Bifurcation analysis of the generalized stretch-twist-fold flow

Applied Mathematics and Computation 229 (2014) 16–26

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Bifurcation analysis of the generalized stretch-twist-fold flow Jianghong Bao ⇑, Qigui Yang Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, PR China

a r t i c l e

i n f o

Keywords: Stretch-twist-fold flow Chaos Hopf bifurcation Heteroclinic bifurcation Lyapunov exponent

a b s t r a c t Based on the stretch-twist-fold flow, a generalized stretch-twist-fold flow is introduced. By choosing an appropriate bifurcation parameter, Hopf bifurcations occur in this system when the bifurcation parameter exceeds a critical value. The formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are presented. In addition, the paper also investigates the bifurcations of the heteroclinic orbits for this system. The existence and its associated existing regions are given for two heteroclinic orbits, respectively. Finally, some numerical simulations for justifying the theoretical analysis are presented. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Since the discovery of the Lorenz chaotic system, chaos has been developed and intensively studied in the past four decades. Recently this study about chaos has concentrated on not only proposing new and interesting chaotic systems, but also enhancing complex dynamics and topological structure based on the existing chaotic systems [1–6]. The stretch-twist-fold (STF) was devised to represent the stretch-twist-fold mechanism of the magnetic field generation that is believed to be most conducive to the fast dynamo action in magnetohydrodynamics [7,8]. Although there is quite extensive literature about the STF flow [9–13], up to now nothing has been known about the bifurcations of the STF flow. Also, Hopf bifurcations do not exist in the system. To further investigate the bifurcations of the system, we introduce a new system containing the STF flow, which not only preserves the original dynamic properties, but also possesses some new characteristics so as to benefit systematic studies and help reveal the most essential dynamical behavior of the classic STF flow. In addition, since the stretch-twist-fold action represents the optimal reinforcement of stretched field available in a three-dimensional domain [7], the results from the new system are possibly helpful for finally exposure of a magnetic field structure in a plasma contained in a domain bounded by a perfect conductor. In this paper, by introducing two parameters to the original STF flow, we propose a generalized stretch-twist-fold flow. It is interesting to find that the control parameters to the original STF flow, may be able to change its some dynamic properties. Hopf bifurcations do not exist in the original STF flow, but they exist in the generalized STF flow. The original STF flow is conservative whereas the generalized STF flow is no longer conservative. The original STF flow is a class of three-dimensional incompressible steady flows in a sphere, but the generalized STF flow may be not in a sphere when the parameters vary. We also investigate the heteroclinic bifurcations of the new system, which include the heteroclinic bifurcations of the original system as a special case. By choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcations occur in the generalized STF flow when the bifurcation parameter exceeds a critical value and presents the formulae for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions by applying the normal form theory [14,15]. We also study ⇑ Corresponding author. E-mail address: [email protected] (J. Bao). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.037

17

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

the bifurcations of the heteroclinic orbits by the generalized Melnikov method [16]. The existence and its associated existing regions are given for two heteroclinic orbits, respectively. Finally, some numerical simulations are performed to justify the theoretical analysis. The rest of this paper is organized as follows. In Section 2, we present the generalized stretch-twist-fold flow and report the existence of chaos with different initial conditions. In Section 3, by using the normal form theory, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are analyzed in detail. Section 4 explores the bifurcations of the heteroclinic orbits by the generalized Melnikov method. In Section 5, some numerical simulations are presented to illustrate the theoretical analysis. And Section 6 concludes the paper. 2. The generalized stretch-twist-fold flow By introducing two parameters to the original STF flow [9], the following generalized stretch-twist-fold flow is obtained:

8 dx > < dt ¼ az  8ðb þ 1Þxy; dy ¼ 11x2 þ 3y2 þ z2 þ bxz  3c2 ; dt > : dz ¼ ax þ 2yz  bxy; dt

ð2:1Þ

where b; c; a and b are real parameters, determining the chaotic behaviors and bifurcations of the system. In the original STF flow, a and b are positive real parameters and related to the ratios of intensities of the stretch, twist and fold ingredients of the flow. For a  0 and b P 0, system (2.1) always has two isolated equilibria Pþ ½0; c; 0 and P  ½0;  c; 0. When a > 0, there are other equilibria. This system is invariant under the transformation

