Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system

Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

Contents lists available at SciVerse ScienceDirect

Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns

Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system J.H. Yang ⇑, H. Zhu School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, PR China

a r t i c l e

i n f o

Article history: Received 24 May 2012 Received in revised form 19 September 2012 Accepted 22 September 2012 Available online 5 October 2012 Keywords: Vibrational resonance Fractional-order damping Time delay feedback

a b s t r a c t We develop an analytical technique to investigate the phenomenon of vibrational resonance in a fractional-order Duffing system with linear time delay feedback and driven by both low frequency and high frequency periodic signals. At first, the theoretical predication of the response amplitude at the low-frequency is obtained by the method of direct separation of slow and fast motions. Then, the bifurcation analysis is carried out based on three kinds of resonance behaviors. Further, influences of the high frequency signal, the fractional-order damping and the delay parameter on the vibrational resonance are discussed by both theoretical and numerical simulations. If the value of the fractional-order is a controllable parameter, the monotonicity of the response amplitude versus the value of the fractional-order depends on the amplitude of the high-frequency signal. If the delay parameter is a controllable parameter, the response amplitude with respect to the delay parameter presents periodic or quasi-periodic pattern, and it is similar to that in the integer-order differential system with linear time delay feedback. The good agreement between the analytical and numerical results indicates the validity of the theoretical predications. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction In the last decade, the response problem of a nonlinear system that excited by both low frequency and high frequency periodic signals has attracted more and more attention. It is because the weak low-frequency signal, which usually viewed as the useful signal, can be effectively enhanced by an appropriate high-frequency signal in a nonlinear system. The corresponding famous phenomenon named vibrational resonance (VR) was originally proposed by Landa and McClintock [1], in which the response of a bistable system to a weak low-frequency signal can be improved excellently by adjusting another excitation-including high-frequency signal. Since then, VR has been investigated in various kinds of systems by theoretical and experimental treatments [2–10]. Recently, more and more interesting results of VR in the delayed system [11–16], fractional-order system [17] and complex networks [18–20] have been reported. Compared with VR in the ordinary differential system, the fractional-order damping is a key factor that induces different bifurcation and resonance behaviors [17]. The research about fractional-order calculus is a hot topic in many engineering and scientific areas such as rheology, automatic control, signal and image processing, bioengineering, mechanics, electrochemistry, etc. There are several definitions for the fractional-order differentiation, but the most popular definitions are the Riemann–Liouville (RL) definition, the Caputo definition, and the Grünwald–Letnikov (GL) definition [21]. The RL definition is in the form ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J.H. Yang). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.09.023

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

" # Z t a m d f ðtÞ d 1 f ð sÞ ; ¼ d s dta dt m Cðm  aÞ 0 ðt  sÞamþ1

1317

ð1Þ

where m  1 < a < m, m 2 N, and C() is the gamma function. The symbol a denotes the value of the fractional-order that is usually chosen in the range 0 < a < 2. This definition ensures some nice and useful mathematical properties, but the initial conditions are considered to have no physical meaning in many situations. It is because the derivative of a constant is not identical to zero under this definition. To avoid these difficulties, an alternative definition is introduced by Caputo as a

d f ðtÞ 1 ¼ dta Cðm  aÞ

Z

t

f ðmÞ ðsÞ ðt  sÞamþ1

0

ds:

ð2Þ

Due to the existences of the integral and the gamma function, the Caputo definition is difficult in numerical calculations. The GL definition is well well-known for the discretization, and it is given by

   a k d f ðtÞ 1X j a ¼ lim ð1Þ f ðkh  jhÞ; dt a t¼kh h!0 ha j¼0 j

ð3Þ

where the binomial coefficient is

 

a

 

a

¼ 1;

0

j

¼

aða  1Þ . . . ða  j þ 1Þ j!

for j P 1:

