � Re B2ð Þw1 þ Re B2ð Þw2ð Þ�Im B2ð Þw1 þ Re B2ð Þw2

� �þ Oðjwjl2 þ jwj4Þ;

_w3

_w4

� �¼ x�s�

w4

�w3

� �þ l

Re B1ð Þw3 þ Re B1ð Þw4

�Im B1ð Þw3 þ Re B1ð Þw4

� �

þ 2q21 þ q2

2

� � Re B2ð Þw3 þ Re B2ð Þw4ð Þ�Im B2ð Þw3 þ Re B2ð Þw4

� �þ Oðjwjl2 þ jwj4Þ:

If we use double polar coordinates

w1 ¼ q1 cos v1; w2 ¼ q1 sin v1;

w3 ¼ q2 cos v2; w4 ¼ q2 sin v2;

then we get

_q1 ¼ ðlReðB1Þ þ ðq21 þ 2q2

2ÞReðB2ÞÞq1

þ Oðjðq1; q2Þjl2 þ jðq1; q2Þj4Þ;_q2 ¼ ðlReðB1Þ þ ðq2

2 þ 2q21ÞReðB2ÞÞq2

þ Oðjðq1; q2Þjl2 þ jðq1; q2Þj4Þ;_v1 ¼ �x�s� � ImðB1Þl� ImðB2Þq2

1 � 2ImðB2Þq22

þ Oðjðq1; q2Þjl2 þ jðq1; q2Þj4Þ;_v2 ¼ �x�s� � ImðB1Þl� 2ImðB2Þq2

1 � ImðB2Þq22

þ Oðjðq1; q2Þjl2 þ jðq1; q2Þj4Þ:

Introducing the periodic-scaling parameter x and letting

z1ðtÞ ¼ w1ðsÞ þ iw2ðsÞ;z2ðtÞ ¼ w3ðsÞ þ iw4ðsÞ

with

s ¼ ð1þ rÞx�s�½ ��1t;

we obtain

ð1þ rÞ _z1 ¼ �iz1ðtÞ þl

x�s�ðReðB1Þ � iReðB1ÞÞz1ðtÞ

þ 1

x�s�ðjz1j2 þ 2jz2j2ÞðReðB2Þ � iReðB2ÞÞz1ðtÞ

þ Oðjzjl2 þ jzj4Þ

¼ �iz1ðtÞ þl

x�s��B1z1ðtÞ þ

1

x�s��B2z1ðtÞ

� ðjz1j2 þ 2jz2j2Þ þ Oðjzjl2 þ jzj4Þ:

Similarly, we get an equation for z2(t). Thus, ignoring the

terms Oðl2jzjÞ and Oðjzj4Þ, we get the normal form

ð1þrÞ _z1¼�iz1ðtÞþl

x�s��B1z1ðtÞþ

1

x�s��B2z1ðtÞðjz1j2þ2jz2j2Þ;

ð1þrÞ _z2¼�iz2ðtÞþl

x�s��B1z2ðtÞþ

1

x�s��B2z2ðtÞð2jz1j2þjz2j2Þ:

(41)

Let g : C�C�R! C�C be given so that �gðz1; z2; lÞis the right-hand side of (41). Then (41) can be written as

ð1þ xÞ _zþ gðz; lÞ ¼ 0

According to Ref. 39 (pp. 296–297, Theorems 6.3 and 6.5),

the bifurcations of small-amplitude periodic solutions of (41)

are completely determined by the zeros of equation,

� ið1þ rÞzþ gðz; lÞ ¼ 0; (42)

and their (orbital) stability is determined by the signs of three

eigenvalues of the following matrix:

Dgðz; 0Þ � ið1þ rÞId

By (41), (42) is equivalent to

irz1 þl

x�s��B1z1 þ

1

x�s��B2ðjz1j2 þ 2jz2j2Þz1 ¼ 0;

irz2 þl

x�s��B1z2 þ

1

x�s��B2ðjz2j2 þ 2jz1j2Þz2 ¼ 0:

(43)



To apply the results of Ref. 39, we write (43) as

Az1

z2

� �þ B

z21�z1

z22�z2

!¼ 0

with

A ¼ A0 þ ANðjz1j2 þ jz2j2Þ; B ¼ B0

for some complex numbers A0, AN, and B0 given by

A0 ¼ �l

x�s��B1 � ix;

AN ¼ �2

x�s��B2;

B0 ¼1

x�s��B2:

(44)

It follows from (25), (32), (39), and (44) that for k > e;

ReðAN þ B0Þ ¼ �1

x�s�Re �B2ð Þ ¼ ex�ReðdÞ > 0;

Reð2AN þ B0Þ ¼ �3

x�s�Re �B2ð Þ3ex�ReðdÞ > 0;

and

ReB0 ¼1

x�s�Re �B2ð Þ ¼ �ex�ReðdÞ < 0:

In addition, from (12), (25), and (38), we obtain that

Re�B1

x�s�

� �¼ �

1þ x2�

� �s�

ReðdÞ

¼x� 1þ x2

��

x2� � 1

� �C

> 0; for s� ¼ sþ2j;

< 0; for s� ¼ s�2j:

(

By the results of Ref. 39 (pp. 383, Theorem 3.1), we have

the following theorem on the properties of bifurcation at the

critical values s62j :

Theorem 4.2: Assuming that k > e and the assumptions

(A1) and (A2) hold, then the bifurcations are supercritical

(respectively, subcritical) at the critical values sþ2j (respec-

tively, s�2j), and then on the center manifold the phase-locked

oscillations are orbitally asymptotically stable and the mir-

ror-reflecting waves and standing waves are unstable.

