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The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator
Accepted Manuscript Title: The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator Author: T.L.M. Djomo Mbong M. Siewe Siewe C. Tchawoua PII: DOI: Reference:
S0093-6413(16)30243-9 http://dx.doi.org/doi:10.1016/j.mechrescom.2016.10.004 MRC 3121
To appear in: Received date: Revised date: Accepted date:
14-9-2015 18-6-2016 16-10-2016
Please cite this article as: T.L.M. Djomo Mbong, M. Siewe Siewe, C. Tchawoua, The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator, (2016), http://dx.doi.org/10.1016/j.mechrescom.2016.10.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator 1
2
T. L. M. Djomo Mbong1,2 , M. Siewe Siewe1,∗ , C. Tchawoua1
Laboratory of Mechanics, Materials and Structures, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon African Centre of Excellence in Information and Communication Technologies (CETIC), Higher National Polytechnic School, University of Yaounde I, Yaounde, Cameroon ∗ Corresponding author and email:[email protected], [email protected], [email protected]
us
cr
ip t
In this work, we make the general study of the vibratory dynamics of a special fractional quintic oscillator discovered in complex media. First of all, we consider that the system is excited by a combination of both lowfrequency force and high-frequency force. Then, we analyse the occurrence of vibrational resonance, where the response consists of a slow motion and a fast motion respectively with low and high frequencies. Through this, we obtain an approximate analytical expression of the response amplitude and we determine the values of the low frequency and the amplitude of the high-frequency force at which vibrational resonance occurs. The theoretical predictions are found to be in good agreement with numerical results. Moreover, for fixed values of the system parameters, varying the order of the fractional derivatives can introduce new vibrational resonance phenomena. We found that low value of the fractional derivative order favor the occurrence of the first vibrational resonance with cross-well motions. PACS numbers:
0.2
(a)
da x db x + ω02 x + β x3 + γ x5 = f cos ωt + F cos Ωt (2) a +d dt dt b where 1 < a < 2, b = a/2 and ω Ω [20–22]. The potential of the system in the absence of damping and external force is V (x) =
1 2 2 1 4 1 6 ω x + βx + γx 2 0 4 6
(3)
V(x)
0
−0.2
(1)
where f (x(t)) is a linear function of x(t). He showed that the case with 1 < a < 2 corresponds to attenuated oscillation phenomenon while the case with 2 < a < 3 corresponds to amplified oscillation phenomenon. Previous researches on this special oscillator for complex media, were focused on the determination of the unperturbed solution [13–19]. In the present paper, we consider the same system, but disturbed by a biharmonic force. Secondly, we assume that the function f (x(t)) is a nonlinear function in order to have a quintic oscillator [20]. Thus, the classical quintic oscillator becomes a special quintic oscillator written as:
1.5 0.1
−0.1
d
Ac ce pt e
da x db x + f (x(t)) = 0 a +d dt dt b
V(x)
M
In complex media such as glasses, liquid crystals, polymers, and biopolymers, the dynamical variable of interest often obeys fractional differential equations [1–6]. Fractional differential equations have many applications in applied science and engineering [7–11]. Fractional differential equations have been investigated in pure sciences, such as pure mathematics [12]. In [13], A. Tofighi proposed a new damped fractional oscillator of the form
Depending on the values of the system parameters ω02 , β and γ, V (x) can be a double-well or a triple-well potential [20]. For some values, the different shapes of the system potential are given in Fig.1.
an
I. INTRODUCTION
1 0.5 0
−1
0 x
1
−2 −1 (b)
0 x
1
2
FIG. 1: (a) Double-well potential of the quintic oscillator for ω02 = −1.0, β = γ = 1.0 and (b) Triple-well potential for ω02 = 3.0, β = −4.0, γ = 1.0 .
In the system (2), the case where a = 2 has been studied in [20]. It has been shown the multiple occurrences of vibrational resonance phenomenon because of the presence of the biharmonic force. The effect of the nonlinear damping on vibrational resonance has been studied in [23]. This nonlinear phenomenon has also been investigated for fractional differential equations. In [24], Yang and Zhu showed the effect of a fractional damping on vibrational resonance while in [25], J.H. Yang et al. studied vibrational resonance in a Duffing system with a fractional generalized delayed feedback. However, it appears that little attention in the literature has been paid to a resonance analysis of the special fractional quintic oscillator, which is why it is the focus of this paper. The plan of the paper is as follows. In Sec.II, inspired by the theoretical tool in [20, 23], the theoretical analysis for the response amplitude of the output will be carried out for the
Page 1 of 9
2 nonlinear fractional system. According to the analytical results, the impact of the order of the fractional derivative on the vibrational resonance will be discussed in both the doublewell case (Fig.1(a)) and triple-well case (Fig.1(b)). We end in sec.III with conclusion.
where
2 aπ bπ + dΩb cos + ω02 + μ2 = Ωa cos 2 2 2 aπ bπ a b Ω sin + dΩ sin 2 2
and
II. VIBRATIONAL RESONANCE
a
0
d X d X 2 4 X+ + d b + ω0 2 + 3βψav + 5γψav dt a dt 2 3 5 β + 10γψav X 3 + γX 5 + βψav + γψav = f cos ωt (4)
Ac ce pt e
d
b
da ψ db ψ + ω02 ψ + 3βX 2 (ψ − ψav ) + a +d b dt dt 2 3 + β ψ 3 − ψav + 5γX 4 (ψ − ψav ) + 3βX ψ 2 − ψav 2 3 10γX 3 ψ 2 − ψav + 10γX 2 ψ 3 − ψav + 4 5 4 5 5γX ψ − ψav + γ ψ − ψav 2 = F cos Ωt. (5) +10γX 2 ψav
2π 0
ψ i dτ with i = 1, 2, 3, 4, 5.
We seek the approximate solution of ψ in the linear equation a
+ dΩ sin bπ b + dΩ cos 2
(9) +
ω02
Then by substituting the expression of ψ in the equation of the slow motion, we obtain
cr
where
C1 = ω02 +
3βF 2 15γF 4 + , 2μ2 8μ4
C2 = β +
5γF 2 μ2
(11)
Eq.(10) can be used like the equation of the motion of a particle plunged in an effective potential defined by: Vef f (X) =
1 1 1 C1 X 2 + C2 X 4 + γX 6 . 2 4 6
(12)
The equilibrium points of this effective potential are given by: −C2 + C22 − 4C1 γ ∗ ∗ X1 = 0, X2,3 = ± , 2γ −C2 − C22 − 4C1 γ ∗ X4,5 = ± (13) 2γ
M
By substituting x = X + ψ in Eq.(2), we respectively obtain the equations of slow motion and fast motion given by:
1 2π
2
us
The system is subjected to two periodic forces f cos ωt and g cos Ωt with Ω ω, so its movement will be a combination of a slow motion X(t) with frequency ω and of a fast motion ψ(t, τ = Ωt) with frequency Ω. We express the average value 2π 1 of the variable of the fast motion as ψav = 2π ψdτ = 0.
i where ψav =
cos
2 aπ
bπ 2
da X db X + C1 X + C2 X 3 + γX 5 = f cos ωt (10) a +d dt dt b
Theoretical description of vibrational resonance
a
Ωa
an
A.