ð4Þ

In general, these three kinds of definition are equivalent for a wide class of functions [22]. Until now, there are huge volumes of literatures about fractional-order calculus or VR in various disciplines. However, to our knowledge, the work of VR in the fractional-order system is very little [17]. It arouses our interest in studying VR in different fractional-order models. In the present work, we wish to investigate VR in a fractional-order Duffing system with linear time delay feedback. The system will be studied is described by a

2

d xðtÞ dt

2

þd

d xðtÞ 2 3 a  x0 xðtÞ þ bx ðtÞ þ cxðt  sÞ ¼ f cosðxtÞ þ F cosðXtÞ: dt

ð5Þ

Here, the GL definition is used. The parameters satisfy d > 0, x20 > 0, b > 0, c > 0, s P 0, f << 1 and x << X. When the delay is free, i.e., s = 0, the potential of the system is VðxÞ ¼ 12 ðc  x20 Þx2 þ 14 bx4 , and it has a double-well when c  x20 < 0. In the following analysis, only the double-well case will be considered. The bifurcation and resonance induced by the fractionalorder damping and the time delay feedback are our focus. The paper is structured as follows. In Section 2, the theoretical analysis will be carried out and the approximate solution of the response amplitude at the low-frequency will be obtained. Then, the bifurcation and resonance analysis will be given according to the analytical results. In Section 3, the theoretical predications will be verified by the numerically computed values, and the effects of the fractional-order damping and the delay parameter on VR will be discussed in detail. In the last section, the conclusions of this work will be summarized. 2. Theoretical predications In the ordinary differential system, due to x << X, the method of direct separation of slow and fast motions is successfully used in solving the approximate response amplitude of the system at the low-frequency x [23]. In this section, we will use this method to get the theoretical predications of the response amplitude in the fractional system. 2.1. The response amplitude at the frequency x Assuming the solution of Eq. (5) is in the form x(t) = X(t) + W(t), where X and w are slow and fast motions with period 2p/x and 2p/X, respectively, one has 2

d XðtÞ

a

2

þ

d WðtÞ

a

d XðtÞ d WðtÞ 3 2 3 2 2 2 a þd a  x0 XðtÞ  x0 WðtÞ þ bX ðtÞ þ bW ðtÞ þ 3bXðtÞW ðtÞ þ 3bX ðtÞWðtÞ dt dt dt dt þ cXðt  sÞ þ cWðt  sÞ ¼ f cosðxtÞ þ F cosðXtÞ: 2

2

þd

ð6Þ

Seeking the solution of w(t) in the equation a

2

d WðtÞ d WðtÞ þd  x20 WðtÞ þ cWðt  sÞ ¼ F cosðXtÞ: dt a dt2

ð7Þ

For t ? 1, one easily obtains the stationary solution

WðtÞ ¼

F

l

cosðXt þ /Þ;

ð8Þ

1318

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

where

l2 ¼ ½c cosðXsÞ þ dXa cosðap=2Þ  x20  X2 2 þ ½c sinðXsÞ  dXa sinðap=2Þ2 and

c sinðXsÞ  dXa sinðap=2Þ : c cosðXsÞ þ dXa cosðap=2Þ  x20  X2

/ ¼ tan1

Substituting Eq. (8) into Eq. (6) and averaging all terms in the interval [0, 2p/X], it yields a

2

d XðtÞ d XðtÞ þd þ C 1 XðtÞ þ bX 3 ðtÞ þ cXðt  sÞ ¼ f cosðxtÞ; dt a dt2

ð9Þ

2

where C 1 ¼ 3bF  x20 . 2l2 When f = 0 and s = 0, all the equilibrium points of Eq. (9) are

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X  ¼  ðC 1 þ cÞ=b:

X 0 ¼ 0;

ð10Þ ⁄

The slow oscillations may occur around the stable steady states. Considering the deviation Y(t) of X(t) from X , Y(t) = X(t)  X⁄, one has a

2

d YðtÞ d YðtÞ þd þ x2r YðtÞ þ 3bX  Y 2 ðtÞ þ bY 3 ðtÞ þ cYðt  sÞ ¼ f cosðxtÞ; dt a dt2

