Published: 15 August 2015

Dynamical analysis of fractional-order Mathieu equation

Shaofang Wen1
Yongjun Shen2
Xianghong Li3
Shaopu Yang4
Haijun Xing5
1Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang, China
2, 4, 5Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, China
3Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China
Corresponding Author:
Yongjun Shen
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Abstract

The dynamical characteristics of Mathieu equation with fractional-order derivative is analytically studied by the Lindstedt-Poincare method and the multiple-scale method. The stability boundaries and the corresponding periodic solutions on these boundaries for the constant stiffness δ0=n2 (n = 0, 1, 2, …), are analytically obtained. The effects of the fractional-order parameters on the stability boundaries and the corresponding periodic solutions, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient (ELDC) and the equivalent linear stiffness coefficient (ELSC). The comparisons between the transition curves on the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. The following analysis is focused on the effects of the fractional parameters on the stability boundaries located in the δ-ε plane. It is found that the increase of the fractional order p could make the ELDC larger and ELSC smaller, which could result into the rightwards and upwards moving of the stability boundaries simultaneously. It could also be concluded the increase of the fractional coefficient K1 would make the ELDC and ELSC larger, which could move the transition curves to the left and upwards at the same time. These results are very helpful to design, analyze or control this kind of system, and could present beneficial reference to the similar fractional-order system.

1. Introduction

In 1695, the French mathematician Hospital and German mathematician Leibniz put forward the concept of fractional-order calculus for the first time. In the following centuries, the theory about fractional-order calculus including the definition, characteristics, calculation of fractional-order calculus and the relationship with integer-order calculus, etc, had been developing rapidly and a series of meaningful results had been obtained [1-7].

In recent years, fractional-order calculus was paid more and more attention from researchers in different fields and became an international hot topic. A lot of mathematicians, physicists, chemists, dynamicists, and engineers in some relevant fields had applied this mathematical tool to solve the problems they met. For example, Shen and Yang et al. [8-11] investigated several linear and nonlinear fractional-order oscillators by averaging method, and found that the fractional-order derivatives had both damping and stiffness effects on the dynamical response in those oscillators. Gorenflo et al. [12], Jumarie [13], Ishteva et al. [14] and Agnieszka et al. [15] respectively studied the definitions and numerical methods of fractional-order calculus for Grünwald-Letnikov, Riemann-Liouville and Caputo. Chen and Zhu [16] reviewed the analytical methods and control strategies for quasi-integrable Hamiltonian systems with fractional-order derivatives, and pointed out some possible developing issues. They [17-20] also studied some nonlinear fractional-order system with different kinds of noise, and obtained some important statistic properties of the fractional-order system. Wang et al. [21-22] investigated a linear single degree-of-freedom oscillator with fractional-order derivative, and obtained the composition solution of its initial value problem without external excitation. Li et al. [23] discussed the properties of three kinds of fractional derivatives and sequential property of the Caputo derivative is also derived. Li et al. [24-26] had done a lot of researches in the mathematical theory of fractional-order calculus, and also established some efficient numerical algorithms. Wahi and Chatterjee [27] studied a special linear single degree-of-freedom oscillator with fractional-order derivative by averaging method, and analyzed the effects of the fractional-orders derivative. Xu et al. [28] used Lindstedt-Poincaré method and the multiple-scale approach to investigate fractional-order Duffing oscillator subjected to random excitation.

In addition to the theoretical research, fractional calculus had also been used to solve engineering problems. Comparing with the traditional integer-order system, the fractional-order system is much closer to the real nature of the world, and has more advantages, such as strong ability of anti-noise, good robustness, high control precision and so on. Accordingly fractional-order calculus has the significant value for the engineering field.

