Applied Mathematics and Computation 225 (2013) 333–344

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Hopf bifurcation analysis and amplitude controlof the modified Lorenz system

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.09.057

⇑ Corresponding author.E-mail address: wxd959@ujs.edu.cnl (X. Wang).

Xuedi Wang ⇑, Lianwang Deng, Wenli ZhangNonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China

a r t i c l e i n f o

Keywords:Modified Lorenz systemHopf bifurcationBifurcation controlLimit cycleAmplitude control

a b s t r a c t

This paper is concerned with the Hopf bifurcation analysis and amplitude control of themodified Lorenz system. The Hopf bifurcation of system is investigated by utilizing theHopf bifurcation theory and the center manifold theorem firstly. Then the direction andstability of limit cycle emerging from Hopf bifurcation are determined by the designed con-troller. Moreover, the amplitude of limit cycle emerging from Hopf bifurcation is controlledby a nonlinear feedback controller. Finally, numerical simulations are given to verifytheoretical analysis.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Hopf bifurcation analysis of nonlinear dynamical system has been paid close attention widely over the last decade.Several methods for Hopf bifurcation analysis have been presented recently. In [1] the bifurcation control problem wasanalyzed by using state feedback and the stability of the limit cycle emerging from Hopf bifurcation was determined byits characteristic exponents, Kang and others [2,3] investigated the bifurcation problem by using normal forms and invari-ants. The center manifold theorem was used in [4] to study the system dynamics on a two dimensional manifold. Theydecompose the original system into two subsystems: a two-dimensional, non-hyperbolic system and a n-2 dimensional,hyperbolic one, then they expressed the hyperbolic part in the Brunovski form. In general, Hopf bifurcation control is meanto designing a controller what can change the bifurcation behavior of the original system. Controller can be designed to shiftthe location of bifurcation point, to change the direction and stability of Hopf bifurcation, etc. In particular, a typical objectiveis to monitor the amplitude of limit cycle. On the one hand, we can decrease amplitude to restrain the vibration behavior ofsystem; on the other hand, we can obtain a desirable effect via increasing amplitude. Therefore, Hopf bifurcation control andamplitude control with various objectives have been implemented in mechanics, biology, and economy [5–9].

In this paper, we consider a modified Lorenz system as follows:

x�¼ �ay;

y�¼ ax� exz;

z�¼ xy� bz;

8>><>>: ð1Þ

where a, b and e are all positive real parameters. When a = 2, b = 1, e = 1, the Lyapunov exponent is 0.0022, �0.2009, �0.8014respectively, therefore the system (1) is still a chaotic system (Fig. 1). In this paper we will investigate the Hopf bifurcationanalysis and the amplitude control of system (1) by using different controllers.

Fig. 1. The Lyapunov exponent of modified Lorenz system (1): 0:0022;�0:2009;�0:8014.

334 X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344

The rest of this paper is organized as follows. In Section 2, the Hopf bifurcation control of the modified Lorenz system withcontroller is studied. In Section 3, the amplitude of limit cycle of the origin system and controlled system is analyzed. InSection 4, numerical simulations are given. Finally, a conclusion is drawn in Section 5.

2. Hopf bifurcation analysis

2.1. Preliminaries

Four essential theorems are included in this section. The center manifold theorem reduces the dimension of the system totwo dimensions, and the Hopf bifurcation theorem establishes conditions to determine the direction and stability of the limitcycle emerging from Hopf bifurcation. In addition, the Theorems 3 and 4 give results about Hopf bifurcation control indifferent conditions.

Theorem 1 (The local Center Manifold Theorem, Perko [4]). Let f 2 XrðtÞ, where t is an open subset of Rn containing the originand r P 1. Suppose that f ð0Þ ¼ 0 and Df ð0Þ has c eigenvalues with zero real parts and s eigenvalues with negative real parts,where c þ s ¼ n. Then the system n