ð11Þ

where x2r ¼ C 1 þ 3bX 2 . Because f << 1, one can ignore the nonlinear terms in Eq. (11) to solve the solution of Y(t). According to the linear equation a

2

d YðtÞ d YðtÞ þd þ x2r YðtÞ þ cYðt  sÞ ¼ f cosðxtÞ; dt a dt2

ð12Þ

one gets Y(t) = ALcos(xt + h), where

f AL ¼ pffiffiffi ; S S ¼ ½x2r þ c cosðxsÞ þ dxa cosðap=2Þ  x2 2 þ ½c sinðxsÞ  dxa sinðap=2Þ2

ð13Þ

and

h ¼ tan1

c sinðxsÞ  dxa sinðap=2Þ : x þ c cosðxsÞ þ dxa cosðap=2Þ  x2 2 r

ð14Þ

pffiffiffi The response amplitude of the system at the low-frequency x is defined by Q ¼ AL =f ¼ 1= S. When S arrives at its local minimum, Q reaches the local maximum and the resonance occurs. According to Eq. (13), if the signal amplitude F is a controllable parameter, the resonance appears when

x2r ¼ x2  c cosðxsÞ  dxa cos ap=2Þ:

ð15Þ

Further, if a or s is the controllable parameter, the resonance is predicted by the equation dS/da = 0 or dS/ds = 0, respectively. 2.2. Bifurcation analysis In this subsection, we make the variable F as a controllable parameter. In Eq. (10), the stable steady states X  exist when C1 + c < 0, and it indicates

F < Fc ¼



2l2 2 ðx0  cÞ 3b

1=2 :

ð16Þ

Otherwise, only one stable steady state X 0 exists when

 F P Fc ¼

1=2 2l2 2 : ðx0  cÞ 3b

ð17Þ

The value F = Fc is the bifurcation point that induces the change in the number of the stable steady states. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi When F < Fc, the slow oscillation occurs around the stable steady states X  . Substituting X  ¼  ðC 1 þ cÞ=b into Eq. ð1Þ (15), one obtains the critical value F VR at which the resonance appears,

1319

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

ð1Þ

F VR ¼



l2

3b

½2x20  3c  x2 þ dxa cosðap=2Þ þ c cosðxsÞ

1=2

< Fc:

ð18Þ

Moreover, the parameters satisfy the condition

x2 þ 3c  2x20  c cosðxsÞ < dxa cosðap=2Þ < x2 þ c  c cosðxsÞ:

ð19Þ ⁄

⁄

When F P Fc, the slow oscillation appears around the stable steady state X = 0. Substituting X = 0 into Eq. (15), the other ð2Þ resonance occurs at the point F VR , ð2Þ

F VR ¼

 2 1=2 2l > Fc: ½x20 þ x2  dxa cosðap=2Þ  c cosðxsÞ 3b

ð20Þ

Here, the parameters satisfy the condition

dxa cosðap=2Þ < x2 þ c  c cosðxsÞ:

ð21Þ

Hence, we have the following conclusions: ð1Þ ð2Þ (1) If the condition in Eq. (19) is satisfied, the resonance occurs at F VR and F VR with the increase of F, and the two corresponding peak values of Q are identical,

Q ð1Þ max ¼

1 : jc sinðxsÞ  dxa sinðap=2Þj

ð22Þ

At F = Fc, the response amplitude Q arrives at the local minimum. (2) If the parameters satisfy

dxa cosðap=2Þ 6 x2 þ 3c  2x20  c cosðxsÞ;

ð23Þ

ð2Þ F VR ,

the resonance occurs at and the corresponding peak value of Q is also (3) If the parameters satisfy

Q ð1Þ max

.

dxa cosðap=2Þ P x2 þ c  c cosðxsÞ; both

ð1Þ F VR

and

ð2Þ F VR

ð24Þ

are inexistent, and the resonance occurs at the turning point F = Fc. Here, the corresponding peak value of Q is