Ω sin
φ = − tan−1
b
ip t
The phenomenon of vibrational resonance possibly appears when a system is excited by two periodic signals in which one frequency is very large compared to the other signal. The dynamic system that we study in this paper meets all these conditions. Thus, vibrational resonance can occur in our system.
aπ
(8)
b
d ψ d ψ + d b + ω02 ψ = F cos Ωt. dt a dt
(6)
da Y db Y + α1 Y + α2 Y 2 + α3 Y 3 + α4 Y 4 a +d dt dt b +γY 5 = f cos ωt
(14)
where
By solving this equation, we obtain F ψ = cos (Ωt + φ) . μ
These equilibrium points depend on the control parameters F , a and b. They will vary according to the values of those parameters and the effective potential will change. V (x) is (i) a double-well potential for ω02 < 0, γ > 0, β-arbitrary or ω02 < 0, γ < 0, β > 0 with β 2 > 4ω02 γ and (ii) a triplewell potential for ω02 > 0, γ > 0, β < 0 with β 2 > 4ω02 γ. In the same way, Vef f (x) is (i) a double-well potential for C1 < 0, γ > 0, C2 -arbitrary or C1 < 0, γ < 0, C2 > 0 with C22 > 4C1 γ and (ii) a triple-well potential for C1 > 0, γ > 0, C2 < 0 with C22 > 4C1 γ. To perturb the system in its slow motion, we express Y = X−X ∗ and substitute this in Eq.(10) to obtain:
(7)
α1 = C1 + 3C2 X ∗ 2 + 5γX ∗ 4 , α3 = C2 + 10γX ∗ 2 , α2 = 3C2 X ∗ + 10γX ∗ 3 , α4 = 5γX ∗ . (15)
Page 2 of 9
3 For f 1, we suppose that |Y | 1 and we neglect the nonlinear terms in the Eq.(14). Thus when t → ∞, yield Y (t) = AL cos (ωt + ϕ) and we can define the response amplitude of the system as: 0
1 AL =√ f S
F
Q=
80
40
(16)
where S=
ωr2
0 0.5
2 bπ + dω cos + ω cos 2 2 2 aπ bπ + dω b sin + ω a sin 2 2 a
aπ
b
0.75
1
b
FIG. 2: Evolution of the bifurcation value with the order of the fractional derivative.
√ and ωr = α1 is the natural frequency of the linear version of the equation of motion of slow motion in the absence of the external force f cos ωt (it is called resonant frequency of the low-frequency oscillation). We see that the response amplitude depend on a and b representing the order of the fractional derivatives.
a monostable system, for slow motion, occurs for high value of the high-frequency force amplitude.
cr
us
Ac ce pt e
d
M
In this section, we analyze the vibrational resonance in the different shape of the system potential V (x). From the theoretical expression of Q given by Eq.(16), we can determine the values of the control parameter (F ,ω)at which vibrational resonance occurs (FV R ,ωV R ). Resonance will occur when Q will thus take its maximum value (Qmax ), so when S will thus take its minimum value under the con d2 S
dition SF F |F =FV R = > 0. (for FV R ) and dF 2 F =FV R
d2 S
> 0. (for ωV R ) Sωω |ω=ωV R = dω 2 ω=ωV R
From the theoretical expression of Q given by Eq.(16), we can determine the values of the control parameters (F , ω) at which vibrational resonance occurs (FV R , ωV R ). By regarding ω as the first control parameter, in order to obtain the lowfrequency at which vibrational resonance occurs (ωV R ), we solve the nonlinear equation Sω = 0 given by:
an
B. Resonance with the potential V (x)
ip t
(17)
1.
The double-well case
This case occurs when ω02 < 0, γ > 0 and β-arbitrary. Since the phenomenon of resonance will be strongly depend on the form of the effective potential, it is necessary to emphasize the conditions on the form of this potential. Thus we express: −β + β 2 + (10γ |ω02 | /3) F0 = μ (18) 5γ/2
aπ
+ dbω b cos
(19)
By taking account the fact that a = 2b and by using the Cardan method for cubic equation [26], we obtain: - For δ > 0:
(1)
ωV R
√ 13 √ 13 bπ −q + δ −q − δ d b = + − cos 2 2 2 2 (20)
- For δ < 0, the low-frequency at which vibrational resonance occurs is the positive one between the expressions below :
F0 , Vef f is a double well potential in X2∗ and X3∗ > F0 , Vef f is a single-well potential at X1∗ . There-
For F < and for F fore, F = F0 is a bifurcation point which makes the system (10) transit from a bistable to a monostable system. The evolution of this bifurcation point with the order of the fractional derivative is given in Fig.2. From this figure, we see that as the fractional derivative order a increases, the bifurcation point F0 also increases. Thus, for high value of the order of the fractional derivative a, the transition from a bistable to
bπ ωr2 + aω 2a 2 2 (a − b) π 2 2b a+b +d bω + d (a + b) ω cos = 0. 2 aω a cos
1 −p cos arccos (η) − 3 3
1 2π −p b cos arccos (η) + = 2 − 3 3 3
1 4π −p b cos arccos (η) + = 2 − 3 3 3 (2) ωV R
(3) ωV R (4) ωV R
=
b
2
d cos 2 d cos 2 d cos 2
bπ 2 bπ 2 bπ 2
(21)
Page 3 of 9
4
1.5
1
d=0.5 d=1.0 d=1.5
1 a=1.9
(a)
50
F
100
0 1
150 (b)
Q
(a)
1.5 a
2
1
d=0.6 1.5 a
0 1
4 (b)
2
ip t
FIG. 4: (a) Evolution of the response amplitude Q versus lowfrequency ω with d = 0.5 and for different values of the order of the fractional derivative a; (b) Evolution of the response amplitude maximum Qmax (ωV R ) versus the order of the fractional derivative a for different values of damping d.
with 2 Qs = nT
nT
x(t) sin(ωt)dt, 0
2 Qc = nT
nT x(t) cos(ωt)dt. 0
(24) where T = 2π/ω and n = 200; x(t) is given from Eq.(2). From this figure, we see that analytical results are in good agreement with numerical simulations. Thereafter, we take F like control parameter. For F > F0 , Vef f is a single-well potential and ωr2 = C1 . In this case, FV R is respectively given by: −β + β 2 − (10γ (ω02 − ω 2 ) /3) FV R = μ . (25) 5γ/2
M
0 0
2 ω
d
FIG. 3: Evolution of the low-frequency at resonance ωV R : (a) versus F with d = 0.5 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with F = 7.0 and for different values of damping d.
Ac ce pt e
In Fig.3(b), for F = 7.0, we show the effect of the order of the fractional derivative a on ωV R for different values of the linear damping coefficient d. From this figure, it is clear that for weak value of a (near the value 1), ωV R does’nt exist, so vibrational resonance does’nt occurs while for high value of a (near the value 2), as this parameter increases, ωV R also increases. Globally, whatever the value of the order of the fractional derivative vibrational resonance phenomenon occurs at most one time. From Eq.(16), we can plot the analytical curves of the response amplitude Q of the system (2) and results are given in Fig.4. From Fig.4(a), we notice that we have only one peak on each curve, thus, we rejoin the previous idea by saying that there is occurrence of one vibrational resonance. In the same figure, we see that as the order of the fractional derivative increases, the response amplitude increases near the resonances but decreases far from them. This result is confirm by Fig.4(b), where we plot the response amplitude at resonance versus a. In Fig.4(a), continuous curves are obtained by analytical results while the dotted ones are obtained by numerical simulations using the formula [23–25]: Q = Q2s + Q2c /f, (23)
For F < F0 , Vef f is a double-well potential and ωr2 = α1 . In this case, it is very difficult to obtain an analytical expression of FV R . Therefore, we determine the value of F at resonance numerically. Because of higher nonlinearities, FV R is determined by maximizing Q numerically [27] . Thus, we obtain values of FV R for various values of ω ranging between [0.0;2.0] and a between [1.0;2.0]. Figs.5(a)-(b) are given. 90
120 a=2.0 60 a=1.9 0 0 (a)
d=1.7 FVR
a=2.0
0 0
d=0.3 d=0.4 d=0.5
an
VR
2
ω
ωVR
a=1.8
0.5
FVR
2
3
1
us
In Fig.3(a), we plot the evolution of ωV R versus the highfrequency force amplitude F for different values of the order of the fractional derivative a. We notice that, for each value of a, for low values of F , ωV R decreases and increases for high values of F . Between the two evolutions, there is a gap where vibrational resonance does’nt occurs. For a = 2.0 (case of the classical quintic oscillator), the gap is F ∈ [63.6; 68.1], for a = 1.9, the gap is F ∈ [50.10; 55.22] and for a = 1.8, the gap is F ∈ [39.40; 44.78]. Thus, we conclude that as a decreases, the gap of value of F where vibrational resonance does’nt occurs, are near to zero.
