triple-zero bifurcation in van der pol’s oscillator with delayed feedback

Commun Nonlinear Sci Numer Simulat 17 (2012) 5229–5239

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Commun Nonlinear Sci Numer Simulat

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Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback q

Xing He a, Chuandong Li a,⇑, Yonglu Shu b

a College of Computer Science, Chongqing University, Chongqing 400030, PR Chinab College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China

a r t i c l e i n f o

Article history:Received 26 June 2011Received in revised form 30 November 2011Accepted 4 May 2012Available online 17 May 2012

Keywords:Triple-zero bifurcationBogdanov–Takens bifurcationZero-Hopf bifurcation

1007-5704/$ - see front matter � 2012 Elsevier B.Vhttp://dx.doi.org/10.1016/j.cnsns.2012.05.001

q This research is supported by the National NatuFunds for the Central Universities of China (Project⇑ Corresponding author.

E-mail address: [email protected] (C. Li).

a b s t r a c t

In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zerobifurcation is investigated by using center manifold reduction and the normal form methodfor retarded functional differential equation. We get the versal unfolding of the norm formsat the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation,Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numericalsimulations are given to support the analytic results.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Since van der Pol’s equation was first introduced in a mathematic model of ordinary differential equation, there has beenincreasing interest in investigating the dynamical behaviors of van der Pol’s equation due to its wide application, such asoscillation in electrical circuits, engineering sciences and many physical problem. Some important results have beenreported [5,6,13].

Recently, some researchers have considered the effect of time delay in van der Pol’s equation as most practical implemen-tations of feedback have inherent delays. It is shown that the presence of time delay can change the amplitude of limit cycleoscillations [2,7–12,15–17,19,20]. In this paper, we consider a van der Pol’s equation with delayed feedback in the followingmodel [8]

€xþ e x2 � 1� �

_xþ x ¼ eg x t � sð Þð Þ; x 2 R; e > 0; ð1Þ

where g 0ð Þ ¼ 0; g0 0ð Þ ¼ k; g00 0ð Þ – 0. This model exhibits a diversity of local bifurcations at the origin, it is shown that themodel not only undergoes codimension one bifurcation, such as pitchfork bifurcation, transcritical bifurcation [8] and Hopfbifurcation [9], but also exhibits codimension two singularities, such as Bogdanov–Takens bifurcation [15] and fold-Hopfbifurcation [16,17]. When the parameters s ¼ s0 ¼

ffiffiffi2p

; e ¼ e0 ¼ffiffiffi2p

; k ¼ k0 ¼ffiffi2p

2 , triple-zero bifurcation of codimension-three also occurs in this model. In fact, with above parameters, the linear part of the system (1) has triple zero eigenvaluesand the remaining roots have negative real parts [8].

The study carried out in the present paper may contribute to understand the codimension-three singularity in van derPol’s oscillator with delayed feedback. We use s; e; k as bifurcation parameters. System (1) exhibits codimension-three sin-gularity when three parameters vary in a neighborhood of the critical values. By using the normal form method for retarded

. All rights reserved.

ral Science Foundation of China Grant Nos. 60974020 and 11171360 and the Fundamental ResearchNo. CDJZR10 18 55 01).

http://dx.doi.org/10.1016/j.cnsns.2012.05.001

mailto:[email protected]

http://dx.doi.org/10.1016/j.cnsns.2012.05.001

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5230 X. He et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 5229–5239

functional differential equation [1,3,4,14], we obtain the normal form to study its dynamical behaviors. It is shown that peri-odic solution, quasi-periodic solution and chaotic solution can be found near the triple-zero point.

This paper is organized as follows. In the next section, the preliminaries relevant to the normal forms with parameters forretarded functional differential equation are presented. In Section 3, we analyze triple-zero singularity in van der Pol’s oscil-lator with delayed feedback and get the versal unfolding of triple-zero bifurcation. In Section 4, the normal form are used topredict local triple-zero bifurcation diagram. In Section 5, some numerical simulations are given to support the analyticresults. Section 6 summarizes the main conclusions.