�¼ f ðnÞ can be written in diagonal form

x�¼ Axþ F1ðx; yÞ; y

�¼ Bxþ F2ðx; yÞ;

where ðx; yÞ 2 Rc � Rs;A 2 Rc�c has c eigenvalues with zero real parts, B 2 Rs�s has s eigenvalues with negative real parts, Fið0Þ ¼ 0and DFið0Þ ¼ 0, for i ¼ 1;2. Furthermore, there exists a c-dimensional invariant center manifold Wc

locð0Þ tangent to the centereigenspace Ec at the origin, where Wc

locð0Þ ¼ ðx; yÞ 2 Rc � Rs y ¼ hðxÞ forj�

x 2 Ndð0Þg, Ndð0Þ is a neighborhood of the origin, withradius d, and h 2 XrðNdð0ÞÞ satisfies

DhðxÞðAxþ F1ðx; hðxÞÞÞ � BhðxÞ � F2ðx;hðxÞÞ ¼ 0:

Finally, the flow on the center manifold is defined by the differential equations

x�¼ Axþ F1ðx; hðxÞÞ; for x 2 Ndð0Þ:

Theorem 2 (Hopf Bifurcation Theorem, Guckenheimer and Holmes [10]). Suppose that the system

x�¼ f ðx;lÞ; x 2 Rn; l 2 R;

has an equilibrium point ðx0;l0Þ such that

(A1) Dxf ðx0;l0Þ has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real parts.(A2) Let k1ðlÞ, k2ðlÞ be the eigenvalues of Dxf ðx0;l0Þ which are imaginary at l ¼ l0, such that Ref@kðlÞ

@l jl¼l0g ¼ d–0.

X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344 335

Then there is a unique two-dimensional center manifold passing through ðx0;l0Þ 2 Rn � R and a smooth system of coordinates forwhich the Taylor expansion of degree three on the center manifold with polar coordinates form: r

�¼ ðdlþ ar2Þr; h

�¼ xþ clþ br2.

If a–0, then there is a surface of periodic solutions in the center manifold which has quadratic tangency with the eigenspace ofkðl0Þ, �kðl0Þ agreeing to second order with the paraboloid. l ¼ �ar2=d. If a < 0; then these periodic solutions are stable, whileif a > 0; they are repelling limit cycles.There exists an expression for bidimensional systems to find the first Lyapunov coefficient a [11]. Consider the system

x�¼ Jxþ FðxÞ; ð2Þ

where

J ¼0 �xx 0

� �; FðxÞ ¼

F1ðxÞF2ðxÞ

� �; Fð0Þ ¼ 0 and DFð0Þ ¼ 0:

Then a ¼ 116x ðR1 þxR2Þ, where

R1 ¼ F1x1x2 ðF1x1x1 þ F1x2x2 Þ � F2x1x2 ðF2x1x1 þ F2x2x2 Þ � F1x1x1 F2x1x1 þ F1x2x2 F2x2x2 ; R2 ¼ F1x1x1x1 þ F1x1x2x2 þ F2x1x1x2 þ F2x2x2x2 :

There exists another way to express R2. If FðxÞ ¼ 12 Aðx; xÞ þ 1

6 Bðx; x; xÞ þ � � �, where

A ¼Q 1

Q 2

� �; B ¼

C11 C21

C21 C22

� �With Q i;Cij 2 R2�2; then R2 ¼ trðBÞ ¼ trðC11 þ C22Þ;

with trð�Þ ¼ traceð�Þ.

Theorem 3 [12]. Consider system (2) and suppose that DFð0Þ ¼ JH 00 JS

� �, with JH ¼

0 �x0

x0 0

� �2�2

and JS 2 Rðn�2Þ�ðn�2Þ

is Hurwitz. If k1 and k2 given by [12] are different from zero respectively and Gð0Þ ¼ b1

b2

� �with b1 ¼ 0, then there exists b1

and b2 such that system (2) with the control law u ¼ b1tþ b2ðz21 þ z2

2Þ undergoes a Hopf bifurcation. Moreover, the signs of dand a given by [12], the stability and direction of the limit cycle emerging from Hopf bifurcation are determined by selecting b1

and b2.