1 Q ð2Þ max ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ½c cosðxsÞ þ dxa cosðap=2Þ  c  x2 2 þ ½c sinðxsÞ  dxa sinðap=2Þ2

ð25Þ

Consequently, the parameters induce the change in the number of the stable steady states and then lead to different resonance behaviors. 3. Numerical simulations To verify the validity of theoretical predications, numerical results are calculated in this section. For the numerical simulation, the response amplitude at the frequency x is calculated from

Q¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q 2sin þ Q 2cos f

ð26Þ

with

Q sin ¼

2 kT

Z

kT

xðtÞ sinðxtÞdt;

Q cos ¼

0

2 kT

Z

kT

xðtÞ cosðxtÞdt;

0

where T = 2p/x and k is a positive integer. To obtain the numerical results, the time series x(t) should be computed at first. In this paper, the GL definition is used to discretize the equation. For the reason of simplicity, we always chose k = 100, f = 0.1, x20 ¼ 1, b = 1 and c = 0.1 in the flowing simulations. 3.1. Fractional-order a induces bifurcation Firstly, we make the amplitude of the high frequency signal as the controllable parameter. Letting W = dxa cos (ap/2), W 1 ¼ x2 þ 3c  2x20  c cosðxsÞ, W2 = x2 + c  c cos (xs), in Fig. 1 (a), the condition in Eq.(19) is always satisfied when ð2Þ a 2 (0, 2). As a result, the double resonance appears at F ð1Þ VR and F VR with the increase of F. These predications are clearly shown in Fig. 1 (b)–(d). The good agreement between the approximate analytical results with the numerical simulations indicates ð1Þ the validity of the theoretical predications. In this figure, when the value of a turns from 0.4 to 1.4, F VR becomes smaller ð2Þ while F VR becomes larger gradually. The value of a does not induce the change in the number of the stable steady states. Hence, the bifurcation does not occur in this case.

1320

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

2

4

(a)

3

W2 Q

W

1 0 -1

Q(1) max

(b) α =0.4

2 1

W1 0

2

0.5

1

1.5

α

0

2

α =1.0

F(2) VR Fc 100 F

50

150

(d)

2

Q(1) max

α =1.4

Q

Q

1.5

0

3

Q(1) max

(c)

F(1) VR

1 0.5

1 F(1) VR 0

50

Fc 100 F

F(2) VR 150

0

F(1) VR 0

F(2) VR Fc100 F

50

150

ð1Þ

200

ð2Þ

Fig. 1. (a) The region corresponding to the double resonance. (b)–(d) The double resonance occurs at F VR and F VR . The solid lines are theoretical predications while dotted lines are numerical results. The simulation parameters are x = 1, X = 10, d = 0.65, s = p/2.

Different resonance behaviors that are induced by the fractional-order a are given in Fig. 2. In Fig. 2(a), the regions corresponding to double and single resonances are given. When 0 < a < ac, the condition in Eq. (19) is validity. It results ð1Þ ð2Þ in the double resonance occurring at F VR and F VR . However, when ac 6 a < 2, the condition in Eq. (23) is satisfied. It leads ð2Þ to the single resonance takes place at F VR . These theoretical predications are verified in Fig. 2(b)–(d). When a = 0.5 and 1.0, one has W1 < W < W2 and it means the double resonance. However, when a = 1.6, one has W < W1, and the single resoð2Þ nance appears at F VR . The response amplitude Q reaches the local minimum at Fc. If the parameters are chosen as those in Fig. 3, the double resonance and two kinds of single resonance are demonstrated. In Fig. 3(a), when a 2 (0, a1], W P W2, the condition in Eq. (24) is satisfied. The single resonance occurs at F = Fc, as is shown ð1Þ ð2Þ in Fig. 3(b). When a 2 (a1, a2), W1 < W < W2, the double resonance appears at F VR and F VR , e.g., a = 1 in Fig. 3(c). When