2. Preliminaries

This section presents the triple-zero bifurcation theory of normal form with parameters for functional differential equa-tions [1,3,4,14]. We consider an abstract retarded functional differential equation with parameters in the phase spaceC ¼ C �s;0½ �; Rnð Þ

_xðtÞ ¼ LðlÞxt þ Gðxt ;lÞ; ð2Þ

where xt 2 C is defined by xt hð Þ ¼ x t þ hð Þ, �s 6 h 6 0, the parameter l 2 Rp is a parameter vector in a neighborhood V ofzero, L lð Þ : V ! L C;Rnð Þ is Ck�1, and G : C � Rp ! Rn is Ck k P 2ð Þ with G 0;lð Þ ¼ 0; DG 0;lð Þ ¼ 0.

Define L ¼ Lð0Þ and Fðxt ;lÞ ¼ Gðxt ;lÞ þ L lð Þ � L 0ð Þð Þxt , then system (2) can be rewritten as

_xðtÞ ¼ Lxt þ Fðxt;lÞ: ð3Þ

Then the linear homogeneous retarded functional differential equation of Eq. (3) can be written as

_x tð Þ ¼ Lxt: ð4Þ

Since L is a bounded linear operator, L can be represented by a Riemann–Stieltjes integral

Lu ¼Z 0

�sdgðhÞuðhÞ; 8 u 2 C; ð5Þ

by the Riesz representation theorem, where g hð Þ h 2 �s;0½ �ð Þ is an n� n matrix function of bounded variation. Let A0 be theinfinitesimal generator for the solution semigroup defined by system (4) such that

A0u ¼ _u; D A0ð Þ ¼ u 2 C1 �s;0½ �;Rnð Þ : _u 0ð Þ ¼Z 0

�sdgðhÞu hð Þ

� �ð6Þ

and its adjoint

A�0w ¼ � _w; D A�0� �

¼ w 2 C1 0; s½ �;Rn��

: _w 0ð Þ ¼ �Z 0

�sdg hð Þw �hð Þ

� �: ð7Þ

Define the bilinear form between C and C0 ¼ C 0; s½ �;Rn��

by

w;uh i ¼ w 0ð Þu 0ð Þ �Z 0

�s

Z h

0w n� hð Þdg hð Þu nð Þdn; 8 w 2 C 0;8u 2 C: ð8Þ

Assume that L has triple zero eigenvalues and all other eigenvalues have negative real parts. Let K be the set of eigenvalueswith zero real part and P be the generalized eigenvalues space associated with K and P� the space adjoint with P. Then C canbe decomposed as

C ¼ P � Q where Q ¼ u 2 C : u;wh i ¼ 0; 8w 2 P�f g;

with dim P ¼ 3. Choose the bases U and W for P and P� such that

W;Uh i ¼ I; _U ¼ UJ; _W ¼ �JW;

where I is the m�m identity matrix and J ¼0 1 00 0 10 0 0

0@

1A. Following the ideas in [3,4], we consider the enlarged phase space

BC

BC ¼ u : �s;0½ � ! Rn : u is continuous on½�s; 0Þ; 9 limh!0�

u hð Þ 2 Rn� �

:

Then the elements of BC can be expressed as w ¼ uþ x0a, u 2 C; a 2 Rn and

x0ðhÞ ¼0;�s 6 h < 0;I; h ¼ 0;

�ð9Þ

X. He et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 5229–5239 5231

where I is the identity operator on C. The space BC has the norm uþ x0aj j ¼ uj jC þ aj jRn . The definition of the continuousprojection p : BC ! P by

p uþ x0að Þ ¼ U W;uð Þ þW 0ð Þa½ � ð10Þ

allows us to decompose the enlarged phase space BC ¼ P � Kerp. Let x ¼ Uzþ y. Then system (2) can be decomposed as

_z ¼ JzþW 0ð ÞF Uzþ y;lð Þ;

_y ¼ AQ1 yþ I � pð Þx0F Uzþ y;lð Þ; z 2 R3; y 2 Q 1;