Theorem 4 [12]. Consider system (2) and suppose that DFð0Þ ¼ JH 00 JS

� �, with JH ¼

0 �x0

x0 0

� �2�2

and JS 2 Rðn�2Þ�ðn�2Þ

is Hurwitz. If k1 and k2 given by [12] are different from zero respectively and Gð0Þ ¼ b1

b2

� �with b1–0, then there exist

L ¼ ða1;a2Þ and b ¼ ðb1; b2Þ such that system (2) with the control law u ¼ tða1z1 þ a2z2Þ þ b3z31 þ b4z3

2 undergoes a Hopf

bifurcation. Moreover, the signs of d and a given by [12], the stability and direction of the limit cycle emerging from Hopf

bifurcation are determined by selecting L and b.

2.2. Statement of the controlled system

In order to control the Hopf control bifurcation behaviors of system (1), a controller is determined as ðb11 þ xÞu, whereb11–0. The controlled system is written as

x�¼ �ayþ ðb11 þ xÞu;

y�¼ ax� exz;

z�¼ xy� bz;

8>><>>: ð3Þ

For convenience of computation, we rewrite this system in another form:

n�¼ FðnÞ þ GðnÞu; ð4Þ

where n 2 R3 is the state variable and u 2 R is the control input, the vector fields FðnÞ;GðnÞ are sufficiently smooth.Furthermore,

n�¼

x�

y�

z�

0BB@

1CCA; FðnÞ ¼

F1ðnÞF2ðnÞ

� �; F1ðnÞ ¼

�ayax� exz

� �; F2ðnÞ ¼ ðxy� bzÞ; Fð0Þ ¼ 0;

336 X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344

and

GðnÞ ¼G1ðnÞG2ðnÞ

� �; G1ðnÞ ¼

b11 þ x

0

� �; G2ðnÞ ¼ ð0Þ; Gð0Þ ¼

b11

00

0B@

1CA:

Now we assume that n ¼ ðgx Þ;g 2 R2;x 2 R1, then we can obtain that

DFð0Þ ¼JH 00 JS

� �; JH ¼

0 �a

a 0

� �; JS ¼ �b; Gð0Þ ¼

b11

00

0B@

1CA ¼ b1

b2

� �; b1 2 R2; b2 2 R1;

DGð0Þ ¼ M ¼M1 M2

M3 M4

� �; M1 ¼

1 00 0

� �; M2 ¼

00

� �; M3 ¼ ð0 0Þ; M4 ¼ ð0Þ:

The expansion of system (4) around n ¼ 0 yields

g�¼ JHgþ F21ðg;xÞ þ F31ðg;xÞ þ � � � þ ðb1 þM1gþ � � �Þu;

x�¼ JSxþ F22ðg;xÞ þ F32ðg;xÞ þ � � � þ ðb2 þ � � �Þu;

ð5Þ

where

F2jðg;xÞ ¼12½Fjggð0;0Þðg;gÞ þ 2Fjgxð0;0Þðg;xÞ þ Fjxxð0;0Þðx;xÞ�;

F3jðg;xÞ ¼16

Fjgggð0; 0Þðg;g;gÞ þ � � �

Next, the objective is design a control law u ¼ uðg;lÞ for the original system (1) occurring a Hopf bifurcation near n ¼ 0 andl ¼ 0, where l is a bifurcation parameter. Thus, we are capable of determining the stability and direction of the limit cycleemerging from Hopf bifurcation.

In this paper, we consider the control law

uðg;lÞ ¼ lða1g1 þ a2g2Þ þ b1g31 þ b2g3

2; ð6Þ

for ai; bi 2 R; i ¼ 1;2; b1–0.

2.3. Center manifold

In this section we will calculate the center manifold at n ¼ 0 and the dynamics on it. This allows us to simplify the analysisby reducing the dimension of controlled system (3) from three to two.

The control law u ¼ uðg;lÞ in (6) is equivalent to

uðg;lÞ ¼ lLgþ 16

C0ðg;g;gÞ; ð7Þ

where L ¼ ða1 a2Þ;C0 ¼ ðC01 C02 Þ, C01 ¼ ð6b1 00 0 Þ and C02 ¼ ð

0 00 6b2

Þ.