The well-known Mathieu equation, first introduced by Mathieu [29] when he studied the vibration of elliptical membranes in 1868, is a linear differential equation with periodic coefficients. Mathieu equation had been applied in physics and engineering fields, and many complex dynamical properties had been found herein. In recent years, many scholars had studied the Mathieu equation and found that the fractional-order system could generate different dynamical properties from the integer-order system. For example, the chaotic behaviors of a nonlinear damped Mathieu system were investigated by applying numerical integration method [30]. Ebaid et al. [31] considered the fractional-order calculus model of damped Mathieu equation and obtained the approximate analytical solution by using two different methods. The transition curves separating the stable and unstable regions in the fractional-order Mathieu equation had been studied by the method of harmonic balance [32-33], and Leung et al. also investigated a general version of fractional-order Mathieu-Duffing equation by harmonic balance method.

In this paper, the fractional-order Mathieu equation with linear damping is studied by the Lindstedt-Poincaré (L-P) method coupled with the multiple-scale method, and the effects of the fractional parameters on the dynamical properties of the fractional-order Mathieu equation are analyzed in detail. The paper is organized as follow. Section 2 presents the stability boundaries and the corresponding approximately analytical periodic solutions on these boundaries for the constant stiffness δ0=n2 (n= 0, 1, 2, …), where the effects of the fractional-order derivative on the system damping and stiffness are formulated as equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC). In section 3, the comparisons between the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. The following analysis is focused on the effects of the fractional parameters on the stability boundaries in the δ-ε plane. At last the detail results are summarized and the conclusions are made.

2. Fractional-order Mathieu equation

In this paper, we shall consider the fractional-order Mathieu equation as:

1
x¨t+2ζx˙t+δ+2εcos2txt+K1Dpxt=0, ε1, 0<ζ1,

where 2ζ, δ, and 2εcos2t are the system linear damping coefficient, constant stiffness coefficient and the periodic time-varying coefficient respectively. Dp[x(t)] is the p-order derivative of x(t) to t with the fractional coefficient K1 (K1>0) and the fractional order p (0p1). There are several definitions for fractional-order derivative, such as Grünwald-Letnikov, Riemann-Liouville and Caputo definitions [1-7]. Here Caputo’s definition is adopted with the form as:

2
Dpxt=1Γ1-p0tx'ut-updu,

where Γy is Gamma function satisfying Γy+1=yΓy.

3. Approximately analytical solution and the stability boundaries

The approximately analytical solution and the stability boundaries are determined under the conditions of small damping coefficient. Using the parameters transformation:

ζ=εμ, μ=O1, K1=εk, k=O1.

Eq. (1) becomes:

3
x¨(t)+2εμx˙(t)+(δ+2εcos2t)x(t)+εkDp[x(t)]=0.

Introducing the fast and slow time scales as follows:

Tr=defεrt, r=0, 1, 2,,

and the differentiation operators:

4
D0=T0, D1=T1, D2=T2

the ordinary derivative can be transformed into the partial derivatives [34] as follows:

5
ddt=D0+εD1+,
d2dt2=D02+2εD0D1+.

The approximate solution of Eq. (3) can be written as:

6a
x=x0(T0,T1,T2)+εx1(T0,T1,T2)+ε2x2(T0,T1,T2)+.

Applying the modified L-P method, we introduce the following variable transformation [34-35]:

6b
δ=δ0+εδ1+ε2δ2+ε3δ3+.

Substituting Eq. (5), Eq. (6a), and Eq. (6b) into Eq. (3) and separating the terms in the yielded equation based on the power of ε, we derive the following equations:

7
O(ε0):D02x0+δ 0 x0=0,
8
Oε1:D02x1+δ 0 x1=-2D0D1x0-2μD0x0-δ 1 x0-2cos2T 0 x0-kD0px0,
9
O(ε2):D02x2+δ 0 x2=-2D0D2x0-2D0D1x1-D12x0-2μD0x1
-2μD1x0-δ 2 x0-δ 1 x1-2cos2T 0 x1-kD0px1-kD1px0.

The solution of Eq. (7) can be written as:

10
x0=AT1,T2eiδ0T0+cc,

where cc denotes the complex conjugate of the preceding term. The approximately analytical solution and the boundary will be presented in the following parts according to the different constant stiffness coefficient δ0.