8<: ð11Þ

for y 2 Q 1 ¼ Q \ C1 � Kerp, where AQ1 is the restriction of A0 as an operator from Q1 to the Banach space Kerp.Employing Taylor’s theorem, system (11) becomes

_z ¼ Jzþ RjP21j f 1

j z; y;lð Þ;

_y ¼ AQ1 yþ RjP21j f 2

j z; y;lð Þ;

8<: ð12Þ

where f ij z; y;lð Þ i ¼ 1;2ð Þ denotes the homogeneous polynomials of degree j in variables z; y;lð Þ. For J ¼

0 1 00 0 10 0 0

0@

1A, the

non-resonance conditions are naturally satisfied. According to normal form theory developed in [5], system (12) can betransformed into the following normal form on the center manifold

_z ¼ Jzþ 12

g12 z;0;lð Þ þ h:o:t: ð13Þ

For a normed space Z, denoted by V6j Zð Þ the linear space of homogeneous polynomials of z;lð Þ ¼ z1; z2; z3;l1;l2;l3

� �with

degree j and with coefficients in Z and define Mj to be the operator in V6j R3 � Kerp� �

with the range in the same spaceby

Mj p;hð Þ ¼ M1j p;M2

j h� �

; ð14Þ

where

M1j p ¼ M1

j

p1

p2

p3

0BB@

1CCA ¼

@p1@z1

z2 þ @p1@z2

z3 � p2

@p2@z1

z2 þ @p2@z2

z3 � p3

@p3@z1

z2 þ @p3@z2

z3

0BBBB@

1CCCCA;

M2j h ¼ M2

j h z;lð Þ ¼ Dzh z;lð ÞJx� AQ1 h z;lð Þ:

Using M1j , we have the following decompositions

V6j R3� �

¼ Im M1j

� �� Im M1

j

� �� c; V6

j R3� �

¼ Ker M1j

� �� Ker M1

j

� �� c: ð15Þ

By the above decompositions, g12 z;0;lð Þ can be expressed as

g12 z;0;lð Þ ¼ Project

Im M12ð Þð Þc f 1

2 z;0;lð Þ: ð16Þ

The base of V62 R3 � Kerp� �

is composed of the following 63 elements. The detailed expressions can be found in [14].

zizj

0

0

0BB@

1CCA;

lilj

0

0

0BB@

1CCA;

lizi

0

0

0BB@

1CCA;

0

zizj

0

0BB@

1CCA;

0

lilj

0

0BB@

1CCA;

0

lizi

0

0BB@

1CCA;

0

0

zizj

0BB@

1CCA;

0

0

lilj

0BB@

1CCA;

0

0

lizi

0BB@

1CCA; i; j ¼ 1;2;3


and the images of these elements under M12 are

z1z2

00

0@

1A; z2z3

00

0@

1A; z2

2 þ z1z3

00

0@

1A; z2

3

00

0@

1A; liz2

00

0@

1A; liz3

00

0@

1A; �z2

1

2z1z2

0

0@

1A; �z2

2

2z2z3

0

0@

1A;

�z23

00

0@

1A; �lilj

00

0@

1A; �z1z2

z22 þ z1z3

0

0@

1A; �z1z3

z2z3

0

0@

1A; �z2z3

z23

0

0@

1A; �liz1

liz2

0

0@

1A; �liz2

liz3

0

0@

1A;

�liz3

00

0@

1A; 0

�z21

2z1z2

0@

1A; 0

�z22

2z2z3

0@

1A; 0

�z23

0

0@

1A; 0

�lilj

0

0@

1A; 0

�z1z2

z22 þ z1z3

0@

1A; 0

�z1z3

z2z3

0@

1A;

0�z2z3

z23

0@

1A; 0

�liz1

liz2

0@

1A; 0

�liz2

liz3

0@

1A; 0

�liz3

0

0@

1A; i; j ¼ 1;2;3:

Therefore, a basis of Im M12

� �� ccan be taken as the set composed by the elements

00z2

1

0B@

1CA;

00z2

2

0B@

1CA;

00

liz1

0B@

1CA;

00

liz2

0B@

1CA;