2

2

(a) W2

1

Q(1) max

W

1 0 -1

0.5 0

1.5

0.5

1

α

1.5α c

W1 2

0

F(1) VR 50

F(2) VR Fc 100

(d)

2

α =1.0

150

Q(1) max

α =1.6

1.5

1

0.5

0

α =0.5

2.5

(c)

Q(1) max

Q

(b)

1.5

1 F(1) VR 0

50

Fc 100

0.5

F(2) VR 150

0

Fc 0

100

F(2) VR 200

300

F

ð1Þ

ð2Þ

Fig. 2. (a) The regions corresponding to double resonance and single resonance. (b) and (c), the double resonance occurs at F VR and F VR . (d) The single ð2Þ resonance occurs at F VR . The solid lines are theoretical predications while dotted lines are numerical results. The simulation parameters are x = 1, X = 10, d = 1.0, s = p/2.

1321

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

2

1.5

(a)

W2

0

W1

-1 -2

α2

α1 0

0.5

1

1.5

α

0.8

Q(2) max

1 α =0.4 0.5 0

2

0

50

1.5

(c)

Q(1) max

0.7

Fc 100 F

150

(d)

1

Q(1) max

α =1.5

Q

0.6 α =1.0

Q

(b)

Q

W

1

0.5

0.5

F(1) VR 0

50

0

Fc 100 F(2) 150 VR F

F(2) VR

Fc 0

100

200

300

F ð1Þ

Fig. 3. (a) The regions corresponding to single resonance and double resonance. (b) The single resonance occurs at Fc. (c) The double resonance occurs at F VR ð2Þ ð2Þ and F VR . (d) The single resonance occurs at F VR . The solid lines are theoretical predications while dotted lines are numerical results. The simulation parameters are x = 1.0, X = 10, d = 1.5, s = p/2.

a 2 [a2, 2), W < W1, the resonance occurs at F ð2Þ VR , and Fc is a critical value at which the response amplitude Q arrives at the local minimum, as is shown in Fig. 3(d). 3.2. Delay parameter s induces bifurcation If the fractional-order a is fixed, the change of the delay parameter s can also induce different resonance behaviors. The delay-induced bifurcation in VR was investigated in the integer-order differential system [11–16]. Here, we would like to study it in the fractional-order differential system. In Fig. 4, the influence of the delay parameter on the resonance behaviors parameter s is given. In Fig. 4(a), in the interval s 2 (0, 2p/x), the regions corresponding to different resonance behaviors are shown. When the delay parameter s is in the 1.5

(a)

1

2

W2

Q

W

0.5 0 -0.5 -1

1.5 Q(1) max

τ1 2

τ

4

0

τ2 6

Q(2) max

1 0.5

W1 0

(b)

1.5

τ=1.0 0

50

Fc 100 F

150

1.5

(c)

Q(2) max

(d) 1

Q

Q

1 0.5

0.5

τ=3.5 0

0

F(1) VR 50

τ=6.28

F(2) VR

Fc 100 F

150

0

0

50

Fc 100 F

150

Fig. 4. (a) The regions corresponding to single resonance and double resonance. (b) and (d), The single resonance occurs at Fc. (c) The double resonance ð1Þ ð2Þ occurs at F VR and F VR . The solid lines are theoretical predications while dotted lines are numerical results. The simulation parameters are a = 0.45, x = 0.9, X = 10 and d = 1.2.