3.1. δ0=0

According to Eq. (7) and Eq. (10), the solution can be written as:

11
x0=a=const,

where a is an arbitrary constant and determined by the initial condition. Substituting Eq. (11) into Eq. (8) one could obtain:

D02x1=-aδ1+2cos2T0.

In order to eliminate the secular terms in x1, one must put:

12a
δ1=0.

The solution of Eq. (8) is:

12b
x1=a2cos2T0.

Based on the formula for fractional derivative [36], Dpeiλt could be approximately reduced to:

13
Dpeiλt=(iλ)peiλt.

According to Eq. (13) and the Euler formula, one can obtain:

14
kD0px1=kD0pa2cos2T0=a2kD0pei2T0+e-i2T02=ka4[(2i)pei2T0+-2i)pe-i2T0.

Substituting Eq. (11), Eq. (12), Eq. (13) and Eq. (14) into Eq. (9), we obtain:

15
D02x2=-aδ2+12+2μasin2T0-a2cos4T0-14ka[(2i)pei2T0+-2i)pe-i2T0.

To eliminate the secular terms, we arrive at:

16
-aδ2+12=0,

and:

16a
δ2=-12.

The solution of Eq. (9) is derived as:

16b
x2=-μa2sin2T0+a32cos4T0+116ka(2i)pei2T0+116ka(-2i)pe-i2T0,

where the formula:

17
ip=(eiπ2)p=eipπ2=cospπ2+isinpπ2,

is used in Eq. (16b). One can get another form of Eq. (16b):

18
x 2 =a2-μsin2T0+2pk4cospπ2+2T0+a32cos4T0+Oε3.

Substituting Eq. (12a) and Eq. (16a) into Eq. (6b), the transition curve of the stability boundary near to δ0=0 is given by:

19
δε=-ε22+Oε3.

Substituting Eq. (11), Eq. (12b) and Eq. (18) into Eq. (6a), one could obtain the corresponding periodic solution with period π on this curve:

20
x=a+εa21+2pK 1 4cospπ2cos2t-ζ+2pK 1 4sinpπ2sin2t
+ε2a32cos4t+Oε3.

From Eq. (19), we can know that the result of the transition curve is independent of the fractional coefficient K1 and the fractional order p. In order to analyze the effects of the fractional-order derivative, we must obtain the third-order approximate solution:

21
O(ε3):D02x3+δ 0 x3=-2D1D0x2-2D1D2x0-D12x1-2D2D0x1-2cos2T 0 x2
-δ 3 x0-δ 2 x1-δ 1 x2-2μD0x2-2μD1x1-2μD2x0
-kD0px2-kD1px1-kD2px0.

As the similar procedure, we arrive at:

-δ3a-2pka8cospπ2=0

and:

22
δ3=-2pk8cospπ2,

based on the eliminating procedure for the secular term.

In the case δ0=0, the third-order approximate solution could be established as:

23
Δ0=-ε22+Oε4,

where Δ0=δ+ε22pK18cospπ2, which is defined as the equivalent linear stiffness coefficient (ELSC) in the case δ0=0.

3.2. δ0=1

In the case δ0=1, the solution of Eq. (7) can be written as:

24
x0=AeiT0+cc.

Using Eq. (13), one can get:

25
kD0px0=kD0pAeiT0+cc=kipAeiT0+cc.

Substituting Eq. (24) and Eq. (25) into Eq. (8), we can obtain:

26
D02x1+x1=-2iD1A-2iμA-δ1A-A--kipAeiT0-Aei3T0+cc.

After eliminating the secular terms, we arrive at:

27
-2iD1A-2iμA-δ1A-A--kipA=0

where D1A stands for the derivative with the respect to the slow time scale T1, and A- means the conjugate of A. In this paper, we want to obtain the periodic solutions, so that the hypothesis D1A= 0 is reasonable.