00

liz3

0B@

1CA;

00

z1z2

0B@

1CA;

00

z1z3

0B@

1CA; i; j ¼ 1;2;3:

Then we have

12

g12 z;0;lð Þ ¼

0k1z1 þ k2z2 þ k3z3 þ g1z2

1 þ g2z22 þ g3z1z2 þ g4z1z3 þ h:o:t:

: ð17Þ

3. Triple-zero bifurcation

From [8], system (1) undergoes a triple-zero bifurcation if s ¼ s0 ¼ffiffiffi2p

; e ¼ e0 ¼ffiffiffi2p

; k ¼ k0 ¼ffiffi2p

2 . In this section, we aregoing to obtain the normal forms of the reduced equations using center manifold theory and the normal form method forfunctional differential equation.

Let x1 ¼ x; x2 ¼ _x, then system (1) can be written as

_x1 ¼ x2;

_x2 ¼ �x1 � e x21 � 1

� �x2 þ eg x1 t � sð Þð Þ:

(ð18Þ

Rescaling the time by t ! t=s to normalize the delay, system (18) becomes

_x1 ¼ sx2;

_x2 ¼ �sx1 � es x21 � 1

� �x2 þ es kx1 t � 1ð Þ þ 1

2 g00 0ð Þx21 t � 1ð Þ

� �þ h:o:t:

(ð19Þ

Let s ¼ffiffiffi2pþ l1; e ¼

ffiffiffi2pþ l2; k ¼

ffiffi2p

2 þ l3. Then system (19) can be written as

_x1 ¼ ðffiffiffi2pþ l1Þx2;

_x2 ¼ �ðffiffiffi2pþ l1Þx1 þ ð2þ

ffiffiffi2p

l1 þffiffiffi2p

l2Þx2 þ ðffiffiffi2pþ l1 þ l2 þ 2l3Þx1ðt � 1Þ þ G2

2 þ h:o:t;

(ð20Þ

where G22 ¼ g00ð0Þx2

1ðt � 1Þ.From Section 2, we let

g hð Þ ¼ Ad hð Þ þ Bd hþ 1ð Þ; ð21Þ

where

A ¼ 0ffiffiffi2p

�ffiffiffi2p

2

!; B ¼

0 0ffiffiffi2p

0

;

and define

L0u ¼Z 0

�1dg hð Þu hð Þ; 8 u 2 C: ð22Þ


The infinitesimal generator

A0u ¼_u �1 6 h < 0;R 0�1 dg hð Þu hð Þ; h ¼ 0:

(ð23Þ

Rewrite system (20) as

_xt ¼ L lð Þxt þ G xt ;lð Þ þ h:o:t ¼ L0 þ L1 lð Þð Þxt þ G xt;lð Þ þ h:o:t; ð24Þ
where
L0xt ¼

ffiffiffi2p

x2 0ð Þ

�ffiffiffi2p

x1 0ð Þ þ 2x2 0ð Þ þffiffiffi2p

x1 �1ð Þ

0@

1A; G xt ;lð Þ ¼

0

G22

!;

L1 lð Þxt ¼l2x2 0ð Þ

�l1x1 0ð Þ þffiffiffi2p

l1 þffiffiffi2p

l2

� �x2 0ð Þ þ l1 þ l2 þ 2l3

� �x1 �1ð Þ

0@

1A:

System (20) can be transformed into

_xt ¼ L0xt þ F xt ;lð Þ þ h:o:t; ð25Þ

where F xt ;lð Þ ¼ L1 lð Þxt þ G xt ;lð Þ, and the bilinear form on C� � C is

w;uh i ¼ w 0ð Þu 0ð Þ þZ 0

�1w nþ 1ð ÞBu nð Þdn; ð26Þ

where U hð Þ ¼ u1 hð Þ;u2 hð Þ;u3 hð Þð Þ 2 C, W sð Þ ¼w1ðsÞw2ðsÞw3ðsÞ

0@

1A 2 C� are respectively the bases for the center space P and its dual

space P�. In the following, we will find U hð Þ and W sð Þ based on the techniques developed by [14,18].