2 a=2.0 a=1.9 a=1.8 a=1.7
Qmax(ωVR)
2 bπ , 2 bπ d 4p3 −q −27 2 , q= cos , δ = q + η= × 3 2 p 27 12 2 bπ . −6ωr2 cos (bπ) + 6ωr2 (22) −3d2 + d cos 2 3d2 d2 − cos p = ωr2 cos (bπ) + 2 4
cr
where
60
d=1.0
30 d=0.3
a=1.8 1 ω
0 1
2 (b)
1.5 a
2
FIG. 5: Evolution of the high-frequency force amplitude at resonance FV R : (a) versus ω with d = 0.5 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with ω = 1.0 and for different values of damping d.
In Fig.5(a), we represent the value of F at which vibrational resonance occurs (FV R ). From this figure, we see that for low
Page 4 of 9
5
an
−3 −1
F
100
0 0
150
20 F
(b)
b
40
Qmax(FVR)
2
1.5
1
0.5 1
(c)
d=0.5 d=0.7 d=0.9 d=1.1
1.5 a
2
FIG. 6: (a)-(b): Evolution of the response amplitude Q versus highfrequency force amplitude F with d = 0.5 and for different values of the order of the fractional derivative a; (c): Evolution of the response amplitude maximum Qmax (FV R ) versus the order of the fractional derivative a for different values of damping d.
the fractional derivative a (i.e. near the value 2.0). For this case, we obtain two maximums with the same magnitude, inducing the occurrence of two vibrational resonances for this range of value of a. In Fig.6(b), we consider the case of weak values of the order of the fractional derivative a (i.e. near the value 1.0). For this case, we obtain only one maximum, in-
(c)
Dbx
0
(b)
0 x
1
0 x
1
3
0
−3 −1
3
−3 −1
1
Dbx
D x
M
0.4
d
(a)
50
a=1.3 a=1.2 a=1.1 a=1.0
Ac ce pt e
0 0
Q
Q
1
0 x
3
0.8 a=2.0 a=1.9 a=1.8 a=1.7
us
b
0
cr
3
(a)
2
ip t
ducing the occurrence of a single vibrational resonances for this range of value of a. For all cases ((a) and (b)), numerical simulations are in agreement with analytical results. In each case, we plot the response amplitude versus F for different values of a. We notice that as the order of the fractional derivative increases, the response amplitude increases. This result is confirm in Fig.6(c), where we plot the maximum value of Q with a. On this figure, we see the increases of the response amplitude with the increases of the order of the fractional derivative. We compare the change in the actual motion x(t). We fix F = 10; Note that, this value is near and less than the value of FV R at the first resonance whatever the value of a. For this value of F , we plot in Fig.7, the phase portraits of the actual motion for four values of a (i.e. a = 1.00, 1.34, 1.56, 2.00).
D x
values of ω (i.e. between 0 and 1.8), there is the existence of two values of FV R for a single value of ω, while for high value of this one (i.e. greater 1.8), there is the existence of only one value of FV R for a single value of ω. Therefore, there is occurrence of two resonances for low values of ω while for high values of ω, there is occurrence of only one resonance. On the same figure, we see that as a increases the values of FV R also increase. This result is confirm in Fig.5(b), where we plot FV R versus a. We see on this figure that as a increases, FV R also increases. On the same curve, there is an important result; that is, the order of the fractional derivative brings a new resonance for some values. We see that for low value of this parameter, we have one resonance while, as a continues to increases a new resonance occurs and we globally have apparition of two vibrational resonances phenomena due to the existence of two values of FV R for a single fractional derivative order given. Always in Fig.5(b), we remark that the linear damping does’nt have an effect on the values of FV R in the case of classical quintic oscillator (i.e. a = 2.0) because whatever the value of d, all curves intersect at a = 2.0. It is not the case when a = 2.0. For this case, we see that as d increases, the value of F at resonance also increases. We confirm all these results by plotting both analytical and numerical curves of the response amplitude Q versus F in Fig.6. In Fig.6(a), we consider high values of the order of
0 x
0
−3 −1
1 (d)
FIG. 7: Phase portraits of actual motion with F = 10.0. Initial conditions are taken in the left-well potential (i.e. (x(0), Db x(0)) = (−0.78615, 0.0)). (a): For b = 0.5; (b): For b = 0.67; (c): For b = 0.78 and (d): For b = 1.0.
On each figure, the vertical straight line, is the location of the double-well potential barrier. For b = 0.5 (Fig.7(a)), the orbit is expanded in the two regions (x < 0 and x > 0) separated by the double-well potential barrier, inducing the presence of a cross-well motion. Therefore, for this value of a, the first vibrational resonance will occur with a cross-well motion. In Fig.7(b)-(d), respectively for b = 0.67, 0.78, 1.0, the size of the orbits decreases and are finally located only in region x < 0 inducing the absence of the cross-well motion. Thus, we conclude that, for weak value of the fractional derivative order, there is occurrence of the first vibrational resonance with cross-well motion while, for high value of this parameter, the first resonance occurs without the cross-well motion. For a best comprehension of the notion of cross-well motion in this espacial quintic oscillator, we plot the average time spend by each orbit in the region x < 0 before cross to the re-
Page 5 of 9
6 gion x > 0. This time is called the Mean Residence Time (τM R ). In Fig.8, we give the evolution of τM R with the fractional derivative order a. From this figure, we notice that the
150 (1) 0
F0 , F0
(2)
F
(2)
(1)
F0
50
0.25
τ
MR
0.5
100
0 1
ip t
cr
4
d
The triple-well case
Ac ce pt e
This case occurs when ω02 > 0, γ > 0, β < 0 with β 2 > 4ω02 γ. Since the phenomenon of resonance will be strongly depend on the form of the effective potential, it is necessary to emphasize the conditions on the form of this potential for the triple-well case. Thus, we express:
(1,2) F0
=μ
−β ∓
β 2 − (10γω02 /3) 5γ/2
(26)
(1)
∗ For F < F0 , Vef f is a triple-well potential in X1,2,3 ; for (1)
(2)
∗ F ∈ [F0 ; F0 ], Vef f is a double-well potential in X2,3 and (2)
for F > F0 , Vef f is a single-well potential at X1∗ . There(1) (2) fore, F = F0 and F = F0 are bifurcation points which make the system (10) transit from a tristable to a bistable and from a bistable to a monostable system respectively. The evolution of these bifurcation points with the order of the fractional derivative is given in Fig.9. From this figure, we see that as the fractional derivative order a increases, the bifurca(1,2) tion points F = F0 also increase. Thus, for high value of the fractional derivative order a, the transition from tristable to a bistable or from a bistable to a monostable system for
4
a=2.0
M (a)
VR
a=1.9
2
0 0
ω
times spend by the orbit in each well are equal for low value of the fractional derivative order a (for a ∈ [1.00; 1.15]). For 1.15 < a < 1.60, the orbit spend much more time in the leftwell than in the right-well because the mean residence time in L the left-well (τM R ) is greater than the mean residence time in R the right-well (τM R ); Thus, for 1.00 < a < 1.60 it a crosswell motion phenomenon. However, at a > 1.6, the orbit R L is entire in the left-well because τM R is zero while τM R is different to zero: there is no occurrence of a cross-well motion phenomenon. Thus, according to the value of the fractional derivative order a, there is appearance of vibrational resonance with or without a cross-well motion.
slow motion, occurs for high amplitudes of the high-frequency force. As in the case of the double-well, we plot the evolution of low-frequency at resonance ωV R versus the high-frequency force amplitude and versus the fractional derivative order for the triple-well potential. Results are given in Fig.10.
us
FIG. 8: Mean Residence Time (τM R ) versus a. Red color is the mean residence time of the orbit in the left well while the blue color is the mean residence time of the orbit in the right well. Initial conditions are (x(0) = −0.78615, Db x(0) = 0.0).
2.
2
FIG. 9: Evolution of the bifurcation values with the fractional derivative order.
1.6
a
an
1.3
ωVR
0 1
1.5 a
d=0.5 d=1.0 d=1.5
a=1.8
100 F
2
0 1
200 (b)
1.5 a
2
FIG. 10: Evolution of the low-frequency at resonance ωV R : (a) versus F with d = 0.5 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with F = 7.0 and for different values of damping d.