Substituting A=a/2-b/2i and Eq. (17) into Eq. (27), and after separating the real and imaginary parts of Eq. (27), we obtain:

28
1+δ1+kcospπ2a+2μ+ksinpπ2b=0,2μ+ksinpπ2a+1-δ1-kcospπ2b=0.

The necessary conditions for existing nonzero solution about (a,b) in Eq. (28) is:

29
det1+δ1+kcospπ22μ+ksinpπ22μ+ksinpπ21-δ1-kcospπ2=0.

That is:

30
1+δ1+kcospπ21-δ1-kcospπ2-2μ+ksinpπ22=0.

Then, we can get:

31
δ1=±1-2μ+ksinpπ22-kcospπ2.

In this case, the particular solution of Eq. (8) is:

32
x1=A8ei3T0+cc.

Substituting Eq. (24) and Eq. (32) into Eq. (9), we can obtain:

33
D02x2+x2=-2iD2A-D12A-2μD1A-δ2A-A8-kD1pAeiT0
+-34iD1A-34iμA-δ1A8-(3i)pkA8ei3T0-A8ei5T0+cc.

After eliminating the secular terms, one can get:

34
-2iD2A-D12A-2μD1A-δ2A-A8-kD1pA=0.

As the similar approach, we can suppose D1A=0, D2A=0 and D1pA=0. From Eq. (34) we can obtain:

35
δ2=-18.

In this case, the particular solution of Eq. (9) is:

36
x2=i3μA32+δ1A64+ip3pkA64ei3T0+A192ei5T0+cc.

Substituting Eq. (31) and Eq. (35) into Eq. (6b), the two transition curves emanating from δ0=1 are given by:

37
Δ1=1±ε2-C12-18ε2+Oε3,

where Δ1=δ+K1cos(pπ/2) and C1=2ζ+K1sin(pπ/2). Here Δ1 and C1 are defined as the equivalent linear stiffness coefficient and the equivalent linear damping coefficient respectively in the case δ0=1.

Substituting Eq. (24), Eq. (32) and Eq. (36) into Eq. (6a), one could obtain the corresponding periodic solution with period 2π on these curves:

38
x=AeiT0+εA8ei3T0+ε2i3μA32+δ1A64+ip3pkA64ei3T0+ε2A192ei5T0+cc+Oε3.

Substituting Eq. (17), Eq. (37) and A=a/2-b/2i into Eq. (38), one can get another form of Eq. (38) with the original parameters:

39
x=acost+bsint+ε81+δ8a+3ζb4cos3t+1+δ8b-3ζa4sin3t
+3pK1ε64cospπ2a+sinpπ2bcos3t+cospπ2b-sinpπ2asin3t
+ε64±ε2-C12-Δ1acos3t+bsin3t+ε2192acos5t+bsin5t+Oε3.

3.3. δ0=4

In order to analyze the effect of damping ratio near δ=4, one should assumed ζ=O(ε2). Using the transformation:

ζ=ε2μ^, μ^=O1, K1=ε2k^, k^=O1.

Eq. (1) becomes:

40
x¨(t)+2ε2μ^x˙(t)+(δ+2εcos2t)x(t)+ε2k^Dp[x(t)]=0.

Eq. (8) and Eq. (9) become:

41
Oε1:D02x1+δ0x1=-2D0D1x0-δ1x0-2cos2T0x0,
42
O(ε2):D02x2+δ 0 x2=-2D0D2x0-2D0D1x1-D12x0-2μ^D0x0-k^D0px0
-δ 2 x0-δ 1 x1-2cos2T0x1.

In the case δ0=4, the solution of Eq. (7) can be written as:

43
x0=Aei2T0+cc.

Substituting Eq. (43) into Eq. (41), we get:

44
D02x1+4x1=-4iD1A-δ1Aei2T0-Aei4T0+cc.

To eliminate the secular terms, we arrive at -4iD1A-δ1A=0. Due to D1pA=0, one can get δ1=0. The solution of Eq. (41) is:

45
x1=A12ei4T0-A4+cc.