Theorem 1. There are bases for the center space P and its dual space P�, respectively.

U hð Þ ¼ u1 hð Þ;u2 hð Þ;u3 hð Þð Þ ¼1 h 1

2 h2

0ffiffi2p

2

ffiffi2p

2 h

0@

1A; �1 6 h 6 0;

W sð Þ ¼ col w1 sð Þ;w2 sð Þ;w3 sð Þð Þ ¼

2740� 3

2 s� 3s2 3ffiffi2p

80 � 3ffiffi2p

4 sþ 3ffiffi2p

2 s2

32þ 6s 3

ffiffi2p

4 � 3ffiffiffi2p

s

�6 3ffiffiffi2p

0BBB@

1CCCA; 0 6 s 6 1;

0 1 00 1

such that W; Uh i ¼ I; _U ¼ UJ; _W ¼ �JW, where J ¼ 0 0 10 0 0

@ A.

Proof. For J ¼0 1 00 0 10 0 0

0@

1A; k ¼ 0 is an eigenvalue of A0 with algebraic multiplicity = 3 and geometric multiplicity = 1. So we

obtain

A0u1 ¼ 0; A0u2 ¼ u1; A0u3 ¼ u2 ð27Þ

and A0u1 ¼ 0 is equivalent to

_u1 hð Þ ¼ 0; �1 6 h < 0;R 0�1 dg hð Þu1 hð Þ ¼ 0; h ¼ 0;

(ð28Þ

from which we obtain u1 hð Þ ¼ u01 2 R2 n 0f g and Aþ Bð Þu0

1 ¼ 0. Set u01 ¼

10 , and we get u1 hð Þ ¼ 1

0 .Moreover, A0u2 ¼ u1 is equivalent to

_u2 hð Þ ¼ u01; �1 6 h < 0;R 0

�1 dg hð Þu2 hð Þ ¼ u01; h ¼ 0;

8<: ð29Þ

which implies that u2 hð Þ ¼ u01hþu0

2 and Bþ Ið Þu01 ¼ Aþ Bð Þu0

2. Choose u02 ¼

0ffiffi2p

2

, and we obtain u2 hð Þ ¼ hffiffi

2p

2

.

In addition, A0u3 ¼ u2 is equivalent to

_u3 hð Þ ¼ u2 hð Þ; �1 6 h < 0;R 0�1 dg hð Þu3 hð Þ ¼ u0

2; h ¼ 0;

8<: ð30Þ


which means that u3 hð Þ ¼ 12 u0

1h2 þu0

2hþu03 and Bþ Ið Þu0

2 � 12 Bu0

1 ¼ Aþ Bð Þu03. Let u0

3 ¼00

, and we obtain u3 hð Þ ¼

12 h2ffiffi

2p

2 h

!. Let U hð Þ ¼ u1 hð Þ;u2 hð Þ;u3 hð Þð Þ. Then we have _U ¼ UJ.

Similarly, we have

A�0w3 ¼ 0; A�0w2 ¼ w3; A�0w1 ¼ w2; ð31Þ

since A�0w3 ¼ 0 is equivalent to

_w3 sð Þ ¼ 0; 0 6 s < 1;R 0�1 dg sð Þw3 �sð Þ ¼ 0; s ¼ 0;

(ð32Þ

from which we obtain w3 sð Þ ¼ w03 ¼ a; bð Þ 2 R2 n 0f g and w0

3 Aþ Bð Þ ¼ 0, which givesffiffiffi2p

aþ 2b ¼ 0. Set a ¼ffiffiffi2p

m; b ¼ �m. AsW;Uh i ¼ I, we obtain

w3 sð Þ;u3 hð Þh i ¼ w3 0ð Þu3 0ð Þ þZ 0

�1w3 nþ 1ð ÞBu3 nð Þdn ¼ 1; ð33Þ

we have m ¼ �3ffiffiffi2p

. Hence w3 sð Þ ¼ �6;3ffiffiffi2p� �

.