In Fig.10(a), we plot the evolution of ωV R versus the highfrequency force amplitude F for different values of the order of the fractional derivative a. We notice that, as in the double-well case, there is a gap where vibrational resonance does’nt occurs. For a = 2.0 (case of the classical quintic oscillator), the gap is F ∈ [145; 150.6], for a = 1.9, the gap is F ∈ [113.3; 119.6] and for a = 1.8, the gap is F ∈ [88.7; 95.3]. Thus, we conclude that as, a decreases, the gap of value of F where vibrational resonance does’nt occurs, are near to zero. In Fig.10(b), we show the effect of the order of the fractional derivative a on ωV R for different value of the linear damping coefficient d. From this figure, it is clear that for weak value of a (near the value 1), ωV R does’nt exist, so vibrational resonance does’nt occurs while for high value of a (near the value 2), as this parameter increases, ωV R also increases. Globally, whatever the value of the order of the fractional derivative vibrational resonance phenomenon occurs at most one time. We can confirm these results by plotting the response amplitude of the system. The evolution of this amplitude versus the low-frequency is given in Fig.11(a). From this figure, we
Page 6 of 9
7
50
0 1 (a)
2.5 ω
0 1
5 (b)
1
(b)
d
2
3
ip t
cr
us
Q
d=0.5 d=0.8 d=1.2
0.35
0 0
1.5 a
2
a=1.4 80 F
160
FIG. 13: Response amplitude of the system versus F for the triplewell case.
M
(a)
0.4
0 0
2
a=1.9
an
0 0
Qmax
Q
0.4
1.5 a
40
also increases. This result is shown in Fig.12(b). The appari-
0.8
a=2.00 a=1.95 a=1.90
a=1.5 a=1.1 a=1.3
FIG. 12: Evolution of the high-frequency force amplitude at resonance FV R : (a) versus a with ω = 1.0 and for different values of the damping d; (b) the damping d for different values of the order of the fractional derivative a.
0.7
0.8
80
d=3.0 d=1.5 d=0.3
FVR
100
FVR
notice the presence of one maximum for each curve. Therefore, the system exhibit at most one resonance. We also remark that as the fractional derivative order decreases, the amplitude of the system also decreases. Thus, we can predict that for certain values of a, the response amplitude will not have a local maximum. From this figure, the dashed curves represent the numerical curve while the curve with straight line is for analytical result. We see that there is a good agreement between the two analysis. Fig.11(b) presents the same behavior. From this figure, we plot the maximum value of the response amplitude (at resonance) versus the fractional derivative order a. We notice that, for low value of this coefficient, the maximum value of the response amplitude does’nt exist; so, there is not an occurrence of vibrational resonance phenomenon. For high value of a, Qmax meaning of occurrence of the vibrational resonance phenomenon. We can say that, for this range of value, as a increases, as vibrational resonance will occurs with high amplitude (because Qmax increases).
d
FIG. 11: Evolution of the low-frequency at resonance ωV R : (a) versus F with d = 1.0 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with F = 7.0 and for different values of damping d.
Ac ce pt e
Thereafter, we take F like control parameter. For F > (2) F0 , Vef f is a single-well potential and ωr2 = C1 . In this (1) (2) case, FV R is given by Eq.(25). For F0 < F < F0 , Vef f (1) is a double-well potential and for F < F0 , Vef f is a triple2 well potential. For these cases, ωr = α1 and it is very difficult to obtain an analytical expression of FV R . Therefore, we determine the value of F at resonance numerically. Because of higher nonlinearities, FV R is determined by maximizing Q numerically. Thus, we obtain values of FV R for various values of a ranging between [1.0;2.0] and d between [0.0;3.0]. Figs.12(a)-(b) are given. In Fig.12(a), where we plot FV R versus a for different values of the damping coefficient, we see that as a increases, FV R also increases. On the same curve, there is an important result; that is, the order of the fractional derivative brings a new resonance for some values. For low values of this parameter, we have two resonances (i.e. two values of FV R ). As a continues to increases a new resonance occurs and we globally have apparition of three vibrational resonances phenomena due to the existence of three values of FV R for a single fractional derivative order given. Always in Fig.12(a), we remark that the linear damping have an effect on the values of FV R . We see that as d increases, the value of F at resonance
tion of a new vibrational resonance is confirm in Fig.13. We see that by representing the response amplitude of the system for a = 1.4, we have two maximum; synonym of two peaks of resonance. When a = 1.9, the curve in red has three maximum inducing three occurrences of resonance. Globally, as in the case of the double-well potential, in the triple-well, the fractional derivative order can bring a new phenomenon of vibrational resonance.
III.
CONCLUSION
In this work, we made the general study of the vibratory dynamics of a special fractional quintic oscillator discovered in complex media. For the quintic form of the restoring force, we showed that this media can be a bistable or a tristable system. Then, we supposed that natural forces in this type of model could be regarded as the combination of two periodic forces of high and low frequency which play the role of forcing of the system. The excitation by these two external periodic forces f cosωt and gcosΩt with Ω ω allowed us to study vibrational resonance. We noticed that, according to the values of the parameters of control, resonance could occur at most two times for a double-well potential and at most three times for a triple-well potential. We showed that the fractional derivative order has an important role in this phenomenon because, according the value of this parameter, vibrational resonance
Page 7 of 9
8 system. Thus, the mechanism of vibrational resonance in this special oscillator considered in this paper is quite different from that of the classical quintic oscillator.
[1] K.B. Oldham and J. Spanier, ”The fractional Calculus”, Academic Press, New York, NY, USA, (1974). [2] I. Podlubny, ”Fractional Differential Equations”, Academic Press, San Diego Calif, USA, (1999). [3] R. Hilfer, ”Application of Fractional Calculus in Physics”, World Scientific, Singapore, (2000) [4] B. J. West, M. Bologna, and P. Grigolini, ”Physics of Fractal Operators”, Springer, NewYork, NY, USA, (2003). [5] G. M. Zaslavski, ”Hamiltonian Chaos and Fractional Dynamics”, Oxford University Press, Oxford, UK, (2005). [6] R. Metzler and J. Klafter,”Therandomwalksguidetoanomalousdiffusion:afractionaldynamicsapproach”, Physics Report 339, 1-77, (2000). [7] R.Metzler, E. Barkai and J.Klafter, ”Anomalousdiffusionand relaxation close to thermal equilibrium: a fractional FokkerPlanckequationapproach, Physical Review Letters 82, 35633567, (1999). [8] G. Failla and A. Pirrotta, ”On the stochastic response of a fractionally damped Duffing oscillator”, Communications in Nonlinear Science and Numerical Simulation 17, 5131-5142, (2012). [9] M. DiPaola, A. Pirrotta and A. Valenza, ”Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results”, Mechanics of Materials 43, 799-806, (2011). [10] C. Celauro, C. Fecarotti, A. Pirrotta and A. C. Collop, ”Experimental validation of a fractional model for creep/recovery testing of asphalt mixtures”, Construction and Building Materials 36, 458-466, (2012). [11] Y. Li and Y. Chen, ”When is a Mittag-Leffler function a Nussbaum function”, Automatica 45, 1957-1959, (2009). [12] J. R. Wang, L. Lv and W. Wei, ”Differential equations of fractional order α ∈ (2, 3) with boundary value conditions inabstractBanachspaces”, Mathematical Communications 17, 371387, (2012). [13] A. Tofighi, ”An Especial Fractional Oscillator”, International Journal of Statistical Mechanics 2013, 1-5, (2013). [14] R. Gorenflo and F. Mainardi, ”Fractional calculus: integral and differential equations of fractional order”, Springer, New York, NY, USA, (1997).