Substituting Eq. (43) and Eq. (45) into Eq. (42), we get:

46
D02x2+4x2=-4iD2A-D12A-4iμ^A-2pipk^A-δ2A+A6-A-4ei2T0
-2iD1A3ei4T0-A12ei6T0+cc.

To eliminate the secular terms, we arrive at:

-4iD2A-D12A-4iμ^A-2pipk^A-δ2A+A6-A-4=0.

As the same procedure in the case δ0=1, we get:

det-2pk^cospπ2+δ2+112-2pk^sinpπ2+4μ^-2pk^sinpπ2+4μ^2pk^cospπ2+δ2-512=0.

That will result into:

47
δ2=16-2pk^cospπ2±116-[2pk^sin(pπ2)+4μ^]2.

Substituting the above results into Eq. (6b), the two transition curves emanating from δ0=4 are given by:

48
Δ2=4+ε26±ε416-4C22+Oε3,

where:

49
Δ2=δ+2pK1cospπ2, C2=2ζ+2p-1K1sinpπ2,

Δ2, C2 are defined as the equivalent linear stiffness coefficient (ELSC) and the equivalent linear damping coefficient (ELDC) respectively in the case δ0=4.

Substituting the above results into Eq. (6a), one could establish the corresponding periodic solution with period π on these curves:

50
x=Aei2T0+εA12ei4T0-A4+ε2A384ei6T0+cc+Oε3.

Transforming the exponent form into trigonometric one, we can obtain:

51
x=acos2t+bsin2t-εa4-a12cos4t-b12sin4t+ε2384acos6t+bsin6t+Oε3.

By analyzing the ELSC and the ELDC in three cases near δ0=n2 (n= 0, 1, 2), the general forms of the ELSC and the ELDC are given by:

52a
Δ=δ+δ0pK1cospπ2,
52b
C=2ζ+δ0p-1K1sinpπ2.

Here we can analyze some peculiar cases. If taking K1=0, Eq. (1) becomes the classic Mathieu equation. The stability boundaries and the corresponding periodic solutions near δ0=n2 (n= 0, 1, 2) are identical with the results in the classic monograph [34-35]. If taking p=1 or p=0, the fractional-order derivative will become the linear damping or the linear stiffness. The stability boundaries and the corresponding periodic solutions are identical with the results in the references [2-4] about the Mathieu equation. Therefore, it is indirectly verified that the correctness of this method and the results.

From the three cases, it can be concluded that the transition curve of the second-order approximate solution near δ0=0 is independent of the fractional-order derivative. But the fractional parameters will affect the transition curve of the third-order approximate solution near δ0=0 by the forms of the ELSC and the ELDC. It could also be found that the fractional-order parameters will affect the transition curves of the second-order approximate solution near δ0=1 and δ0=4 by the forms of the ELSC and the ELDC. In the three cases near δ0=n2 (n= 0, 1, 2), the ELSC and the ELDC could be arrived at the general forms shown in Eq. (52). Some parts of the corresponding periodic solutions on these boundaries can also be established by the forms of the ELSC and the ELDC.

From Eq. (52), it could be concluded that fractional-order parameters K1 and p have important influence on the ELSC and the ELDC. It is easy to find that the equivalent linear damping and stiffness coefficients are all monotonically increasing function of the fractional coefficient K1. The more important is the influence of the fractional order p on the ELSC and the ELDC. When p0, the ELDC will arrive at the minimum value as 2ζ, and the ELSC is δ+K1, so that the fractional-order derivative is completely equivalent to the linear stiffness. On the contrary, when p1, the ELDC will be the maximum value as 2ζ+K1, and the ELSC is δ, so that the fractional-order derivative is completely equivalent to the linear damping.

4. Numerical simulations

4.1. Comparison between the approximate analytical solution and the numerical results

In order to verify the validity of the method and the results in this paper, numerical results of the original equation are presented to compare the differences between the numerical simulations and the approximate analytical solutions as the next step.