By the definition of A�0; A�0w2 ¼ w3 is equivalent to

� _w2 sð Þ ¼ w03; 0 6 s < 1;R 0

�1 dg sð Þw3 �sð Þ ¼ w03; s ¼ 0;

8<: ð34Þ

which implies that w2 sð Þ ¼ �u03sþ w0

2 and w03 Bþ Ið Þ ¼ w0

2 Aþ Bð Þ, setting w02 ¼ n1;n2ð Þ, which gives n1 þ

ffiffiffi2p

n2 ¼ 3. AsW;Uh i ¼ I, we obtain


�1w2 nþ 1ð ÞBu3 nð Þdn ¼ 0: ð35Þ

we calculate n2 ¼ 3ffiffi2p

4 ; n1 ¼ 32. Hence

w2 sð Þ ¼ 32þ 6s;

3ffiffiffi2p

4� 3

ffiffiffi2p

s

!:

Moreover, A�0w1 ¼ w2, is equivalent to

� _w1 sð Þ ¼ w2 sð Þ; 0 6 s < 1;R 0�1 dg sð Þw1 sð Þ ¼ w0

2; s ¼ 0;

8<: ð36Þ

which means that w1 sð Þ ¼ w01 � sw0

2 þ 12 s2w0

3 and w01 Aþ Bð Þ ¼ w0

2 Bþ Ið Þ � 12 w0

3B. Setting w02 ¼ k1; k2ð Þ, which gives k1 þ

ffiffiffi2p

k2 ¼ 34.

As W;Uh i ¼ I, we obtain


�1w1 nþ 1ð ÞBu3 nð Þdn ¼ 0: ð37Þ

we have k1 ¼ 2740 ; k2 ¼ 3

ffiffi2p

80 . Hence

w1 sð Þ ¼ 2740� 3

2s� 3s2;

3ffiffiffi2p

80� 3

ffiffiffi2p

4sþ 3

ffiffiffi2p

2s2

!:

Let W ¼ col w1 sð Þ;w2 sð Þ;w3 sð Þð Þ. Then we have W; Uh i ¼ I; _U ¼ UJ; _W ¼ �JW. h

Let x ¼ Uzþ y, namely

x1 hð Þ ¼ z1 þ hz2 þ12

h2z3 þ y1 hð Þ; x2 hð Þ ¼ffiffiffi2p

2z2 þ

ffiffiffi2p

2hz3 þ y2ðhÞ: ð38Þ

Then

x1 0ð Þ ¼ z1 þ y1 0ð Þ; x1 �1ð Þ ¼ z1 � z2 þ12

z3 þ y1 �1ð Þ; x2 0ð Þ ¼ffiffiffi2p

2z2 þ y2 0ð Þ: ð39Þ


From Section 2, system (20) can be decomposed as

_z ¼ JzþW 0ð ÞF Uzþ y;lð Þ þ h:o:t;

_y ¼ AQ1 yþ I � pð Þx0F Uzþ y;lð Þ þ h:o:t; z 2 R3; y 2 Q1:

8<: ð40Þ

On the center manifold, system (40) can be written as8
_z1 ¼ z2 þ 27
40 F12 þ 3

ffiffi2p

80 F22 þ h:o:t;

_z2 ¼ z3 þ 32 F1

2 þ 3ffiffi2p

4 F22 þ h:o:t;

_z3 ¼ �6F12 þ 3

ffiffiffi2p

F22 þ h:o:t;

>><>>: ð41Þ

where

F12 ¼

ffiffiffi2p

2l2z2; F2

2 ¼ l2z1 þ 2l3z1 � 2l3z2 þ12l1z3 þ

12l2z3 þ l3z3 þ g00 0ð Þ z1 � z2 þ

12

z3

2

:

Following the computation of the normal form for functional differential equations introduced by Section 2, we get the nor-mal form with versal unfolding on the center manifold

_z1 ¼ z2;

_z2 ¼ z3;

_z3 ¼ k1z1 þ k2z2 þ k3z3 þ g1z21 þ g2z2

2 þ g3z1z2 þ g4z1z3 þ h:o:t;