[15] B. N. N. Achar, J. W. Hanneken, T. Enck and T. Clarke, ”Dynamics of the fractional oscillator”, Physica A 297, 361367, (2001). [16] B. N. N. Achar, J. W. Hanneken, and T. Clarke, ”Damping characteristicsofafractionaloscillator”, Physica A 339, 311319, (2004). [17] V. E. Tarasov and G. M. Zaslavsky, ”Dynamics with low-level fractionality”, Physica A 368, 399415, (2006). [18] A. Tofighi, ”The intrinsic damping of the fractional oscillator”, Physica A 329, 2934, (2003). [19] A. Tofighi and H. N. Pour, ”ε-expansion and the fractional oscillator, Physica A 374, 4145, (2007). [20] S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A. F. Sanjuan, ”Analysis of vibrational resonance in a quintic oscillator”, Chaos 19, 043128 (2009). [21] K. Abirami, S. Rajasekar and M. A. F. Sanju´an, ”Vibrational resonance in the Morse oscillator”, Pramana-Journal of Physics 81, 127-141, (2013). [22] S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A. F. Sanjuan, ”Vibrational resonance in an asymmetric Duffing oscillator”, Int. J. Bifurcat. Chaos 21, 275-286, (2011). [23] T. L. M. Djomo Mbong, M. Siewe Siewe and C. Tchawoua, ”The effect of nonlinear damping on vibrational resonance and chaotic behavior of a beam fixed at its two ends and prestressed”, Communications in Nonlinear Science and Numerical Simulation 22, 228243, (2015). [24] J. H. Yang and H. Zhu, ”Vibrational resonance in Duffing systems with fractional-order damping”, Chaos 22, 013112, (2012). [25] J. H. Yang, M. A. F. Sanjuan, C. J. Wang and H. Zhu, ”Vibrational Resonance in a Du?ng System with a Generalized Delayed Feedback”, Journal of Applied Nonlinear Dynamic 2, 397408, (2013). [26] W. Dunham, ”Cardano and the solution of the cubic”, Ch.6 in Journey through Genius: The Great Theorems od Mathematics, New-York, Wiley, 133-154, (1990). [27] J. D. Hoffman, ”Numerical methods for engeneers and scientists”, Marcel Dekker, Second edition, (2001).
Ac ce pt e
d
M
an
us
cr
ip t
does not occur or does occur one/two times for the doublewell case or two/three times for the triple-well case. Globally, we concluded that the presence of the fractional derivative order coefficient can introduce new resonance phenomena in the
Page 8 of 9
Highlights
• We present an especial fractional quintic oscillator
ip t
• For high values of the fractional derivative order, an additional resonance occurs • Fractional derivative order increases the high-frequency amplitude at the resonance
Ac
ce p
te
d
M
an
us
cr
• The damping has an effect on the high-frequency force amplitude at the resonance
Page 9 of 9
S0093-6413(16)30243-9 http://dx.doi.org/doi:10.1016/j.mechrescom.2016.10.004 MRC 3121
To appear in: Received date: Revised date: Accepted date:
14-9-2015 18-6-2016 16-10-2016
Please cite this article as: T.L.M. Djomo Mbong, M. Siewe Siewe, C. Tchawoua, The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator, (2016), http://dx.doi.org/10.1016/j.mechrescom.2016.10.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator 1
2
T. L. M. Djomo Mbong1,2 , M. Siewe Siewe1,∗ , C. Tchawoua1
Laboratory of Mechanics, Materials and Structures, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon African Centre of Excellence in Information and Communication Technologies (CETIC), Higher National Polytechnic School, University of Yaounde I, Yaounde, Cameroon ∗ Corresponding author and email:[email protected], [email protected], [email protected]
us
cr
ip t
In this work, we make the general study of the vibratory dynamics of a special fractional quintic oscillator discovered in complex media. First of all, we consider that the system is excited by a combination of both lowfrequency force and high-frequency force. Then, we analyse the occurrence of vibrational resonance, where the response consists of a slow motion and a fast motion respectively with low and high frequencies. Through this, we obtain an approximate analytical expression of the response amplitude and we determine the values of the low frequency and the amplitude of the high-frequency force at which vibrational resonance occurs. The theoretical predictions are found to be in good agreement with numerical results. Moreover, for fixed values of the system parameters, varying the order of the fractional derivatives can introduce new vibrational resonance phenomena. We found that low value of the fractional derivative order favor the occurrence of the first vibrational resonance with cross-well motions. PACS numbers:
0.2
(a)
da x db x + ω02 x + β x3 + γ x5 = f cos ωt + F cos Ωt (2) a +d dt dt b where 1 < a < 2, b = a/2 and ω Ω [20–22]. The potential of the system in the absence of damping and external force is V (x) =
1 2 2 1 4 1 6 ω x + βx + γx 2 0 4 6
(3)
V(x)
0
−0.2
(1)
where f (x(t)) is a linear function of x(t). He showed that the case with 1 < a < 2 corresponds to attenuated oscillation phenomenon while the case with 2 < a < 3 corresponds to amplified oscillation phenomenon. Previous researches on this special oscillator for complex media, were focused on the determination of the unperturbed solution [13–19]. In the present paper, we consider the same system, but disturbed by a biharmonic force. Secondly, we assume that the function f (x(t)) is a nonlinear function in order to have a quintic oscillator [20]. Thus, the classical quintic oscillator becomes a special quintic oscillator written as:
1.5 0.1
−0.1
d
Ac ce pt e
da x db x + f (x(t)) = 0 a +d dt dt b
V(x)
M
In complex media such as glasses, liquid crystals, polymers, and biopolymers, the dynamical variable of interest often obeys fractional differential equations [1–6]. Fractional differential equations have many applications in applied science and engineering [7–11]. Fractional differential equations have been investigated in pure sciences, such as pure mathematics [12]. In [13], A. Tofighi proposed a new damped fractional oscillator of the form
Depending on the values of the system parameters ω02 , β and γ, V (x) can be a double-well or a triple-well potential [20]. For some values, the different shapes of the system potential are given in Fig.1.
an
I. INTRODUCTION
1 0.5 0
−1
0 x
1
−2 −1 (b)
0 x
1
2
FIG. 1: (a) Double-well potential of the quintic oscillator for ω02 = −1.0, β = γ = 1.0 and (b) Triple-well potential for ω02 = 3.0, β = −4.0, γ = 1.0 .
In the system (2), the case where a = 2 has been studied in [20]. It has been shown the multiple occurrences of vibrational resonance phenomenon because of the presence of the biharmonic force. The effect of the nonlinear damping on vibrational resonance has been studied in [23]. This nonlinear phenomenon has also been investigated for fractional differential equations. In [24], Yang and Zhu showed the effect of a fractional damping on vibrational resonance while in [25], J.H. Yang et al. studied vibrational resonance in a Duffing system with a fractional generalized delayed feedback. However, it appears that little attention in the literature has been paid to a resonance analysis of the special fractional quintic oscillator, which is why it is the focus of this paper. The plan of the paper is as follows. In Sec.II, inspired by the theoretical tool in [20, 23], the theoretical analysis for the response amplitude of the output will be carried out for the
Page 1 of 9
2 nonlinear fractional system. According to the analytical results, the impact of the order of the fractional derivative on the vibrational resonance will be discussed in both the doublewell case (Fig.1(a)) and triple-well case (Fig.1(b)). We end in sec.III with conclusion.
where
2 aπ bπ + dΩb cos + ω02 + μ2 = Ωa cos 2 2 2 aπ bπ a b Ω sin + dΩ sin 2 2
and
II. VIBRATIONAL RESONANCE
a
0
d X d X 2 4 X+ + d b + ω0 2 + 3βψav + 5γψav dt a dt 2 3 5 β + 10γψav X 3 + γX 5 + βψav + γψav = f cos ωt (4)
Ac ce pt e
d
b
da ψ db ψ + ω02 ψ + 3βX 2 (ψ − ψav ) + a +d b dt dt 2 3 + β ψ 3 − ψav + 5γX 4 (ψ − ψav ) + 3βX ψ 2 − ψav 2 3 10γX 3 ψ 2 − ψav + 10γX 2 ψ 3 − ψav + 4 5 4 5 5γX ψ − ψav + γ ψ − ψav 2 = F cos Ωt. (5) +10γX 2 ψav
2π 0
ψ i dτ with i = 1, 2, 3, 4, 5.