In the δ-ε plane, we restrict the two parameters as ε [0 0.4], and δ [–0.2 4.5]. The sample step of ε and δ are selected as 0.01 and 0.005 respectively. Each point of the δ-ε plane is selected as the δ and ε value in Eq. (1) respectively, and used to calculate the numerical results. To determine the stability of each point, we should analyze the amplitude variation with a long time response. Then the stability boundaries near δ0=n2 (n= 0, 1, 2, …) could be determined.

The numerical method is [1-7]:

53
Dpxtlh-pj=0lCjpxtl-j,

where tl=lh is the time sample points, h is the sample step, Cjp is the fractional binomial coefficient with the iterative relationship as:

C0p=1, Cjp=1-1+pjCj-1p.

Here we select h= 0.001 and the total computation time is 500 s.

An illustrative example system is studied herein as defined by the basic system parameters: ζ= 0.005, K1= 0.005, p= 0.5. Based on the stability boundaries determined by Eq. (23), Eq. (37) and Eq. (48), one could analytically obtain those curves on the stability boundaries shown in Fig. 1, where the black solid lines are denoted for the approximate analytical solution. In Fig. 1, the red dots are for the unstable points and the blue dots are for the stable points, where the transition curves separating the red region from the blue region are the stability boundaries by numerical integration. From the observation of Fig. 1, we could conclude that the approximately analytical solution agrees very well with the numerical results and could present satisfactory precision.

Fig. 1The curves of the numerical results and the approximate analytical solution on the stability boundaries

The curves of the numerical results and the approximate analytical solution  on the stability boundaries

4.2. Effects of fractional-order parameters on the stability boundaries

In order to illustrate the effects of fractional-order parameters on the stability boundaries, the fractional order p are selected a set of values as 0, 0.2, 0.5, 0.8 and 1 respectively. The transition curves near δ0=1, δ0=4 and δ0=0 are shown in Fig. 2, Fig. 3 and Fig. 4 respectively. From the observation of the changes about the curves in Fig. 2 to Fig. 4, it is found that the increase of the fractional order p could make the ELDC larger, which would result into the upwards moving of the stability boundaries. That is to say, the unstable region becomes smaller and the stable region becomes larger. Simultaneously, it could also be found that the increase of the fractional order p would make the ELSC smaller, which could result into the rightwards moving of the stability boundaries. Therefore, it could be concluded that fractional order p has important influence on the stability boundaries.

Fig. 2The effects of the fractional order p on the stability boundaries for δ0= 1 where ζ= 0.05 and K1= 0.05

The effects of the fractional order p on the stability boundaries for δ0= 1 where ζ= 0.05 and K1= 0.05

Fig. 3The effects of the fractional order p on the stability boundaries for δ0= 4 where ζ= 0.005 and K1= 0.005

The effects of the fractional order p on the stability boundaries for δ0= 4 where ζ= 0.005 and K1= 0.005

Fig. 4The effects of the fractional order p on the stability boundaries for δ0=0 where ζ= 0.5 and K1= 0.5

The effects of the fractional order p on the stability boundaries for δ0=0 where ζ= 0.5 and K1= 0.5

Fig. 5The effects of the fractional coefficient K1 on the stability boundaries for δ0= 1 where ζ= 0.05 and p= 0.5

The effects of the fractional coefficient K1 on the stability boundaries for δ0= 1 where ζ= 0.05 and p= 0.5

Fig. 6The effects of the fractional coefficient K1 on the stability boundaries for δ0= 4 where ζ= 0.005 and p= 0.5

The effects of the fractional coefficient K1 on the stability boundaries for δ0= 4 where ζ= 0.005 and p= 0.5

Fig. 7The effects of the fractional coefficient K1 on the stability boundaries for δ0=0 where p= 0.5

The effects of the fractional coefficient K1 on the stability boundaries for δ0=0 where p= 0.5

The fractional coefficient K1 are selected some other different values, and the results are shown in Fig. 5 to Fig. 7. From the observation of those curves in Fig. 5 to Fig. 7, it is found that the increase of the fractional coefficient K1 could make the ELSC larger, which would move the transition curves to the left. Simultaneously, it could be also found that the increase of the fractional coefficient K1 will make the ELDC lager, which could move the transition curves to the upwards and make the unstable region smaller. Therefore, it could be concluded that the fractional coefficient K1 has also important influence on the stability boundaries.