Substituting (7) into in (5), we obtain the closed-loop system

g�¼ JHgþ f3ðg;x;lÞ;

x�¼ JSxþ f4ðg;x;lÞ;

ð8Þ

where

f3ðg;x;lÞ ¼ lb1Lgþ F21ðg;xÞ þ F31ðg;xÞ þ16

C0ðg;g;gÞb1 þ lLgðM1gþM2gÞ

¼ lb1Lgþ0�eg1g2

� �þ 1

6C0ðg;g;gÞb1 þ lLg

g1

g2

� �;

ð9Þ

and

f4ðg;x;lÞ ¼ lb2Lgþ F22ðg;xÞ þ F32ðg;xÞ þ16

C0ðg;g;gÞb2 þ lLgðM3gþM4gÞ

¼ lb2Lgþ 16

C0ðg;g;gÞb2 þ12

0 11 0

� �ðg;gÞ:

ð10Þ

X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344 337

So

g�¼

0 �a

a 0

� �gþ lb1Lgþ

0�eg1g2

� �þ 1

6C0ðg;g;gÞb1 þ lLg

g1

g2

� �; ð11Þ

x�¼ �bxþ lb2Lgþ 1

6C0ðg;g;gÞb2 þ

12

0 11 0

� �ðg;gÞ: ð12Þ

Next work focus on finding a1;a2; b1 and b2. The approach we use here is the center manifold theory.

2.3.1. Quadratic termsWe can rewrite (8) as an extended system

g�

l�

x�

0BB@

1CCA ¼

JH 0 00 0 00 0 JS

0B@

1CA

glx

0B@

1CAþ

f3ðg;x;lÞ0f4ðg;x;lÞ

0B@

1CA: ð13Þ

The system (13) has a two-dimensional center manifold through the origin

x ¼ hðg;lÞ ¼ 12gT Hg; ð14Þ

and hð0;0Þ ¼ 0, Dhð0; 0Þ ¼ 0. Substituting (14) into (13) and using the chain rule, we obtain

@hðg;lÞ@g

½JHgþ f3ðg;l; hðg;lÞÞ� � JShðg;lÞ � f4ðg;l;hðg;lÞÞ � 0: ð15Þ

Before calculate H in (14), we are going to calculate the quadratics terms of f4ðg;l;hðg;lÞÞ.

From (12), f4ðg;l;hðg;lÞÞ ¼ 12

0 11 0

� �ðg;gÞ ¼ 1

2 gT Ng, where N ¼ 0 11 0

� �.

Then

gT H� �

JHg� JS12gT Hg

� �� 1

2gT Ng � 0:

Assume that H ¼ ð h1 h2

h3 h4Þ, therefore,� �� � � � � �

gT h1 h2

h3 h4

0 �a

a 0gþ b � 1

2gT h1 h2

h3 h4g ¼ 1

2gT 0 1

1 0g;

h1 h2

h3 h4

� ��

0 �a

a 0

� �þ 1

2b

h1 h2

h3 h4

� �¼

0 12

12 0

!;

ah2 �ah1

ah4 �ah3

� �þ

12 bh1

12 bh2

12 bh3

12 bh4

!¼

0 12

12 0

!: ð16Þ

By direct calculation in (16), we have

H ¼h1 h2

h3 h4

� �¼� 2a

b2þ4a2b

b2þ4a2

bb2þ4a2

2ab2þ4a2

!:

2.3.2. Dynamics on the center manifoldThe dynamics on the center manifold is given by g

�¼ JHgþ f3ðg;l;hðg;lÞÞ. From (9), we know

f3ðg;x;lÞ ¼ lb1Lgþ F21ðg;xÞ þ F31ðg;xÞ þ16

C0ðg;g;gÞb1 þ lLgðM1gþM2gÞ

¼ lb1Lgþ0eg1g2

� �þ 1

6C0ðg;g;gÞb1 þ lLg

g1

g2

� �¼ l/gþ 1

2wðg;gÞ þ 1

6uðg;g;gÞ;

where

/ ¼ b1L;wðg;gÞ ¼ gT F1ggð0; 0Þg ¼ 0; ð17Þ

16uðg;g;gÞ ¼ 1

6C0ðg;g;gÞb1 þ

12gT F1gxð0;0ÞgT Hg: ð18Þ

338 X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344

Therefore, the dynamics on the center manifold can be expressed as

g�¼ ðJH þ l/Þgþ 1

6uðg;g;gÞ: ð19Þ

2.4. Control law

We can get constants ai and bi ði ¼ 1;2Þ by using the Theorem 2. Then the Hopf bifurcation behavior for system (3) nearn ¼ 0 can be controlled.