We seek the approximate solution of ψ in the linear equation a
+ dΩ sin bπ b + dΩ cos 2
(9) +
ω02
Then by substituting the expression of ψ in the equation of the slow motion, we obtain
cr
where
C1 = ω02 +
3βF 2 15γF 4 + , 2μ2 8μ4
C2 = β +
5γF 2 μ2
(11)
Eq.(10) can be used like the equation of the motion of a particle plunged in an effective potential defined by: Vef f (X) =
1 1 1 C1 X 2 + C2 X 4 + γX 6 . 2 4 6
(12)
The equilibrium points of this effective potential are given by: −C2 + C22 − 4C1 γ ∗ ∗ X1 = 0, X2,3 = ± , 2γ −C2 − C22 − 4C1 γ ∗ X4,5 = ± (13) 2γ
M
By substituting x = X + ψ in Eq.(2), we respectively obtain the equations of slow motion and fast motion given by:
1 2π
2
us
The system is subjected to two periodic forces f cos ωt and g cos Ωt with Ω ω, so its movement will be a combination of a slow motion X(t) with frequency ω and of a fast motion ψ(t, τ = Ωt) with frequency Ω. We express the average value 2π 1 of the variable of the fast motion as ψav = 2π ψdτ = 0.
i where ψav =
cos
2 aπ
bπ 2
da X db X + C1 X + C2 X 3 + γX 5 = f cos ωt (10) a +d dt dt b
Theoretical description of vibrational resonance
a
Ωa
an
A.
Ω sin
φ = − tan−1
b
ip t
The phenomenon of vibrational resonance possibly appears when a system is excited by two periodic signals in which one frequency is very large compared to the other signal. The dynamic system that we study in this paper meets all these conditions. Thus, vibrational resonance can occur in our system.
aπ
(8)
b
d ψ d ψ + d b + ω02 ψ = F cos Ωt. dt a dt
(6)
da Y db Y + α1 Y + α2 Y 2 + α3 Y 3 + α4 Y 4 a +d dt dt b +γY 5 = f cos ωt
(14)
where
By solving this equation, we obtain F ψ = cos (Ωt + φ) . μ
These equilibrium points depend on the control parameters F , a and b. They will vary according to the values of those parameters and the effective potential will change. V (x) is (i) a double-well potential for ω02 < 0, γ > 0, β-arbitrary or ω02 < 0, γ < 0, β > 0 with β 2 > 4ω02 γ and (ii) a triplewell potential for ω02 > 0, γ > 0, β < 0 with β 2 > 4ω02 γ. In the same way, Vef f (x) is (i) a double-well potential for C1 < 0, γ > 0, C2 -arbitrary or C1 < 0, γ < 0, C2 > 0 with C22 > 4C1 γ and (ii) a triple-well potential for C1 > 0, γ > 0, C2 < 0 with C22 > 4C1 γ. To perturb the system in its slow motion, we express Y = X−X ∗ and substitute this in Eq.(10) to obtain:
(7)
α1 = C1 + 3C2 X ∗ 2 + 5γX ∗ 4 , α3 = C2 + 10γX ∗ 2 , α2 = 3C2 X ∗ + 10γX ∗ 3 , α4 = 5γX ∗ . (15)
Page 2 of 9
3 For f 1, we suppose that |Y | 1 and we neglect the nonlinear terms in the Eq.(14). Thus when t → ∞, yield Y (t) = AL cos (ωt + ϕ) and we can define the response amplitude of the system as: 0
1 AL =√ f S
F
Q=
80
40
(16)
where S=
ωr2
0 0.5
2 bπ + dω cos + ω cos 2 2 2 aπ bπ + dω b sin + ω a sin 2 2 a
aπ
b
0.75
1
b
FIG. 2: Evolution of the bifurcation value with the order of the fractional derivative.
√ and ωr = α1 is the natural frequency of the linear version of the equation of motion of slow motion in the absence of the external force f cos ωt (it is called resonant frequency of the low-frequency oscillation). We see that the response amplitude depend on a and b representing the order of the fractional derivatives.
a monostable system, for slow motion, occurs for high value of the high-frequency force amplitude.
cr
us
Ac ce pt e
d
M
In this section, we analyze the vibrational resonance in the different shape of the system potential V (x). From the theoretical expression of Q given by Eq.(16), we can determine the values of the control parameter (F ,ω)at which vibrational resonance occurs (FV R ,ωV R ). Resonance will occur when Q will thus take its maximum value (Qmax ), so when S will thus take its minimum value under the con d2 S
dition SF F |F =FV R = > 0. (for FV R ) and dF 2 F =FV R
d2 S
> 0. (for ωV R ) Sωω |ω=ωV R = dω 2 ω=ωV R
From the theoretical expression of Q given by Eq.(16), we can determine the values of the control parameters (F , ω) at which vibrational resonance occurs (FV R , ωV R ). By regarding ω as the first control parameter, in order to obtain the lowfrequency at which vibrational resonance occurs (ωV R ), we solve the nonlinear equation Sω = 0 given by:
an
B. Resonance with the potential V (x)
ip t
(17)
1.
The double-well case
This case occurs when ω02 < 0, γ > 0 and β-arbitrary. Since the phenomenon of resonance will be strongly depend on the form of the effective potential, it is necessary to emphasize the conditions on the form of this potential. Thus we express: −β + β 2 + (10γ |ω02 | /3) F0 = μ (18) 5γ/2
aπ
+ dbω b cos
(19)
By taking account the fact that a = 2b and by using the Cardan method for cubic equation [26], we obtain: - For δ > 0:
(1)
ωV R
√ 13 √ 13 bπ −q + δ −q − δ d b = + − cos 2 2 2 2 (20)
- For δ < 0, the low-frequency at which vibrational resonance occurs is the positive one between the expressions below :
F0 , Vef f is a double well potential in X2∗ and X3∗ > F0 , Vef f is a single-well potential at X1∗ . There-
For F < and for F fore, F = F0 is a bifurcation point which makes the system (10) transit from a bistable to a monostable system. The evolution of this bifurcation point with the order of the fractional derivative is given in Fig.2. From this figure, we see that as the fractional derivative order a increases, the bifurcation point F0 also increases. Thus, for high value of the order of the fractional derivative a, the transition from a bistable to
bπ ωr2 + aω 2a 2 2 (a − b) π 2 2b a+b +d bω + d (a + b) ω cos = 0. 2 aω a cos
1 −p cos arccos (η) − 3 3
1 2π −p b cos arccos (η) + = 2 − 3 3 3
1 4π −p b cos arccos (η) + = 2 − 3 3 3 (2) ωV R
(3) ωV R (4) ωV R
=
b
2
d cos 2 d cos 2 d cos 2
bπ 2 bπ 2 bπ 2
(21)
Page 3 of 9
4
1.5
1
d=0.5 d=1.0 d=1.5
1 a=1.9
(a)
50
F
100
0 1
150 (b)
Q
(a)
1.5 a
2
1
d=0.6 1.5 a
0 1
4 (b)
2
ip t
FIG. 4: (a) Evolution of the response amplitude Q versus lowfrequency ω with d = 0.5 and for different values of the order of the fractional derivative a; (b) Evolution of the response amplitude maximum Qmax (ωV R ) versus the order of the fractional derivative a for different values of damping d.
with 2 Qs = nT
nT
x(t) sin(ωt)dt, 0
2 Qc = nT
nT x(t) cos(ωt)dt. 0
(24) where T = 2π/ω and n = 200; x(t) is given from Eq.(2). From this figure, we see that analytical results are in good agreement with numerical simulations. Thereafter, we take F like control parameter. For F > F0 , Vef f is a single-well potential and ωr2 = C1 . In this case, FV R is respectively given by: −β + β 2 − (10γ (ω02 − ω 2 ) /3) FV R = μ . (25) 5γ/2
M
0 0
2 ω
d
FIG. 3: Evolution of the low-frequency at resonance ωV R : (a) versus F with d = 0.5 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with F = 7.0 and for different values of damping d.