5. Conclusions

The dynamical characteristics of Mathieu equation with fractional-order derivative is studied by the Lindstedt-Poincare method and the multiple-scale method. The stability boundaries and the corresponding periodic solutions on these boundaries for the constant stiffness δ0=n2 (n= 0, 1, 2, …), are analytically obtained. The effects of the fractional-order parameters on the stability boundaries and the corresponding periodic solutions, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient (ELDC) and the equivalent linear stiffness coefficient (ELSC). The comparisons between the transition curves on the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. It has been illustrated that the different fractional-order parameters have important effects on the stability boundaries. These results are very helpful to design, analyze or control this kind of system, and could present beneficial reference to the similar fractional-order system.

References

  • Miller K. S., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
  • Podlubny I. Fractional Differential Equations. Academic, London, 1998.
  • Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
  • Das S. Functional Fractional Calculus for System Identification and Controls. Springer-Verlag, Berlin, 2008.
  • Caponetto R., Dongola G., Fortuna L., Petras I. Fractional Order Systems: Modeling and Control Applications. World Scientific, New Jersey, 2010.
  • Monje C. A., Chen Y. Q., Vinagre B. M., Xue D. Y., Feliu V. Fractional-order Systems and Controls: Fundamentals and Applications. Springer-Verlag, London, 2010.
  • Petras I. Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press, Beijing, 2011.
  • Shen Y. J., Yang S. P., Xing H. J., Gao G. S. Primary resonance of Duffing oscillator with fractional-order derivative. Communications in Nonlinear Science and Numerical Simulation, Vol. 17, Issue 7, 2012, p. 3092-3100.
  • Shen Y. J., Yang S. P., Xing H. J., et al. Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives. International Journal of Non-Linear Mechanics, Vol. 47, Issue 9, 2012, p. 975-983.
  • Shen Y. J., Wei P., Yang S. P. Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dynamics, Vol. 77, Issue 4, 2014, p. 1629-1642.
  • Shen Y. J., Yang S. P., Sui C. Y. Analysis on limit cycle of fractional-order van der Pol oscillator. Chaos, Solitons & Fractals, Vol. 67, 2014, p. 94-102.
  • Gorenflo R., Abdel-Rehim E. A. Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. Journal of Computational and Applied Mathematics, Vol. 205, Issue 2, 2007, p. 871-881.
  • Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of non differentiable functions further results. Computers and Mathematics with Applications, Vol. 51, Issue 9-10, 2006, p. 1367-1376.
  • Ishteva M., Scherer R., Boyadjiev L. On the Caputo operator of fractional calculus and C-Laguerre functions. Mathematical Sciences Research Journal, Vol. 9, Issue 6, 2005, p. 161-170.
  • Agnieszka B. Malinowska, Delfim F. M. Torres Fractional calculus of variations for a combined Caputo derivative. Fractional Calculus and Applied Analysis, Vol. 14, Issue 4, 2011, p. 523-537.
  • Chen L. C., Zhu W. Q. Stochastic dynamics and fractional optimal control of quasi-integrable Hamiltonian systems with fractional derivative damping. Fractional Calculus and Applied Analysis, Vol. 16, Issue 1, 2013, p. 189-225.
  • Chen L. C., Zhu W. Q. The first passage failure of SDOF strongly nonlinear stochastic system with fractional derivative damping. Journal of Vibration Control, Vol. 15, Issue 8, 2009, p. 1247-1266.
  • Chen L. C., Zhu W. Q. Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations. International Journal of Non-linear Mechanics, Vol. 46, Issue 12, 2011, p. 1324-1329.