8><>: ð42Þ

where

k1 ¼ 3ffiffiffi2p

l2 þ 6ffiffiffi2p

l3; k2 ¼ �3ffiffiffi2p

l1 þ3ffiffiffi2p

4l2 �

9ffiffiffi2p

2l3; g1 ¼ 3

ffiffiffi2p

g00ð0Þ;

k3 ¼9ffiffiffi2p

4l1 þ

123ffiffiffi2p

80l2 þ

63ffiffiffi2p

40l3; g2 ¼

63ffiffiffi2p

40g00 0ð Þ; g3 ¼ �

9ffiffiffi2p

2g00 0ð Þ; g4 ¼

63ffiffiffi2p

40g00 0ð Þ:

It is easy to check that

@ k1; k2; k3ð Þ@ l1;l2;l3

� � !

l¼0

¼ �108ffiffiffi2p

–0: ð43Þ

Then the map l1;l2;l3

� �# k1; k2; k3ð Þ is regular. Thus we can get the following result

Theorem 2. If g00 0ð Þ–0, then, on the center manifold, system (20) is equivalent to the normal form (42).

4. Bifurcation diagram

In this section, we discuss the bifurcation diagram of system (20). We consider the truncated normal form of system (42).

_z1 ¼ z2;

_z2 ¼ z3;

_z3 ¼ k1z1 þ k2z2 þ k3z3 þ g1z21 þ g2z2

2 þ g3z1z2 þ g4z1z3;

8><>: ð44Þ

The bifurcation diagram of system (44) can be found in [1]. Here we just briefly list some results.

(i) System (44) undergoes a transcritical bifurcation at the origin on the curve

T ¼ k1; k2; k3ð Þ : k1 ¼ 0f g;

(ii) system (44) undergoes a Hopf bifurcation at the origin on the curve

H1 ¼ k1; k2; k3ð Þ : k3 ¼ �k1

k2; k2 < 0

� �;

(iii) system (44) undergoes a Hopf bifurcation at the non-trivial equilibrium point on the curve

H2 ¼ k1; k2; k3ð Þk3 ¼g4

g1� g1

g3k1 � g1k2

k1;

g1

g3k1 � g1k2> 0

� �;

(iv) system (44) undergoes a Bogdanov–Takens bifurcation at the origin on the curve

B ¼ k1; k2; k3ð Þ : k1 ¼ 0; k2 ¼ 0f g;

(v) system (44) undergoes a zero-Hopf bifurcation at the origin on the curve

−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

x1

x 2

Fig. 1. Phase portraits of system (18) for s ¼ffiffiffi2p

; e ¼ 1:404; k ¼ffiffi2p

2 .

0 50 100 150 200 250 300−1

−0.5

0

0.5

t

x 1

0 50 100 150 200 250 300−1

−0.5

0

0.5

t

x 2

Fig. 2. Waveform plot of the variable x of system (18) for s ¼ 1:514; e ¼ 1:2608; k ¼ffiffi2p

2 .

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x1

x 2

Fig. 3. Phase portraits of system (18) for s ¼ 1:514; e ¼ 1:2608; k ¼ffiffi2p

2 .


−2 −1.5 −1 −0.5 0 0.5 1 1.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

x1

x 2

Fig. 4. Phase portraits of system (18) for s ¼ 1:404; e ¼ 1:413; k ¼ 0:4571.

−1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

x1

x 2

Fig. 5. Phase portraits of system (18) for s ¼ 1:404; e ¼ 1:413; k ¼ 0:5871.


H3 ¼ k1; k2; k3ð Þ : k1 ¼ 0; k3 ¼ 0; k2 < 0f g:

Applying the above results and using the expressions of k1; k2; k3, we obtain the following results.