ðx; y; zÞ ! ðx; y; zÞ; namely, the system has rotation symmetry around the y-axis. The divergence of system (2.1) is r  V ¼ 8by, so the system is no longer conservative. When the parameters b ¼ 1:5; b ¼ 0:001; c ¼ 1:7 and a ¼ 0:1, the three Lyapunov exponents with initial values (0.2, 0.5, 0.5) of system (2.1) are L1 ¼ 0:2246; L2 ¼ 0:0025; L3 ¼ 0:2220, and the corresponding Lyapunov exponents spectrum for a 2 ½5:3; 7 are shown in Fig. 1. When the parameters b ¼ 1; b ¼ 0:1; c ¼ 1 and a ¼ 10000, the Poincaré map with initial values (0.33, 0.5, 0.3) also shows that this system is chaotic, as shown in Fig. 2. 3. Hopf bifurcations in system (2.1) In this section, we employ the normal form theory [14,15] to study the direction, stability and period of bifurcating periodic solutions for system (2.1). System (2.1) always has two isolated equilibria: P ½0;  c; 0 and Pþ ½0; c; 0 for any parameters a; b and b. The eigenvalues of the equilibrium P  ½0;  c; 0 are

0.25 0.2

Lyapunov exponent

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25

5.4

5.6

5.8

6

6.2

α

6.4

6.6

6.8

7

Fig. 1. Lyapunov exponent spectrum of system (2.1) for a 2 ½5:3; 7 with parameters values (b, b, c) = (1.5, 0.001, 1.7).

18

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

1 0.8 0.6 0.4

z

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.5

0

0.5

1

x Fig. 2. Poincaré map of system (2.1) on the x-z plane with the parameters (a; b, b, c) = (10000, 1, 0.1, 1) and initial values (0.33, 0.5, 0.3).

k1 ¼ 6c; k2;3 ¼ ð4b þ 3Þc 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2 ð4b þ 5Þ þ aðcb  aÞ:

The eigenvalues of the equilibrium P þ ½0; c; 0 are

k1 ¼ 6c; k2;3 ¼ ð4b þ 3Þc 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2 ð4b þ 5Þ  aðcb þ aÞ:

We first consider the Hopf bifurcation of system (2.1) at P þ ½0; c; 0. When ðb; cÞ 2

 pffiffiffiffiffiffiffiffiffiffi   b b2 þ16 ðb; cÞ a < c < 0; b ¼  34 , the Jaco8

bian matrix at the equilibrium Pþ ½0; c; 0 has a negative real eigenvalue 6c and a pair of conjugate purely imaginary eigenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values  cab þ a2  4c2 i. Under this condition, the transversality condition

  3 ¼ 4c > 0: k0 b ¼  4 is also satisfied. Accordingly, Hopf bifurcation at P þ occurs. The above analysis is summarized as follows: pffiffiffiffiffiffiffiffiffiffi ffi 2 Theorem 3.1 (Existence of Hopf Bifurcation). Let b 8b þ16 a < c < 0. Then, as b passes through the critical value b0 ¼  34, system (2.1) undergoes a Hopf bifurcation at the equilibrium Pþ ½0; c; 0. Define the following subsets of the Hopf surface U.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8  9  < = pffiffiffi b  b2 þ 16  a b U 1 ¼ ða; b; cÞa > 0; 0 < b < 2; a 0; 0 < b 6 2;  4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8   = < pffiffiffi b  b2 þ 16   U 3 ¼ ða; b; cÞa > 0; 2 6 b; a
b

pffiffiffiffiffiffiffiffiffi ffi 2

b þ16 8

a < c < 0 and b0 ¼  34. For system (2.1),

(i) if ða; b; cÞ 2 U 1 , bifurcating periodic solutions exist for sufficient small b  b0 > 0. Moreover, period solutions of system (2.1) from Hopf bifurcation at Pþ are non-degenerate, supercritical and orbitally stable.