2.4.1. Cubic termsNote that a1 and a2 are contained in the linear part of (19), then the stability of the controlled system (3) at n ¼ 0 can be

determined.From (17) and (18), note that b1 and b2 are contained in the cubic part u, and then we can decompose it in the same way

to the linear part. Observe that

C0ðg;g;gÞb1 ¼ ðC01 C02 Þðg;g;gÞb1 ¼ ððgT C01g;gT C01gÞgÞb1 ¼ u1ðg;g;gÞ;

where

u1 ¼b11C01 b11C02

b21C01 b21C02

� �:

Then

16uðg;g;gÞ ¼ 1

6u0ðg;g;gÞ þ

16u1ðg;g;gÞ;

and

16u0ðg;g;gÞ ¼

12gT F1gxð0;0ÞgT Hg:

So the dynamics on the center manifold is given by

x�¼ Jlxþ 1

6uðx; x; xÞ; ð20Þ

where

Jl ¼ JH þ l/;/ ¼ b1L;uðx; x; xÞ ¼ u0ðx; x; xÞ þu1ðx; x; xÞ: ð21Þ

Now, we will prove the system (20) undergoes a Hopf bifurcation at x ¼ 0;l ¼ 0 by using Theorem 2. To this end, weshould calculate the eigenvalues of Jl and prove that they cross the imaginary axes when l ¼ 0.

From (21),

Jl ¼ JH þ l/ ¼0 �a

a 0

� �þ l

b11

0

� �a1 a2ð Þ ¼

lb11a1 �aþ lb11a2

a 0

� �;

and the corresponding characteristic equation is

jkI � Jlj ¼k� lb11a1 a� lb11a2

�a k

�������� ¼ 0;

that is; k2 � klb11a1 þ a2 � alb11a2 ¼ 0: ð22Þ

When l ¼ 0, we can obtain k1;2 ¼ �ia.Differentiating both sides of Eq. (22) with respect to l, we can obtain

@kðlÞ@l ¼ kb11a1 þ ab11a2

2k� lb11a1:

It is easy to obtain that Ref@k1ðlÞ@l jl¼0g ¼ d–0 ða1–0Þ.

Next we are going to prove that the first Lyapunov coefficient is different to zero. By Theorem 2 we have

l ¼ 116aðR1 þ aR2Þ;

where

R1 ¼ F1x1x2 ðF1g1g1þ F1g2g2

Þ � F2g1g2ðF2g1g1

þ F2g2g2Þ � F1g1g1

F2g1g1þ F1g2g2

F2g2g2; R2 ¼ trðBÞ ¼ trðC11 þ C22Þ:

X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344 339

Then

l ¼ 116aðR1 þ aR2Þ ¼

116aðR1 þ 3 � b � b1Þ ¼

116a

R1 þ38

bb1 ¼ k0 þ38

bb1;

where k0 is a constant.Now it is possible to show that the parameters l and d are given by

l ¼ k0 þ38

bb1;d ¼12

Lb1: ð23Þ

So, form Theorem 4 we can obtain that the system (3) with the control law uðg;lÞ ¼ lða1g1 þ a2g2Þ þ b1g31 þ b2g3

2 under-goes a Hopf bifurcation, and the stability and direction of the limit cycle emerging from Hopf bifurcation are determined.

3. Amplitude control

3.1. Amplitude of limit cycle in the original system

We regard b as the bifurcation parameter for the original system (1). By simple analysis, we know that the equilibriumpoint Oð0; 0;0Þ is a Hopf bifurcation point for any positive real number. Let a ¼ 2; e ¼ 1 and b ¼ b0 ¼ 1. By direct calculation,the eigenvalues of this system at Oð0;0;0Þ are k1;2 ¼ �ia, k3 ¼ �b, and the corresponding eigenvectors are

m1 ¼i

10

0B@

1CA; m2 ¼

�i

10

0B@

1CA; m3 ¼

001

0B@

1CA:

Let a0ð0Þ denote the real part of the derivative of k1 with respect to b at b ¼ b0, we have

a0ð0Þ ¼ Re@k1ðbÞ@b

����b¼b0

( )¼ � 3

10:

Then the system (1) is transformed into a canonical form:

x�¼ �ayþ Q1;

y�¼ axþ Q 2;

z�¼ �bzþ Q 3;

8>><>>: ð24Þ

where Q 1 ¼ 0; Q2 ¼ �exz ; Q 3 ¼ xy.Next we need to calculate the following characteristic quantities similar to that of [13]

g20 ¼ 14

@2Q1@x2 � @2Q1

@y2 þ 2 @2Q2@x@y þ i @2Q2

@x2 � @2Q2@y2 � 2 @2Q1

@x@y

� � ;

g11 ¼ 14

@2Q1@x2 þ @2Q1

@y2 þ i @2Q2@x2 þ @2Q2

@y2

� � ;

G110 ¼ 12

@2Q1@x@z þ

@2Q2@y@z þ i @2Q2

@x@z �@2Q1@y@z

� � ;

G101 ¼ 12

@2Q1@x@z �

@2Q2@y@z þ i @2Q2

@x@z þ@2Q1@y@z

� � ;

ð25Þ

w11 ¼ � 1�4b

@2Q3@x2 þ @2Q3

@y2

� ;

w20 ¼ 14ð2aiþbÞ

@2Q3@x2 � @2Q3

@y2 � 2i @2Q3@x@y

� ;

G21 ¼ 18

@3Q1@x3 þ @3Q1

@x@y2 þ @3Q2@x2@yþ

@3Q2@x3 þ i @3Q2

@x3 þ @3Q2@x@y2 � @3Q1

@x2@y�@3Q1@y3

� � :

By calculation we obtain that

g20 ¼ 0; g11 ¼ 0; G110 ¼12; i:e:; G101 ¼

12; i:e:; w11 ¼ 0; w20 ¼ �

i2ð2iaþ bÞ ; G21 ¼ 0: ð26Þ

Consequently, the curvature coefficient of limit cycle in system(1) is expressed by

r ¼ Reg20g11

2aiþ G110w11 þ

G21 þ G101w20

2

�¼ eb

8ð4a2 þ b2Þ¼ 1

136: ð27Þ

Moreover, the amplitude of limit cycle in system (1) for b > b0 and jb� b0j << 1 is

r <<

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�a0ð0Þ

rjðb� b0Þj

r¼ 6:388

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijb� b0j

q:

340 X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344

3.2. Amplitude control of limit cycle in the controlled system

A nonlinear feedback controller is determined as u1 ¼ �6x3 is added to the first equation of system (24). Thus, system (24)is turned into a controlled system:

x�

y�

z�

0BB@

1CCA ¼

0 �a 0a 0 00 0 �b

0B@

1CA

x

yz

0B@

1CAþ

0�exzxy

0B@

1CAþ

�6x3

00

0B@

1CA ¼

�ayþ Q1 þ u1

axþ Q2

�bzþ Q 3

0B@

1CA: ð28Þ

According to the Ref. [13], we know C1111 ¼ �6.

Next we calculate the curvature coefficient r1 of limit cycle in system (28) by using the same computation process of(25)-(27), we get r1 ¼ 1

136� 94 ¼ �2:243. The amplitude of limit cycle in system (28) for b < b0; and jb� b0j << 1 is

r1 <<ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� a0 ð0Þ

r1ðb� b0Þ

q¼ 0:1338

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijb� b0j

p.

Meanwhile, according to the result of characteristic quantities above, the criticality of the Hopf bifurcation, the directionand stability of the limit cycle emerging from Hopf bifurcation can be judged. The detailed judgment conditions are shown inthe chart below.

Chart 1.

a0ð0Þ

Stabilitycoefficients r

Direction of thebifurcation

360

0.15

0.10

0.05

0.05

0.10

0.15x

Fig. 2

Stability of the emerging periodicsolutions

370 380 390 400t

a. The periodic time series of xðtÞ.