Theorem 3. If g00 0ð Þ – 0, for sufficiently small l1;l2;l3,

(i) System (20) undergoes a transcritical bifurcation at the origin on the curve T ¼ l1;l2;l3

� �: l2 ¼ �2l3

� �,

(ii) System (20) undergoes a Hopf bifurcation at the origin on the curve H1 ¼nðl1;l2;l3Þ : 9

ffiffi2p

4 l1 þ 123ffiffi2p

80 l2þ63ffiffi2p

40 l3 ¼ �3ffiffi2p

l2þ6ffiffi2p

l3

�3ffiffi2p

l1þ3ffiffi2p

4 l2�9ffiffi2p

2 l3; �3

ffiffiffi2p

l1 þ 3ffiffi2p

4 l2 � 9ffiffi2p

2 l3 < 0o

,

(iii) System (20) undergoes a Hopf bifurcation at the non-trivial equilibrium point on the curve H2 ¼nðl1;l2;l3Þ : 9

ffiffi2p

4 l1�3ffiffi2p

80 l2 � 63ffiffi2p

40 l3 ¼3ffiffi2p

l2þ6ffiffi2p

l3

�3ffiffi2p

l1þ36þ3ffiffi2p

4 l2þ36�9ffiffi2p

2 l3; 36þ3

ffiffi2p

4 l2 þ 36�9ffiffi2p

2 l3 < 3ffiffiffi2p

l1

o,

(iv) System (20) undergoes a Bogdanov–Takens bifurcation at the origin on the curve B ¼ l1;l2;l3

� �: l1 ¼ l2 ¼ �2l3

� �,

(v) System (20) undergoes a zero-Hopf bifurcation at the origin on the curve H3 ¼ l1;l2;l3

� �: l3 ¼ � 1

2 l2 ¼�

� 32 l1; l3 > 0g.

−1 −0.5 0 0.5−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x1

x 2


2 .

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x1

x 2


2 .


5. Numerical simulations

In this section, we give an example to verify the theoretical results. At first, we fix g sð Þ ¼ ks� 110 s2 and g00 0ð Þ ¼ � 1

5 – 0.Theorem 2 is satisfied.

(1) On the curve T, transcritical bifurcation occurs. When l2 þ 2l3 < 0, there exist the trivial equilibrium (sink point) anda non-trivial equilibrium (saddle point). When l2 þ 2l3 > 0, the trivial equilibrium becomes a non-trivial equilibrium,and a non-trivial equilibrium becomes the trivial equilibrium. If we fix s ¼

ffiffiffi2p

; e ¼ 1:404; k ¼ffiffi2p

2 , that is to sayl1 ¼ 0; l2 ¼ �0:01; l3 ¼ 0. Fig. 1 verifies this result.

(2) On the curve H2, Hopf bifurcation occurs. If we set s ¼ 1:514; e ¼ 1:2608; k ¼ffiffi2p

2 , that is to sayl1 ¼ 0:1; l2 ¼ �0:1432; l3 ¼ 0, system (18) has two equilibrium points E1 ¼ ð0;0Þ and E2 ¼ ð�0:7976;0Þ. From The-orem 3, a stable limit cycle exists through the Hopf bifurcation near E2. Figs. 2 and 3 verify this result.

(3) In the following, we numerically find the complicated dynamics near the triple-zero point. Fig. 4 shows a stable limitcycle at the origin if s ¼ 1:404; e ¼ 1:413; k ¼ 0:4571. When s ¼ 1:404; e ¼ 1:413; k ¼ 0:5871, double limit cyclesemerge from the origin in Fig. 5. If s ¼ 1:424; e ¼ 1:294; k ¼

ffiffi2p

2 , a chaotic attractor emerges from the origin in Figs. 6and 7 shows a quasi-periodic attractor when s ¼ 1:424; e ¼ 1:214; k ¼

ffiffi2p

2 .


6. Conclusion

A classical van der Pol’s equation with delayed feedback has been studied in the neighborhood of triple-zero codimen-sion-three bifurcation point where the linear part of the system has triple zero eigenvalues. By applying normal form theoryand center manifold reduction, we are able to predict their corresponding local bifurcation diagrams such as transcriticalbifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. There are still abundant and complexdynamical behaviors to be completely and thoroughly investigated and exploited. In the near future, we will further studyglobal bifurcation, such as homoclinic bifurcation and heteroclinic bifurcation near the triple zero point.

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triple-zero bifurcation in van der pol’s oscillator with delayed feedback

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