19

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

(ii) if ða; b; cÞ 2 U 2 [ U 3 , bifurcating periodic solutions exist for sufficient small b  b0 < 0. Moreover, period solutions of system (2.1) from Hopf bifurcation at Pþ are non-degenerate, subcritical and orbitally unstable. (iii) the period and characteristic exponent of the bifurcating periodic solution are:

T¼

  2p  1 þ s2 e2 þ o e4 ; b ¼ b2 e2 þ o e4 : x0

where

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x0 ¼ cab þ a2  4c2 ;

s2 ¼

50a2 bc2 þ 8ba4 þ 9b2 ca3  11ab2 c3 þ b3 c2 a2  400c3 a  104ca3  80bc4 ; 16cð4c2 þ a2 þ cabÞða þ cbÞð5c2 þ a2 þ cabÞ

3aðba þ 4cÞ ; 4ða þ cbÞð5c2 þ a2 þ cbaÞ h i b  b0 2 ¼ þ o ðb  b0 Þ ;

b2 ¼

e2

l2

l2 ¼

3 aðba þ 4cÞ : 32 ða þ cbÞð5c2 þ a2 þ cbaÞc

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 þ16 Proof. Let a < c < 0; x0 ¼ cab þ a2  4c2 ; b ¼ b0 ¼  34 and A ¼ JðPþ Þ, the Jacobian matrix at the equilibrium 8 P þ ½0; c; 0. Then, one has

0 B

þ2ci  xa0þcb

v1 ¼ B @

0

1 C C and A

v2

0 1 0 B C B ¼ @1C A; 0

i which satisfy

Av 1 ¼ ix0 v 1 ;

Av 2 ¼ 6cv 2 :

From system (2.1), define

0 B P ¼ ðRev 1 ; Imv 1 ; v 3 Þ ¼ B @

1

0  axþcb

aþcb

0

0

C 1C A;

0

1

0

2c

0

ð3:1Þ

and

0 1 0 1 0 1 x 0 x1 B C B C B C @ y A ¼ @ c A þ P@ y1 A: z 0 z1 Thus,

8 x_ 1 ¼ x0 y1 þ F 1 ðx1 ; y1 ; z1 Þ; > > < y_ 1 ¼ x0 x1 þ F 2 ðx1 ; y1 ; z1 Þ; > > : z_ 1 ¼ 6cz1 þ F 3 ðx1 ; y1 ; z1 Þ; where

F1 ¼

 2z1 4cay1  x0 ax1 þ 2c2 by1 ; x0 ða þ cbÞ

F2 ¼

z1 ðbx0 x1 þ 2ay1 Þ ; a þ cb

F3 ¼

1 ða þ cbÞ

2

 2 2 2 3c b z1 þ 3y21 c2 b2  44cx0 x1 y1 þ 4y21 cba þ 6z21 bca þ y21 a2  44x21 c2 þ 11x21 a2 þ 11x21 bac  bax0 x1 y1

cx0 b2 x1 y1 þ 44c2 y21 þ 3a2 z21 :

20

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

Furthermore,

!# @2F2 @2F2 ¼ 0; þ @x21 @y21 " ! !# 1 @2F 1 @2F 1 @2F 2 @2F2 @2F2 @2F1 ¼  2  þ2 þi ¼ 0; 4 @x1 @y1 @x1 @y1 @x21 @y21 @x21 @y21 " ! !# 1 @2F 1 @2F 1 @2F 2 @2F2 @2F2 @2F1 ¼  þ2  2 þi ¼ 0; 2 2 2 2 4 @x1 @y1 @x1 @y1 @x1 @y1 @x1 @y1 " ! !# 1 @3F 1 @3F1 @3F2 @3F 2 @3F2 @3F2 @3F1 @3F1 þi ¼ 0: ¼ þ þ þ þ   8 @x31 @x1 @y21 @x21 @y1 @y31 @x31 @x1 @y21 @x21 @y1 @y31

g 11 ¼ g 02 g 20 G21

1 4

"

@2F 1 @2F 1 þ @x21 @y21

!

þi

Meanwhile, one has

1 @2F 3 @2F 3 ¼ þ 4 @x21 @y21

!

3ð4a þ cbÞ ; 2ða þ cbÞ !  88c2 þ 10a2 þ 7cab  3c2 b2 þ ix0 b2 c þ 44c þ ab 1 @2F 3 @2F 3 @2F3 ¼ ¼   2i ; 4 @x21 @x1 @y1 @y21 2ða þ cbÞ2

h11 h20

¼

By solving the following equations

k3 w11 ¼ h11 ; ðk3  2ix0 IÞw20 ¼ h20 ; one obtains

4a þ cb w11 ¼  ; 4cða þ cbÞ   88c2  10a2  7cba þ 3c2 b2 i þ x0 b2 c þ 44c þ ba ; w20 ¼ 4ð3ic þ x0 Þða þ cbÞ2 Furthermore,

G110 ¼

"

1 2

"

@2F1 @2F2 þ @x1 @z1 @y1 @z1

! þi

@2F1 @2F2  @x1 @z1 @y1 @z1

!