1322

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

interval (0, s1] or [s2, 2p/x), one has W P W2. As a result, the single resonance occurs at F = Fc, e.g., Fig. 4(b) and (d). However, ð1Þ ð2Þ when s 2 (s1, s2), one gets W1 < W < W2. It leads to the double resonance occurs at F VR and F VR , as is shown in Fig. 4(c). The single and double resonances induced by the delay parameter s are verified in this figure. In Fig. 5(a), the condition in Eq. (19) is satisfied for any value of s. In other words, the appearance of the double resonance is independent of the delay parameter s. No matter what the delay parameter s is chosen, one always has W1 < W < W2, and ð1Þ ð2Þ the double resonance appears at F VR and F VR . These facts are clearly shown in Fig. 5(b)–(d). In Fig. 6, different resonance behaviors induced by the delay parameter s are given. The double resonance regions in Fig. 6(a) corresponding to the intervals (0, s1) and (s2, 2p/x),while the single resonance region corresponding to the interval ð1Þ ð2Þ [s1, s2]. In Fig. 6(b) and (d), the double resonance occurs at F VR and F VR . In Fig. 6(c), one has W < W1, and the single resonance ð2Þ takes place at F VR . Hence, from Figs. 4–6, it shows that the change of the delay parameter s induces bifurcation in the number of the resonance peaks.

3.3. The response amplitude versus the fractional-order a In this subsection, we discuss the effect of the fractional order damping on the response amplitude. Although the optimal fractional-order a which makes the response amplitude Q reaches the maximal value can be obtained by calculating dS/ da = 0, due to the difficulty in solving the equation, we only give the relevant conclusions according to the numerical results. In Fig. 7(a), the dependence of the response amplitude Q on the parameter F and a is given. For fixed F, if the fractionalorder a is controllable, Q versus a presents different monotonic properties. In Fig. 7(b), for F = 20, Q is an increasing function of the value of a. For F = 80 and 130, Q is a nonlinear function of the fractional-order a. It indicates that the monotonicity of the response amplitude Q with respect to the fractional-order a depends on the value of the signal amplitude F.

3.4. Delay induces periodic and quasi-periodic VR The periodic or quasi-periodic pattern of VR profile induced by the time delay parameter is discussed in Refs. [11–16]. As a further investigation, we study the periodic properties of the response amplitude versus the delay parameter in the fractional-order system. In fact, from the analytical result in Eq. (13), it indicates that Q versus s presents the periodic pattern if X/x is a rational number. Otherwise, if X/x is an irrational value, Q versus s exhibits the quasiperiodic pattern. Here, we verify these predications by both analytical and numerical simulations. In Fig. 8, according to the theoretical predications in Section 2, the three-dimensional curves of Q versus the fractionalorder a and the delay parameter s are shown. The monotonicity of Q versus a depends on the values of s and F is proved again. For the case X/x is a positive integer, the periodic pattern of the VR profile induced by the delay parameter s is demonstrated in Fig. 9. The period equals to that of the low-frequency excitation signal, i.e., Q(s + 2p/x) = Q(s). In Fig. 10, in the case of X/x is irrational, Q(s + 2p/x) nears but is not identical to Q(s). The response amplitude Q versus the delay parameter s presents quasi-periodic pattern. 1.5

W2

(a)

1

(b)

1

Q(1) max

Q

W

0.5 0 -0.5 -1

W1 0

2

τ

4

6

50

Fc 100

F(2) VR 150

F Q(1) max

(d)

0.9

Q(1) max

0.8 Q

Q

F(1) VR

0

0.8 0.7

τ=3.5

0.6 0.5

τ=1.0

0.6

(c)

0.9

0.8

F(1) VR 0

50

Fc 100 F

F(2) VR

0.7

τ=6.28

0.6 150

0.5

F(1) VR 0

50

F(2) VR Fc 100 F ð1Þ

150

ð2Þ

Fig. 5. (a) The region corresponding to the double resonance. (b)–(d) The double resonance occurs at F VR and F VR . The solid lines are theoretical predications while dotted lines are numerical results. The simulation parameters are a = 1.0, x = 0.9, X = 10 and d = 1.2.