V. NUMERICAL CONTINUATION

In this section, we perform some numerical simulations

to illustrate and expand the results obtained above. In the fol-

lowing numerical simulations, we take e ¼ 0:03; k ¼ 0:1such that ðe; kÞ belongs to the shaded region of Fig. 1, where

stability switches occur with increasing the coupling time

delay. The following numerical simulations were performed

by Simulink of MATLAB and the initial functions are chosen as

default of Simulink for delay differential equations with

y1ð0Þ ¼ 0:2; _y1ð0Þ ¼ 0:1; y2ð0Þ ¼ �0:3; _y2ð0Þ ¼ �0:1; y3ð0Þ¼ 0:5; _y3ð0Þ ¼ �0:4.

From (11) and (13), we have

s�10¼:

1:3322; sþ10¼:

4:6172; s�11¼:

8:2315; sþ11¼:

10:3393;

s�12¼:

15:1307; sþ12¼:

16:0615; sþ13¼:

21:7836; s�13¼:

22:0300;

and

sþ20 ¼:

2:2639; s�20 ¼:

4:0801; sþ21 ¼:

8:3267; s�21 ¼:

10:5916;

sþ22 ¼:

14:3896; s�22 ¼:

17:1032; sþ23 ¼:

20:4524;

s�23 ¼:

23:6147; sþ24 ¼:

265153

So, the critical values of the coupling time delay can be

arranged as

s�10 < sþ20 < s�20 < sþ10 < s�11 < sþ21 < sþ11 < s�21 < sþ22

< s�12 < sþ12 < s�22 < sþ23 < sþ13 < s�13 < s�23 < sþ24:

According to Theorem 2.1, the zero equilibrium of system

(2) is asymptotically stable for s 2 s�10; sþ20

� �[ s�20; s

þ10

� �[

s�11; sþ21

� �. Figure 3 illustrate this result for s ¼ 2 2

s�10; sþ20

� �:

According to Theorems 3.1, 4.1, and 4.2, when s is less

than and close to s�10, stable synchronous periodic solutions

bifurcate from the zero solution. When s is greater than and

close to sþ20 or s is less than and close to s�20, stable phase-

locked periodic solutions bifurcate from the zero solution.

The existence of these stable phase-locked periodic solutions

are illustrated in Figs. 4(a) and 4(b) for s ¼ 2:3 greater than

and close to sþ20 and in Figs. 4(e) and 4(f) for s ¼ 2:4 less

than and close to s�20. When s is greater than and close to sþ10

or s is less than and close to s�11, stable synchronous periodic

solutions again bifurcate from the zero solution. The exis-

tence of these stable synchronous periodic solutions is illus-

trated in Figs. 5(a) and 5(b) for s ¼ 4:7 greater than and

close to sþ10 and in Figs. 5(e) and 5(f) for s ¼ 8:1 less than

and close to s�11. These numerical simulations are exactly

consistent with the results obtained in Secs. II–IV. When s is

far away from these critical values, our theoretical analysis

above is not available. However, numerical simulations

show that these stable phase-locked periodic solutions

always exist for all s 2 sþ20; s�20

� �and stable synchronous

periodic solutions always exist for all s 2 sþ10; s�11

� �and that

further the delay is far away from the critical values, the big-

ger the amplitude of the periodic solution is. Figures 5(c)

and 5(d) illustrate the trajectories of these periodic

FIG. 3. (Color online) Trajectories of the oscillators y1 (solid line), y2

(dashed line), and y3 (dotted line): the zero equilibrium is asymptotically sta-

ble for s 2 s�10; sþ20

� �[ s�20; s

þ10

� �[ s�11; s

þ21

� �. Here s ¼ 2 2 s�10; s

þ20

� �.



FIG. 5. (Color online) Trajectories of the oscillators y1 (solid line), y2 (dashed line), and y3 (dotted line) with stable synchronous periodic motion at s ¼ 4:7(a and b), s ¼ 6:5 (c and d), and s ¼ 8:1 (e and f), where s 2 sþ10; s

�11

� �:

FIG. 4. (Color online) Trajectories of the oscillators y1 (solid line), y2 (dashed line), and y3 (dotted line) with stable phase-locked periodic motion at s ¼ 2:3(a and b), s ¼ 3:1 (c and d), and s ¼ 4 (e and f), where s 2 sþ20; s

�20

� �:



oscillations for s ¼ 3:1 far away from the critical values sþ20

and s�20, and Figs. 6(c) and 6(d) illustrate the trajectories of

these periodic oscillations for s ¼ 6:5 far away from the crit-

ical values sþ10 and s�11. The results of these numerical contin-

uation are described as shown in Fig. 7.