  • Chen L. C., Li H. F., Li Z. S., Zhu W. Q. First passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative. Journal of Vibration and Control, Vol. 19, Issue 14, 2013, p. 2154-2163.
  • Chen L. C., Wang W. H., Li Z. S., Zhu W. Q. Stationary response of Duffing oscillator with hardening stiffness and fractional derivative. International Journal of Non-linear Mechanics, Vol. 48, Issue 1, 2013, p. 44-50.
  • Wang Z. H., Hu H. Y. Stability of a linear oscillator with damping force of fractional-order derivative. Science in China G: Physics, Mechanics and Astronomy, Vol. 39, Issue 10, 2009, p. 1495-1502.
  • Wang Z. H., Du M. L. Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system. Shock and Vibration, Vol. 18, Issues 1-2, 2011, p. 257-268.
  • Li C. P., Deng W. H. Remarks on fractional derivatives. Applied Mathematics and Computation, Vol. 187, Issue 2, 2007, p. 777-784.
  • Cao J. X., Ding H. F., Li C. P. Implicit difference schemes for fractional diffusion equations. Communication on Applied Mathematics and Computation, Vol. 27, Issue 1, 2013, p. 61-74.
  • Zeng F. H., Li C. P. High-order finite difference methods for time-fractional subdiffusion equation. Chinese Journal of Computational Physics, Vol. 30, Issue 4, 2013, p. 491-500.
  • Chen A., Li C. P. Numerical algorithm for fractional calculus based on Chebyshev polynomial approximation. Journal of Shanghai Jiaotong University, Vol. 18, Issue 1, 2012, p. 48-53.
  • Wahi P., Chatterjee A. Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dynamics, Vol. 38, Issue 1-4, 2004, p. 3-22.
  • Xu Y., Li Y. G., Liu D., Jia W. T., Huang H. Responses of Duffing oscillator with fractional damping and random phase. Nonlinear Dynamics, Vol. 74, Issue 3, 2013, p. 745-753.
  • Mclachlan N. W. Theory and Application of Mathieu Functions. Oxford University Press, London, 1951.
  • Ge Z. M., Yi C. X. Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems. Chaos, Solitons & Fractals, Vol. 32, Issue 1, 2007, p. 42-61.
  • Ebaid A., ElSayed D. M. M., Aljoufi M. D. Fractional calculus model for damped Mathieu equation: approximate analytical solution. Applied Mathematical Sciences, Vol. 6, Issue 82, 2012, p. 4075-4080.
  • Rand R. H., Sah S. M., Suchorsky M. K. Fractional Mathieu equation. Communications in Nonlinear Science and Numerical Simulation, Vol. 15, Issue 11, 2010, p. 3254-3262.
  • Leung A. Y. T., Guo Z. J., Yang H. X. Transition curves and bifurcations of a class of fractional Mathieu-type equations. International Journal of Bifurcation and Chaos, Vol. 22, 2012, p. 1-13.
  • Nayfeh A. H. Nonlinear Oscillations. Wiley, New York, 1979.
  • Nayfeh A. H. Introduction to Perturbation Techniques. Wiley, New York, 1981.
  • Rossikhin Y. A., Shitikova M. V. On fallacies in the decision between the Caputo and Riemann-Liouville fractional derivative for the analysis of the dynamic response of a nonlinear viscoelastic oscillator. Mechanics Research Communications, Vol. 45, 2012, p. 22-27.

About this article

Received
01 February 2015
Accepted
11 April 2015
Published
15 August 2015
SUBJECTS
Chaos, nonlinear dynamics and applications
Keywords
fractional-order derivative
Mathieu equation
Lindstedt-Poincare method
multiple-scale method
stability boundaries
Acknowledgements

The authors are grateful to the support by National Natural Science Foundation of China (No. 11372198), the Program for New Century Excellent Talents in University (NCET-11-0936), the Cultivation Plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province (LJRC018), the Program for Advanced Talent in the Universities of Hebei Province (GCC2014053), and the Program for Advanced Talent in Hebei (A201401001).