1323

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

1.5

(a)

1

0.8

W2 Q

W

0

W1

-0.5

τ1 2

0

0.8

τ

0.6

4 τ2

0.4

6

Q(1) max

(c)

0.7

0.6

0.6

τ=3.0 0

50

100 F

150

100 F

200

Q(1) max

τ=6.0

F(2) VR Fc100 F

0 F(1) 50 VR

200

150

(d)

0.5 F(2) VR

Fc

F(2) VR

Fc

0 F(1) 50 VR

0.8

0.7

0.5

τ=0.5

0.5

Q

Q

Q(1) max

0.7

0.5

-1

(b)

150

200

ð1Þ

ð2Þ

Fig. 6. (a) The regions corresponding to double resonance and single resonance. (b) and (d), The double resonance occurs at F VR and F VR . (c) The single ð2Þ resonance occurs at F VR . The solid lines are theoretical predications while dotted lines are numerical results. The simulation parameters are a = 1.3, x = 1.0, X = 10 and d = 1.5.

(a) 2.5

1.5

α

2 1

1.5 1

0.5

0.5 0

2

50

(b)

100 F

150

F=80

200

F=20

Q

1.5 1 0.5 0.4

F=130 0.6

0.8

1

α

1.2

1.4

1.6

Fig. 7. (a) The response amplitude Q versus F and a is plotted according to the theoretical result. (b) The response amplitude Q versus a is plotted for different values of F. The solid lines are theoretical predications while dashed lines with dots are numerical results. The simulation parameters are x = 1, X = 10, d = 1.0 and s = p/2.

4. Conclusions In the present work, an analytical technique is developed to investigate the VR phenomenon in a fractional-order system that is excited by both low-frequency and high-frequency periodic signals. Besides, the effect of time delay feedback

1324

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

Fig. 8. The three-dimensional curves of Q versus a and s are plotted according to the theoretical results. The simulation parameters are x ¼ and d = 0.6. (a) F = 20; (b) F = 67; (c) F = 90; (d) F = 115.

0.6

pffiffiffi 3=2, X = 10

α =0.7

(a)

Q

0.55 0.5 0.45

0

0.75

10

20

τ

30

40

α =1.0

(b)

Q

0.7 0.65 0.6 0.55

0

10

3

20

τ

30

40

α =1.5

(c) Q

2.5 2 1.5

0

10

20

τ

30

40

Fig. 9. The periodic VR is induced by the delay parameter s for different fractional-order a. In each subplot, the above lines are theoretical predictions while the bottom lines are numerical results. The simulation parameters are x = 0.9, F = 120, X = 9 and d = 0.6.

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

1325

0.52

α =0.7

(a)

0.5

Q

0.48 0.46 0.44 0.42

0

20

30

40

30

40

30

40

τ

α =1.0

(b)

0.6

Q

10

0.55

0.5 0 1.6

10

20

τ

α =1.5

(c)

Q

1.4

1.2

1

0

10

20

τ

Fig. 10. The quasi-periodic VR is induced by the delay parameter s for different fractional-order a. In each subplot, the above lines are theoretical pffiffiffi predictions while the bottom lines are numerical results. The simulation parameters are x ¼ 2=2, F = 120, X = 9 and d = 0.6.

is also considered. At first, the method of direct separation of slow and fast motions is used to obtain the response amplitude at the low frequency. Then, according to the theoretical predications, the bifurcation behaviors of the response amplitude are studied. These bifurcations are induced by the change in the number of the stable steady states and lead to different resonance behaviors. Based on these results, the conditions corresponding to double resonance and two kinds of single resonance are given. If the amplitude of the high-frequency signal is a controllable parameter, the resonance behaviors are determined by the fractional-order of the damping and the delay parameter together. Furthermore, the monotonic property of the response amplitude with respect to the value of the fractional-order is influenced by the amplitude of the excitation-including high-frequency signal. Finally, the theoretical predications are verified by the numerical calculations. The good agreement between these results indicates the validity of the theoretical analysis. In the fractional-order system, the periodic or quasi-periodic pattern of VR profile with respect to the delay parameter is similar to that in the integer-order system. The study of this paper is important because the two-frequency signals are widely applied in many braches such as communication technology, laser physics, acoustics, neurosciences, physics of the ionosphere, complex networks, etc. Moreover, due to the excellent properties of the fractional-order system in a variety of science and engineering fields, we hope our study will be applicable in solving some problems and arousing more techniques to deal with the fractional-order issues in different disciplines.