When s > sþ21; the zero equilibrium of system (2) loses

its stability permanently. When s is greater than and close to

sþ21, Theorems 3.1, 4.1, and 4.2 show that there are stable

phase-locked periodic orbits bifurcating from the zero equi-

librium. Numerical simulations show that system (2) always

possesses for all s > sþ21. With increasing the coupling time

delay, oscillation patterns of these stable periodic orbits

switch from being phase-locked to synchronous and back to

phase-locked several times and finally becoming synchro-

nous. Numerical simulations also show that these stable peri-

odic orbit changes its oscillation patterns only when s is less

than and very close to the critical values s�2m and s�1n with

m¼ 1, 2, 3, n¼ 2, 3. More precisely, there exist sufficiently

small positive number l1; l2; l3; l4; l5 such that stable

phase-locked periodic orbits exist for

ðsþ21; s�21 � l1Þ [ ðs�12 � l2; s

�22 � l3Þ [ ðs�13 � l4; s

�23 � l5Þ

and stable synchronous periodic orbits exist for

ðs�21 � l1; s�12 � l1Þ [ ðs�22 � l3; s

�13 � l4Þ [ ðs�23 � l5;þ1Þ:

Figure 7 shows the switch from being phase-locked to syn-

chronous when s is less than and close to s�21: Figures 6(a)

and 6(b) show the existence of stable phase-locked periodic

oscillation for s ¼ 10:4 < s�21 � l1 and Figs. 7(c) and 7(d)

show the existence of stable synchronous periodic oscillation

for s ¼ 10:5 > s�21 � l1:

VI. CONCLUSIONS

We have studied the bifurcation, amplitude death, and os-

cillation patterns induced by the coupling time delay in a sys-

tem of three coupled van der Pol, which are coupled through

the damping terms. We have shown that the coupling time

delay can lead to rich dynamics. The stability of zero solution

is determined in the parameter space. The existence of

FIG. 6. (Color online) Regions for stable synchronous

and phase-locked periodic solutions in the plane of the

coupling delay and strength which are separated by

islands of amplitude death.

FIG. 7. (Color online) Trajectories of the oscillators y1 (solid line), y2 (dashed line), and y3 (dotted line) with stable phase-locked periodic motion at s ¼ 10:4(a and b) and stable synchronous periodic motion at s ¼ 10:5 (c and d), where s is less than and close to s�21:



stability switches induced by the coupling time delay is found

in the region of the parameter plane of the damping strength

and the coupling strength. The islands of amplitude death are

also described in the parameter plane of the coupling time

delay and strength for appropriate fixed damping strength.

In the region where stability switches occur, there are

two kinds of critical values s61j and s6

2j of the coupling time

delay leading to Hopf bifurcating periodic solutions. By

using the symmetric bifurcation theory of delay differential

equations due to Wu, we have shown that at the critical val-

ues s61j , there exist a synchronous periodic solution bifurca-

tion and at the critical values s62j , there are eight branches of

asynchronous periodic solutions bifurcating simultaneously

from the zero solution: two phase-locked oscillations, mir-

ror-reflecting waves and three standing waves. The direction

and stability of these bifurcations is based on the normal

form calculations. We have shown that the bifurcation is

subcritical at the critical values s�kj and supercritical at the

critical values sþkj , k¼ 1, 2, j¼ 0, 1, 2, .... We have also

shown that on the center manifold the synchronous periodic

and phase-locked periodic solutions are stable and other mir-

ror-reflecting and standing waves are unstable.

Numerical studies have been employed to support and

extend the obtained theoretical results. For fixed damping

strength and coupling strength, the numerical simulations

illustrate the existence of stable synchronous and phase-

locked periodic solutions near the critical values of the cou-

pling time delay which is in good agreement with our theoreti-

cal analysis results. For the coupling time delay far away from

the critical values, our theoretical analysis is not available.

However, numerical simulations show that with increasing the

coupling time delay from the zero, the dynamics of the system

switches from being synchronous to stability, to being phase-

locked to stability, and so on. When the zero solution ulti-

mately loses its stability, the oscillation patterns of these

stable periodic solutions switch from being phase-locked to

being synchronous (or vice versa) and finally becoming syn-

chronous with increasing the coupling time delay.

ACKNOWLEDGMENTS

The first two authors (Y. Song and J. Xu) would like to

acknowledge the support from the Shanghai Committee of

Science and Technology (No. 10ZR1431900), the National

Nature Science Foundation of China (No. 10971155), and

the State Key Program of National Natural Science of China

(No. 11032009). The third author (T. Zhang) would like to

acknowledge the support from the Australian Research

Council (No. DP0770420) and the National Nature Science

Foundation of China (No. 11001212).

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