Ac ce pt e
In Fig.3(b), for F = 7.0, we show the effect of the order of the fractional derivative a on ωV R for different values of the linear damping coefficient d. From this figure, it is clear that for weak value of a (near the value 1), ωV R does’nt exist, so vibrational resonance does’nt occurs while for high value of a (near the value 2), as this parameter increases, ωV R also increases. Globally, whatever the value of the order of the fractional derivative vibrational resonance phenomenon occurs at most one time. From Eq.(16), we can plot the analytical curves of the response amplitude Q of the system (2) and results are given in Fig.4. From Fig.4(a), we notice that we have only one peak on each curve, thus, we rejoin the previous idea by saying that there is occurrence of one vibrational resonance. In the same figure, we see that as the order of the fractional derivative increases, the response amplitude increases near the resonances but decreases far from them. This result is confirm by Fig.4(b), where we plot the response amplitude at resonance versus a. In Fig.4(a), continuous curves are obtained by analytical results while the dotted ones are obtained by numerical simulations using the formula [23–25]: Q = Q2s + Q2c /f, (23)
For F < F0 , Vef f is a double-well potential and ωr2 = α1 . In this case, it is very difficult to obtain an analytical expression of FV R . Therefore, we determine the value of F at resonance numerically. Because of higher nonlinearities, FV R is determined by maximizing Q numerically [27] . Thus, we obtain values of FV R for various values of ω ranging between [0.0;2.0] and a between [1.0;2.0]. Figs.5(a)-(b) are given. 90
120 a=2.0 60 a=1.9 0 0 (a)
d=1.7 FVR
a=2.0
0 0
d=0.3 d=0.4 d=0.5
an
VR
2
ω
ωVR
a=1.8
0.5
FVR
2
3
1
us
In Fig.3(a), we plot the evolution of ωV R versus the highfrequency force amplitude F for different values of the order of the fractional derivative a. We notice that, for each value of a, for low values of F , ωV R decreases and increases for high values of F . Between the two evolutions, there is a gap where vibrational resonance does’nt occurs. For a = 2.0 (case of the classical quintic oscillator), the gap is F ∈ [63.6; 68.1], for a = 1.9, the gap is F ∈ [50.10; 55.22] and for a = 1.8, the gap is F ∈ [39.40; 44.78]. Thus, we conclude that as a decreases, the gap of value of F where vibrational resonance does’nt occurs, are near to zero.
2 a=2.0 a=1.9 a=1.8 a=1.7
Qmax(ωVR)
2 bπ , 2 bπ d 4p3 −q −27 2 , q= cos , δ = q + η= × 3 2 p 27 12 2 bπ . −6ωr2 cos (bπ) + 6ωr2 (22) −3d2 + d cos 2 3d2 d2 − cos p = ωr2 cos (bπ) + 2 4
cr
where
60
d=1.0
30 d=0.3
a=1.8 1 ω
0 1
2 (b)
1.5 a
2
FIG. 5: Evolution of the high-frequency force amplitude at resonance FV R : (a) versus ω with d = 0.5 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with ω = 1.0 and for different values of damping d.
In Fig.5(a), we represent the value of F at which vibrational resonance occurs (FV R ). From this figure, we see that for low
Page 4 of 9
5
an
−3 −1
F
100
0 0
150
20 F
(b)
b
40
Qmax(FVR)
2
1.5
1
0.5 1
(c)
d=0.5 d=0.7 d=0.9 d=1.1
1.5 a
2
FIG. 6: (a)-(b): Evolution of the response amplitude Q versus highfrequency force amplitude F with d = 0.5 and for different values of the order of the fractional derivative a; (c): Evolution of the response amplitude maximum Qmax (FV R ) versus the order of the fractional derivative a for different values of damping d.
the fractional derivative a (i.e. near the value 2.0). For this case, we obtain two maximums with the same magnitude, inducing the occurrence of two vibrational resonances for this range of value of a. In Fig.6(b), we consider the case of weak values of the order of the fractional derivative a (i.e. near the value 1.0). For this case, we obtain only one maximum, in-
(c)
Dbx
0
(b)
0 x
1
0 x
1
3
0
−3 −1
3
−3 −1
1
Dbx
D x
M
0.4
d
(a)
50
a=1.3 a=1.2 a=1.1 a=1.0
Ac ce pt e
0 0
Q
Q
1
0 x
3
0.8 a=2.0 a=1.9 a=1.8 a=1.7
us
b
0
cr
3
(a)
2
ip t
ducing the occurrence of a single vibrational resonances for this range of value of a. For all cases ((a) and (b)), numerical simulations are in agreement with analytical results. In each case, we plot the response amplitude versus F for different values of a. We notice that as the order of the fractional derivative increases, the response amplitude increases. This result is confirm in Fig.6(c), where we plot the maximum value of Q with a. On this figure, we see the increases of the response amplitude with the increases of the order of the fractional derivative. We compare the change in the actual motion x(t). We fix F = 10; Note that, this value is near and less than the value of FV R at the first resonance whatever the value of a. For this value of F , we plot in Fig.7, the phase portraits of the actual motion for four values of a (i.e. a = 1.00, 1.34, 1.56, 2.00).
D x
values of ω (i.e. between 0 and 1.8), there is the existence of two values of FV R for a single value of ω, while for high value of this one (i.e. greater 1.8), there is the existence of only one value of FV R for a single value of ω. Therefore, there is occurrence of two resonances for low values of ω while for high values of ω, there is occurrence of only one resonance. On the same figure, we see that as a increases the values of FV R also increase. This result is confirm in Fig.5(b), where we plot FV R versus a. We see on this figure that as a increases, FV R also increases. On the same curve, there is an important result; that is, the order of the fractional derivative brings a new resonance for some values. We see that for low value of this parameter, we have one resonance while, as a continues to increases a new resonance occurs and we globally have apparition of two vibrational resonances phenomena due to the existence of two values of FV R for a single fractional derivative order given. Always in Fig.5(b), we remark that the linear damping does’nt have an effect on the values of FV R in the case of classical quintic oscillator (i.e. a = 2.0) because whatever the value of d, all curves intersect at a = 2.0. It is not the case when a = 2.0. For this case, we see that as d increases, the value of F at resonance also increases. We confirm all these results by plotting both analytical and numerical curves of the response amplitude Q versus F in Fig.6. In Fig.6(a), we consider high values of the order of
0 x
0
−3 −1
1 (d)
FIG. 7: Phase portraits of actual motion with F = 10.0. Initial conditions are taken in the left-well potential (i.e. (x(0), Db x(0)) = (−0.78615, 0.0)). (a): For b = 0.5; (b): For b = 0.67; (c): For b = 0.78 and (d): For b = 1.0.
On each figure, the vertical straight line, is the location of the double-well potential barrier. For b = 0.5 (Fig.7(a)), the orbit is expanded in the two regions (x < 0 and x > 0) separated by the double-well potential barrier, inducing the presence of a cross-well motion. Therefore, for this value of a, the first vibrational resonance will occur with a cross-well motion. In Fig.7(b)-(d), respectively for b = 0.67, 0.78, 1.0, the size of the orbits decreases and are finally located only in region x < 0 inducing the absence of the cross-well motion. Thus, we conclude that, for weak value of the fractional derivative order, there is occurrence of the first vibrational resonance with cross-well motion while, for high value of this parameter, the first resonance occurs without the cross-well motion. For a best comprehension of the notion of cross-well motion in this espacial quintic oscillator, we plot the average time spend by each orbit in the region x < 0 before cross to the re-
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6 gion x > 0. This time is called the Mean Residence Time (τM R ). In Fig.8, we give the evolution of τM R with the fractional derivative order a. From this figure, we notice that the
150 (1) 0
F0 , F0
(2)
F
(2)
(1)
F0
50
0.25
τ
MR
0.5
100
0 1
ip t
cr
4
d
The triple-well case
Ac ce pt e
This case occurs when ω02 > 0, γ > 0, β < 0 with β 2 > 4ω02 γ. Since the phenomenon of resonance will be strongly depend on the form of the effective potential, it is necessary to emphasize the conditions on the form of this potential for the triple-well case. Thus, we express:
(1,2) F0
=μ
−β ∓
β 2 − (10γω02 /3) 5γ/2
(26)
(1)
∗ For F < F0 , Vef f is a triple-well potential in X1,2,3 ; for (1)
(2)
∗ F ∈ [F0 ; F0 ], Vef f is a double-well potential in X2,3 and (2)
for F > F0 , Vef f is a single-well potential at X1∗ . There(1) (2) fore, F = F0 and F = F0 are bifurcation points which make the system (10) transit from a tristable to a bistable and from a bistable to a monostable system respectively. The evolution of these bifurcation points with the order of the fractional derivative is given in Fig.9. From this figure, we see that as the fractional derivative order a increases, the bifurca(1,2) tion points F = F0 also increase. Thus, for high value of the fractional derivative order a, the transition from tristable to a bistable or from a bistable to a monostable system for
4
a=2.0
M (a)
VR
a=1.9
2
0 0
ω
times spend by the orbit in each well are equal for low value of the fractional derivative order a (for a ∈ [1.00; 1.15]). For 1.15 < a < 1.60, the orbit spend much more time in the leftwell than in the right-well because the mean residence time in L the left-well (τM R ) is greater than the mean residence time in R the right-well (τM R ); Thus, for 1.00 < a < 1.60 it a crosswell motion phenomenon. However, at a > 1.6, the orbit R L is entire in the left-well because τM R is zero while τM R is different to zero: there is no occurrence of a cross-well motion phenomenon. Thus, according to the value of the fractional derivative order a, there is appearance of vibrational resonance with or without a cross-well motion.