@2F2 @2F1  @x1 @z1 @y1 @z1

!# ¼

iðab  8cÞ ; 2x0

!#  4x0 a þ ia 8c þ ba þ b2 c @2F2 @2F1 ¼ þ ; @x1 @z1 @y1 @z1 2x0 ða þ cbÞ   c 30a2 b  288ca þ 3cab2  48c2 b  ix0 8a2 b  104ca þ ab2 c  16bc2 ¼G21 þ ð2G110 w11 þ G101 w20 Þ ¼ : 8cx0 ð3ic þ x0 Þða þ cbÞ

G101 ¼ g 21

1 2

þi

Based the above calculation and analysis, one can compute the following quantities:

    3c 10ba2  96ac þ cb2 a  16c2 b  ix0 8a2 b  104ca þ ab2 c  16bc2 1 1 g 20 g 11  2jg 11 j2  jg 02 j2 þ g 21 ¼ ; 2x0 3 2 16cx0 ð3ci þ x0 Þða þ cbÞ ReC ð0Þ 3 aðba þ 4cÞ l2 ¼  0 1 ¼ ; a ð0Þ 32 ða þ cbÞð5c2 þ a2 þ cbaÞc

C 1 ð0Þ ¼

i

50a2 bc2 þ 8ba4 þ 9b2 ca3  11ab2 c3 þ b3 c2 a2  400c3 a  104ca3  80bc4 ; x0 16cð4c2 þ a2 þ cabÞða þ cbÞð5c2 þ a2 þ cabÞ 3aðba þ 4cÞ : b2 ¼2ReC 1 ð0Þ ¼ 4ða þ cbÞð5c2 þ a2 þ cbaÞ

s2 ¼ 

ImC 1 ð0Þ þ l2 x0 ð0Þ

¼

where

a0 ð0Þ ¼ 4c; x0 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cab þ a2  4c2 ;

x0 ð0Þ ¼ 

8c2

x0

:

When ða; b; cÞ 2 U 1 ; l2 > 0 and b2 < 0. When ða; b; cÞ 2 U 2 [ U 3 ; Furthermore, the period and characteristic exponent are:

T¼

 2p  1 þ s2 e2 þ o e4 ;

x0

 b ¼ b2 e2 þ o e4 :

l2 < 0 and b2 > 0.

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

where e2 angle):

21

h

i 2 0 ¼ bb l þ o ðb  b0 Þ . And the expression of the bifurcating periodic solution is (except for an arbitrary phase 2

ðx; y; zÞT ¼ Pðx1 ; y1 ; z1 ÞT ; where the matrix P is defined as in (3.1),

x1 ¼ Re u;

y1 ¼ Im u;

  z1 ¼ w11 juj2 þ Re w20 u2 þ o juj3 ;

and

  ie2

4pti 4pti g 02 e T  3g 20 e T þ 6g 11 þ O e3 6x0  2pti ¼ee T þ O e3 : 2pti T

u ¼ee

þ

Similarly, we discuss the other equilibrium P  ½0;  c; 0. Define the following subsets of the Hopf surface S.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9   < pffiffiffi ab b2 þ 16  b =   S1 ¼ ða; b; cÞa > 0; 0 < b < 2; 0; 0 < b 6 2; 0 < c < 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9   < pffiffiffi b2 þ 16  b =  S3 ¼ ða; b; cÞa > 0; 2 6 b; 0 < c < a : : ; 8  We have the following theorems, whose proof is similar to the above two theorems. h pffiffiffiffiffiffiffiffiffi ffi b2 þ16b Theorem 3.3 (Existence of Hopf Bifurcation). Let 0 < c < a. Then, as b varies and passes through the critical b0 ¼  34, 8 system (2.1) undergoes a Hopf bifurcation at the equilibrium P ½0;  c; 0. The direction, stability and period of bifurcating periodic solutions at the equilibrium P  ½0;  c; 0 of system (2.1) are as follows:. Theorem 3.4. Let 0 < c <

pffiffiffiffiffiffiffiffiffi ffi 2

b þ16b 8

a and b0 ¼  34. For system (2.1),

(i) if ða; b; cÞ 2 S1 , bifurcating periodic solutions exist for sufficient small b  b0 > 0. Moreover, period solutions of system (2.1) from Hopf bifurcation at P ½0;  c; 0 are non-degenerate, supercritical and orbitally stable. (ii) if ða; b; cÞ 2 S2 [ S3 , bifurcating periodic solutions exist for sufficient small b  b0 < 0. Moreover, period solutions of system (2.1) from Hopf bifurcation at P ½0;  c; 0 are non-degenerate, subcritical and orbitally unstable. (iii) the period and characteristic exponent of the bifurcating periodic solution are:

T¼

  2p  1 þ s2 e2 þ o e4 ; b ¼ b2 e2 þ o e4 ; x0

where

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x0 ¼ cab þ a2  4c2 ; 50a2 bc2 þ 8ba4  9b2 ca3 þ 11ab2 c3 þ b3 c2 a2 þ 400c3 a þ 104ca3  80bc4 s2 ¼ ; 16cð4c2  a2 þ cabÞða  cbÞð5c2 þ a2  cabÞ 3 aðba  4cÞ b2 ¼ ; 4 ða  cbÞð5c2 þ a2  cbaÞ

e2 ¼

b  b0

l2

l2 ¼ 

h i 2 þ o ðb  b0 Þ ;

3 aðba  4cÞ : 32 ða  cbÞð5c2 þ a2  cbaÞc

4. Bifurcations of heteroclinic orbits In order to study the bifurcations of the heteroclinic orbits by the high-dimensional generalization of the Melnikov method [16], we rewrite system (2.1) as

x_ ¼ f ðxÞ þ hðx; lÞ;

ð4:1Þ

22

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

where

0

1 0 1 az 8ðb þ 1Þx y B C B C bxz f ðxÞ ¼ @ 11x2 þ 3y2 þ z2  3c2 A; hðx; lÞ ¼ @ A; l ¼ ða; bÞ; a x  b x y 2y z   where l is a parameter, 0 6 l  1. Its unperturbed system is

x_ ¼ f ðxÞ:

ð4:2Þ

In addition, let p1 ¼ Pþ and p2 ¼ P . Here b > 1. System (4.1) has two hyperbolic equilibria p1 ; p2 and the heteroclinic orbit

C1 ¼ fx1 ðtÞ ¼ ð0; c tanhð2ctÞ; csechð2ctÞÞ : t 2 Rg;

ð4:3Þ

x1 ðtÞ ! p1 (p2 ) as t ! 1(þ1) when c > 0, and  x1 ðtÞ ! p2 (p1 ) as t ! 1(þ1) when c < 0. The eigenvalues of Df ðp1 Þ with  are

k1 ¼ 6c;

k2 ¼ 2c;

k3 ¼ 8cðb þ 1Þ:

The eigenvalues of Df ðp2 Þ are

k1 ¼ 6c;

k3 ¼ 8cðb þ 1Þ:

k2 ¼ 2c;

This implies that the linear variational equation

/_ ¼ AðtÞ/

ð4:4Þ þ



x1 ðtÞÞ [16]. has an exponential dichotomy on both R and R , where AðtÞ ¼ Df ð The linear variational equation of (4.2) with respect to C1 is

0 B /_ ¼ @

8cðb þ 1Þ tanhð2ctÞ

0

0

6c tanhð2ctÞ

0

2c sec hð2ctÞ

0

1

C 2c sec hð2ctÞ A/; 2c tanhð2ctÞ

and it has exactly two linearly independent bounded solutions:

 2 0; 2ðsechð2ctÞÞ ; 2 tanhð2ctÞsechð2ctÞ ;   1 4b ðcosh ð2ct ÞÞ ðcosh ð8ctÞ þ 3 þ 4 cosh ð4ctÞÞ; 0; 0 8  Thus dim T q W u \ T q W s ¼ 2, where q 2 C1 . It is easy to verify that the corresponding adjoint equation w_ ¼ A ðtÞw has exactly one bounded solution:

0

1

4b

ðsec hð2ctÞÞ B C w ¼ @

4 ; 0; 0A: 2 2ðcosh ðct ÞÞ  1 We calculate the Melnikov function as follows:

M1 ¼

Z

þ1

w hl ðx1 ðtÞ; 0Þdt ¼

pffiffiffi

p Cð2bþ52Þ

2 Cð2bþ3Þ

1

 ; 0 ;

where

0 B hl ðx1 ðtÞ; 0Þ ¼ @

c sec hð2ctÞ 0 0 0

1

C 0 A: 0

Obviously, the rank of M 1 is 1. Then we have the following theorem. Theorem 4.1. Suppose l ¼ ða; bÞ 2 U R2þ , and U is an open neighborhood of the origin in R2þ . Then there exists a neighborhood V 1 U of the origin in R2þ and a curve H1 V 1 such that, for l 2 H1 , system (4.1) has hyperbolic saddles p1 ðlÞ; p2 ðlÞ connected by a heteroclinic orbit Cl : x ¼ xðt; lÞ satisfying

    pi ðlÞ  pi  ¼ O l ; i ¼ 1; 2;     xðt; lÞ  x1 ðt Þ ¼ O l ; t 2 R;

and the tangent space of H1 at l ¼ 0 has normal M 1 .

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

23

Similarly, we discuss the other heteroclinic orbit

C2 ¼ fx1 ðtÞ ¼ ð0; c tanhð2ctÞ; c sec hð2ctÞÞ : t 2 Rg;

ð4:5Þ

with  x1 ðtÞ ! p1 (p2 ) as t ! 1(þ1) when c > 0, and  x1 ðtÞ ! p2 (p1 ) as t ! 1(þ1) when c < 0.

0.015 0.01 0.005

x

0 −0.005 −0.01 −0.015

0

50

100

150

200

0.005 0.01

0.015 0.02

t

0.02 0.015 0.01

z

0.005 0 −0.005 −0.01 −0.015 −0.02 −0.02 −0.015 −0.01 −0.005

0

x

0.015 0.01

z

0.005 0 −0.005 −0.01 −0.015 −0.146 −0.148

y

−0.15 −0.02

−0.01

0

0.01

0.02

x

Fig. 3. Waveform diagram and phase diagram for system (2.1) with b=0.7499.

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

x 10−3

2 1.5 1 0.5 x

0 −0.5 −1 −1.5 −2

4

0

50

100

150

t

200

x 10−3

3 2 1

z

0 −1 −2 −3 −4 −4

−3

−2

−1

0

1

x

2

3

4 x 10−3

x 10 −3 1.5 1 0.5

z

24

0 2

−0.5

1

−1

0 −1

−1.5

y

−2

x 10−3

x

Fig. 4. Waveform diagram and phase portraits for system (2.1) for b=0.7501.

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

The linear variational equation of (4.5) with respect to C2 is

0

B /_ ¼ @

0

25

1

8cðb þ 1Þ tanhð2ctÞ

0

0

6c tanhð2ctÞ

C 2c sec hð2ctÞ A/;

0

2c sec hð2ctÞ

2c tanhð2ctÞ

ð4:6Þ

and it has exactly two linearly independent bounded solutions:

 2 0; 2ðsechð2ctÞÞ ; 2 tanhð2ctÞsechð2ctÞ ;   1 4b ðcosh ð2ct ÞÞ ðcosh ð8ctÞ þ 3 þ 4 cosh ð4ctÞÞ; 0; 0 8  Thus dim T q W u \ T q W s ¼ 2, where q 2 C1 . It is easy to verify that the corresponding adjoint equation w_ ¼ A ðtÞw has exactly one bounded solution:

0 ðsec hð2ctÞÞ

1

4b

B C w ¼ @

4 ; 0; 0A: 2 2ðcosh ðct ÞÞ  1 We calculate the Melnikov function as follows:

M2 ¼

Z

þ1

w hl ðx2 ðtÞ; 0Þdt ¼



1



pffiffiffi

p Cð2bþ52Þ

2 Cð2bþ3Þ

 ; 0 ;

where

0 B hl ðx2 ðtÞ; 0Þ ¼ @

c sec hð2ctÞ 0

1

0

C 0 A:

0

0

Clearly, the rank of M 2 is 1. Then we have the following theorem. Theorem 4.2. Suppose l ¼ ða; bÞ 2 U R2þ , and U is an open neighborhood of the origin in R2þ . Then there exists a neighborhood V 2 U of the origin in R2þ and a curve H2 V 2 such that, for l 2 H2 , system (4.1) has hyperbolic saddles p1 ðlÞ; p2 ðlÞ connected by a heteroclinic orbit Cl : x ¼ xðt; lÞ satisfying

    pi ðlÞ  pi  ¼ O l ; i ¼ 1; 2;     xðt; lÞ  x2 ðt Þ ¼ O l ; t 2 R;

and the tangent space of H2 at l ¼ 0 has normal M 2 . 5. Numerical simulations In this section, we present some numerical results of simulating system (2.1) at different values of a; b, b and c. We can determine the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions by the theorems in Section 3. (i) If a ¼ 0:5; b ¼ 0:7071, and c ¼ 0:149, we can calculate

l2 ¼ 0:6268499102; b2 ¼ 0:7472050930; s2 ¼ 84:35134562: It is found that a stable periodic solution with initial values ð0:01; 0:149; 0:001Þ exists for b ¼ 0:7499, as shown in Fig. 3. (ii) If a ¼ 0:5; b ¼ 1:2856, and c ¼ 0:1446, we can calculate

l2 ¼ 0:2540710575; b2 ¼ 0:2939093993; s2 ¼ 180:2788521: An unstable periodic solution with initial values ð0; 0:1446; 0:001Þ is obtained for b ¼ 0:7501, as shown in Fig. 4. 6. Conclusions In this paper, we introduce the generalized stretch-twist-fold flow and have investigated its complex dynamical behaviors in the parametric space. It is very interesting that the system is not conservative any more and Hopf bifurcations exist in the system while the original STF flow does not possess these properties. The paper proves that Hopf bifurcations occur when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation and stability of the bifurcating peri-

26

J. Bao, Q. Yang / Applied Mathematics and Computation 229 (2014) 16–26

odic solutions are analyzed in detail. We also investigate the bifurcations of the heteroclinic orbits. The existence and its associated existing regions are given for two heteroclinic orbits, respectively. Meanwhile, some numerical simulations for justifying the theoretical analysis are also presented. It is hoped that the investigation of the paper will shed some lights to more systematic studies of the STF flow. Acknowledgments We express our sincere thanks to the anonymous referees for their rigorous comments and valuable suggestions helping to improve the original manuscript. The research is supported by the National Natural Science Foundation of China (No. 11271139), the Science and Technology Planning Project of Guangdong Province, China (No. 2012B061800088) and the Fundamental Research Funds for the Central Universities (No. 2013ZM0115). References [1] Zhouchao Wei, Qigui Yang, Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci, Appl. Math. Comput. 217 (2010) 422– 429. [2] Yongjian Liu, Guoping Pang, The basin of attraction of the Liu system, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2065–2071. [3] Li-Guo Yuan, Du-Xian Nie, Fu. Xin-Chu, Complex orbits in a second-order digital filter with sinusoidal response, Chaos Solitons Fractals 40 (2009) 1660–1667. [4] Junfei Cao, Qigui Yang, Zaitang Huang, Qing Liu, Asymptotically almost periodic solutions of stochastic functional differential equations, Appl. Math. Comput. 218 (2011) 1499–1511. [5] Junwei Wang, Meichun Zhao, Yanbin Zhang, Xiaohua Xiong, Silnikov-type orbits of Lorenz-family systems, Physica A 375 (2007) 438–446. [6] Guirong Jiang, Qigui Yang, Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination, Appl. Math. Comput. 215 (2009) 1035– 1046. [7] Stephen Childress, Andrew D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Springer-Verlag, Berlin Heidelberg, Berlin, 1995. [8] H.K. Moffatt, Stretch, twist and fold, Nature 341 (1989) 285–286. [9] K. Bajer, H.K. Moffatt, On a class of steady confined stokes flows with chaotic streamlines, J. Fluid Mech. 212 (1990) 337–363. [10] Samuel I. Vainshtein, Roald Z. Sagdeev, Robert Rosner, et al, Fractal properties of the stretch-twist-fold magnetic dynamo, Phys. Rev. E 53 (1996) 4729– 4744. [11] D.L. Vainshtein, A.A. Vasiliev, A.I. Neishtadt, Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow, Chaos 6 (1996) 67–77. [12] Jianghong Bao, Qigui Yang, Complex dynamics in the stretch-twist-fold flow, Nonlinear Dyn. 61 (2010) 773–781. [13] Jianghong Bao, Qigui Yang, A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput. 217 (14) (2011) 6526–6540. [14] B. Hassard, N. Kazarinoff, Y. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1982. [15] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998. [16] D. Zhu, Melnikov-type vectors and principal normals, Sci. China Ser. A 37 (1994) 814–822.