Acknowledgements The Project is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2012QNA21) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. The authors are also grateful to the editor and the anonymous reviewers for their useful comments and advice, which are vital for improving the quality of this paper.

1326

J.H. Yang, H. Zhu / Commun Nonlinear Sci Numer Simulat 18 (2013) 1316–1326

References [1] Landa PS, McClintock PVE. Vibrational resonance. J Phys A: Math Gen 2000;33:L433–8. [2] Gitterman M. Bistable oscillator driven by two periodic fields. J Phys A: Math Gen 2001;34:L355–7. [3] Blekhman II, Landa PS. Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation. Int J Non-linear Mech 2004;39:421–6. [4] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan MAF. Analysis of vibrational resonance in a quintic oscillator. Chaos 2009;19:043128. [5] Jeyakumari S, Chinnathambi V, Rajasekar S, Sanjuan MAF. Single and multiple vibrational resonance in a quintic oscillator with monostable potentials. Phys Rev E 2009;80:046608. [6] Rajasekar S, Jeyakumari S, Chinnathambi V, Sanjuan MAF. Role of depth and location of minima of a double-well potential on vibrational resonance. J Phys A: Math Theor 2010;43:465101. [7] Yao C, Zhan M. Signal transmission by vibrational resonance in one-way coupled bistable systems. Phys Rev E 2010;81:061129. [8] Rajasekar S, Abirmmi K, Sanjuan MAF. Novel vibrational resonance in multistable systems. Chaos 2011;21:033106. [9] Baltanás JP, Lo9 pez L, Blechman II, Landa PS, Zaikin A, Kurths J, et al. Experimental evidence, numerics, and theory of vibrational resonance in bistable systems. Phys Rev E 2003;67:066119. [10] Chizhevsky VN, Giacomelli G. Experimental and theoretical study of vibrational resonance in a bistable system with asymmetry. Phys Rev E 2006;73:022103. [11] Yang JH, Liu XB. Delay induces quasi-periodic vibrational resonance. J Phys A: Math Theor 2010;43:122001. [12] Yang JH, Liu XB. Controlling vibrational resonance in a delayed multistable system driven by an amplitude-modulated signal. Phys Scr 2010;82:025006. [13] Yang JH, Liu XB. Controlling vibrational resonance in a multistable system by time delay. Chaos 2010;20:033124. [14] Yang JH, Liu XB. Delay-improved signal propagation in globally coupled bistable systems. Phys Scr 2011;83:065008. [15] Jeevarathinam C, Rajasekar S, Sanjuan MAF. Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback. Phys Rev E 2011;83:066205. [16] Hu D, Yang J, Liu X. Delay-induced vibrational multiresonance in FitzHugh–Nagumo system. Commun Nonlinear Sci Numer Simulat 2012;17:1031–5. [17] Yang JH, Zhu H. Vibrational resonance in Duffing systems with fractional-order damping. Chaos 2012;22:013112. [18] Deng B, Wang J, Wei X, Tsang KM, Chan WL. Vibrational resonance in neuron populations. Chaos 2010;20:013113. [19] Qin Y, Wang J, Men C, Deng B, Wei X. Vibrational resonance in feedforward network. Chaos 2011;21:023133. [20] Yu H, Wang J, Liu C, Deng B, Wei X. Vibrational resonance in excitable neuronal systems. Chaos 2011;21:043101. [21] Monje CA, Chen YQ, Vinagre BM, Xue D, Feliu V. Fractional-order systems and controls. London: Springer; 2010. [22] Podlubny I. Fractional differential equations. San Diego and London: Academic Press; 1999. [23] Blekhman II. Vibrational mechanics. Singapore: World Scientific; 2000.