slow motion, occurs for high amplitudes of the high-frequency force. As in the case of the double-well, we plot the evolution of low-frequency at resonance ωV R versus the high-frequency force amplitude and versus the fractional derivative order for the triple-well potential. Results are given in Fig.10.
us
FIG. 8: Mean Residence Time (τM R ) versus a. Red color is the mean residence time of the orbit in the left well while the blue color is the mean residence time of the orbit in the right well. Initial conditions are (x(0) = −0.78615, Db x(0) = 0.0).
2.
2
FIG. 9: Evolution of the bifurcation values with the fractional derivative order.
1.6
a
an
1.3
ωVR
0 1
1.5 a
d=0.5 d=1.0 d=1.5
a=1.8
100 F
2
0 1
200 (b)
1.5 a
2
FIG. 10: Evolution of the low-frequency at resonance ωV R : (a) versus F with d = 0.5 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with F = 7.0 and for different values of damping d.
In Fig.10(a), we plot the evolution of ωV R versus the highfrequency force amplitude F for different values of the order of the fractional derivative a. We notice that, as in the double-well case, there is a gap where vibrational resonance does’nt occurs. For a = 2.0 (case of the classical quintic oscillator), the gap is F ∈ [145; 150.6], for a = 1.9, the gap is F ∈ [113.3; 119.6] and for a = 1.8, the gap is F ∈ [88.7; 95.3]. Thus, we conclude that as, a decreases, the gap of value of F where vibrational resonance does’nt occurs, are near to zero. In Fig.10(b), we show the effect of the order of the fractional derivative a on ωV R for different value of the linear damping coefficient d. From this figure, it is clear that for weak value of a (near the value 1), ωV R does’nt exist, so vibrational resonance does’nt occurs while for high value of a (near the value 2), as this parameter increases, ωV R also increases. Globally, whatever the value of the order of the fractional derivative vibrational resonance phenomenon occurs at most one time. We can confirm these results by plotting the response amplitude of the system. The evolution of this amplitude versus the low-frequency is given in Fig.11(a). From this figure, we
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7
50
0 1 (a)
2.5 ω
0 1
5 (b)
1
(b)
d
2
3
ip t
cr
us
Q
d=0.5 d=0.8 d=1.2
0.35
0 0
1.5 a
2
a=1.4 80 F
160
FIG. 13: Response amplitude of the system versus F for the triplewell case.
M
(a)
0.4
0 0
2
a=1.9
an
0 0
Qmax
Q
0.4
1.5 a
40
also increases. This result is shown in Fig.12(b). The appari-
0.8
a=2.00 a=1.95 a=1.90
a=1.5 a=1.1 a=1.3
FIG. 12: Evolution of the high-frequency force amplitude at resonance FV R : (a) versus a with ω = 1.0 and for different values of the damping d; (b) the damping d for different values of the order of the fractional derivative a.
0.7
0.8
80
d=3.0 d=1.5 d=0.3
FVR
100
FVR
notice the presence of one maximum for each curve. Therefore, the system exhibit at most one resonance. We also remark that as the fractional derivative order decreases, the amplitude of the system also decreases. Thus, we can predict that for certain values of a, the response amplitude will not have a local maximum. From this figure, the dashed curves represent the numerical curve while the curve with straight line is for analytical result. We see that there is a good agreement between the two analysis. Fig.11(b) presents the same behavior. From this figure, we plot the maximum value of the response amplitude (at resonance) versus the fractional derivative order a. We notice that, for low value of this coefficient, the maximum value of the response amplitude does’nt exist; so, there is not an occurrence of vibrational resonance phenomenon. For high value of a, Qmax meaning of occurrence of the vibrational resonance phenomenon. We can say that, for this range of value, as a increases, as vibrational resonance will occurs with high amplitude (because Qmax increases).
d
FIG. 11: Evolution of the low-frequency at resonance ωV R : (a) versus F with d = 1.0 and for different values of the order of the fractional derivative a; (b) versus the order of the fractional derivative a with F = 7.0 and for different values of damping d.
Ac ce pt e
Thereafter, we take F like control parameter. For F > (2) F0 , Vef f is a single-well potential and ωr2 = C1 . In this (1) (2) case, FV R is given by Eq.(25). For F0 < F < F0 , Vef f (1) is a double-well potential and for F < F0 , Vef f is a triple2 well potential. For these cases, ωr = α1 and it is very difficult to obtain an analytical expression of FV R . Therefore, we determine the value of F at resonance numerically. Because of higher nonlinearities, FV R is determined by maximizing Q numerically. Thus, we obtain values of FV R for various values of a ranging between [1.0;2.0] and d between [0.0;3.0]. Figs.12(a)-(b) are given. In Fig.12(a), where we plot FV R versus a for different values of the damping coefficient, we see that as a increases, FV R also increases. On the same curve, there is an important result; that is, the order of the fractional derivative brings a new resonance for some values. For low values of this parameter, we have two resonances (i.e. two values of FV R ). As a continues to increases a new resonance occurs and we globally have apparition of three vibrational resonances phenomena due to the existence of three values of FV R for a single fractional derivative order given. Always in Fig.12(a), we remark that the linear damping have an effect on the values of FV R . We see that as d increases, the value of F at resonance
tion of a new vibrational resonance is confirm in Fig.13. We see that by representing the response amplitude of the system for a = 1.4, we have two maximum; synonym of two peaks of resonance. When a = 1.9, the curve in red has three maximum inducing three occurrences of resonance. Globally, as in the case of the double-well potential, in the triple-well, the fractional derivative order can bring a new phenomenon of vibrational resonance.
III.
CONCLUSION
In this work, we made the general study of the vibratory dynamics of a special fractional quintic oscillator discovered in complex media. For the quintic form of the restoring force, we showed that this media can be a bistable or a tristable system. Then, we supposed that natural forces in this type of model could be regarded as the combination of two periodic forces of high and low frequency which play the role of forcing of the system. The excitation by these two external periodic forces f cosωt and gcosΩt with Ω ω allowed us to study vibrational resonance. We noticed that, according to the values of the parameters of control, resonance could occur at most two times for a double-well potential and at most three times for a triple-well potential. We showed that the fractional derivative order has an important role in this phenomenon because, according the value of this parameter, vibrational resonance
Page 7 of 9
8 system. Thus, the mechanism of vibrational resonance in this special oscillator considered in this paper is quite different from that of the classical quintic oscillator.
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Ac ce pt e
d
M
an
us
cr
ip t
does not occur or does occur one/two times for the doublewell case or two/three times for the triple-well case. Globally, we concluded that the presence of the fractional derivative order coefficient can introduce new resonance phenomena in the
Page 8 of 9
Highlights
• We present an especial fractional quintic oscillator
ip t
• For high values of the fractional derivative order, an additional resonance occurs • Fractional derivative order increases the high-frequency amplitude at the resonance
Ac
ce p
te
d
M
an
us
cr
• The damping has an effect on the high-frequency force amplitude at the resonance
Page 9 of 9