Criticality of the Hopfbifurcation

b2 ¼ ra0ð0Þ

b� b0

+

� + + Stable Supercritical � � � � Stable Supercritical + + � � Unstable Subcritical � + + + Unstable Subcritical

Note: ‘‘+’’ means positive number, while ‘‘�’’means negative number.

4. Numerical simulations

In this section we will give some numerical simulations to verify the theoretical analysis above. All considerations hereare with respect to the set of parameters: a ¼ 2; e ¼ 1; b ¼ 1.

4.1. The Hopf bifurcation of the original system

The original system (1) undergoes the Hopf bifurcation at Oð0; 0;0Þ, and the amplitude of limit cycle in system (1) isr 6:388. Under the initial values with x½0� ¼ �0:1; y½0� ¼ �0:1; z½0� ¼ �0:1, the phase trajectory and time displacementcurve of system ð1Þ are drawn as shown in Figs. 2a–2d.

360 370 380 390 400t

0.15

0.10

0.05

0.05

0.10

0.15y

Fig. 2b. The periodic time series of yðtÞ.

370 380 390 400t

0.003

0.002

0.001

0.001

0.002

0.003z

Fig. 2c. The periodic time series of zðtÞ.

Fig. 2d. Stable period orbit.

X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344 341

4.2. The Hopf bifurcation control

The controlled system (3) can be written as

x�¼ �ayþ ðb11 þ xÞðla1xþ b1x3Þ

y�¼ ax� exz

z�¼ xy� bz:

8>><>>: ð29Þ

Let a1 ¼ 2; b1 ¼ �6;l ¼ 0:1. By detailed calculation, we can obtain that a0ð0Þ < 0 and r < 0.Therefore, from Chart 1 we know that the Hopf bifurcation at Oð0;0;0Þ is supercritical, the limit cycle exists for l < 0:1

and the limit cycle is stable. Under the initial values with x½0� ¼ �0:1; y½0� ¼ �0:1; z½0� ¼ �0:1, the phase trajectory and timedisplacement curve of system (29) are drawn as shown in Figs. 3a–3e.

Fig. 3a. The periodic time series of xðtÞ.

Fig. 3b. The periodic time series of yðtÞ.

Fig. 3c. The periodic time series of zðtÞ.

Fig. 3d. Supercritical Hopf bifurcation.

342 X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344

Fig. 3e. Stable period orbit.

360 370 380 390 400t

0.02

0.01

0.01

0.02

x

Fig. 4a. The periodic time series of xðtÞ.

360 370 380 390 400t

0.02

0.01

0.01

0.02

y

Fig. 4b. The periodic time series of yðtÞ.

Fig. 4c. The periodic time series of zðtÞ.

X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344 343

Fig. 4d. Stable period orbit.

Fig. 4e. The limit cycles of Hopf bifurcation in original system (1) and controlled system (27).

344 X. Wang et al. / Applied Mathematics and Computation 225 (2013) 333–344

4.3. Amplitude control

From Section 3.2, it is easy to obtain that amplitude of limit cycle in system (28) is r 0:1338. Under the initial valueswith x½0� ¼ �0:1; y½0� ¼ �0:1; z½0� ¼ �0:1, the phase trajectory and time displacement curve of system (28) are drawn asshown in Figs. 4a–4d. Compared with the amplitude of limit cycle in the system (1), it is obvious to know that r1 < r. Thatis, the amplitude of the limit cycle in the controlled system (28) is smaller than that of the original system (1). In addition,Fig. 4e shows that the amplitude of limit cycles of the original system (1) and the controlled system (28). The bigger one islimit cycle of the original system (1), and the other is the limit cycle of the controlled system. Obviously, Numerical resultsare consistent with the above-mentioned theoretical analysis.

5. Conclusions

This paper aims at controlling the Hopf bifurcation behavior and the amplitude of limit cycle emerging from Hopf bifur-cation of the modified Lorenz system. Based on the Hopf bifurcation theory and the center manifold theorem, the Hopf bifur-cation of system (1) with a controller is investigated. Moreover, a nonlinear feedback controller is designed to control theamplitude of limit cycle. By comparing the amplitude of limit cycle of the origin system (1) and controlled system (28),we know that this nonlinear feedback controller is capable of narrowing the amplitude of limit cycle.

References

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