Three-Period Quasi-Periodic Solutions in the Self-Excited Quasi-Periodic Mathieu Oscillator
NAZHA ABOUHAZIM
1, MOHAMED BELHAQ
1,∗, and FAOUZI LAKRAD
21Laboratory of Mechanics, Faculty of Sciences A¨ın Chock, PB 5366 Maˆarif, Casablanca, Morocco;2Institute B of Mechanics, Pfaffenwaldring 9, 70550 Stuttgart, Germany;∗Author for correspondence (e-mail: mbelhaq@hotmail.com)
(Received: 22 January 2004; accepted: 27 August 2004)
Abstract. In this work, we investigate three-period quasi-periodic (QP) oscillations in the vicinity of 2:2:1 resonance in a self- excited QP Mathieu equation using perturbation method. Two successive averaging are performed to reduce the original QP equation to an autonomous amplitude and phase system describing the modulation of the slow flow dynamic. Approximation of three-period QP solution is obtainedviathe study of limit cycle of the reduced autonomous system. The efficiency of the method is illustrated by comparison between analytical approximations and numerical integration. The double reduction procedure, applied in previous works to construct two-period QP solution, can be implemented to approximate excplicit analytical three-period QP solutions.
Key words: quasi-periodic Mathieu equation, self-excitation, three-period quasi-periodicity
1. Introduction
In this paper, we investigate quasi-periodic (QP) oscillations of the self-excited QP Mathieu equation in the form
d
2x
dt
2+ (ω
2+ ρ cos(νt ) + h cos(t ))x + (−α + β x
2) d x
dt = 0. (1)
Here we assume that the QP parametric excitation is the sum of a slow forcing with a frequency
= O(ε) and a forcing having a frequency ν = O(1) resonant with the proper frequency of the system
ω. The other coefficients in Equation (1) are supposed to be small and have positive values. We restrict
our study to the construction of analytical approximations of QP solutions of Equation (1) in the vicinity
of the primary resonances 2:2:1. Equation (1) can serve as a one-mode model to mechanical systems
submitted to a QP parametric excitation and a self-excitation. For instance, in a gear transmission, the
QP parametric excitation can be caused by the QP touth stiffness. The self-excitation can be caused
by the dray friction between the gear teeth. In the case where the parametric excitation is caused by
the periodic touth stiffness, see Schmidt [1]. A self-excited periodic parametric and external excited
system was also studied by Szabelski and Warminski [2]. They investigated periodic and quasi-periodic
solutions and analyzed synchronization phenomenon. Due to the (self-excited) Van der Pol term in the
oscillator (1), two types of solutions may take place in the system. Namely, two- and three-period QP
solutions. To achieve the construction of these solutions, we use the strategy developed by Belhaq and
Houssni [3], which consists in reducing the original QP equation to an autonomous amplitude and phase
system of the slow flow. The stationary solutions of this reduced system correspond to QP solutions of
the original one. This technique was applied successfully to approximate two-period QP solutions of a
weakly damped nonlinear QP Mathieu equation [4]. Rand et al. [5] applied this technique to investigate analytically the regions of stability of the linear QP Mathieu equation (Equation (1) with ρ = h = ε, α = 0, β = 0) in the vicinity of 2:2:1 resonance (ν = 1, = 1 + ε). For other works on the QP Mathieu equation, see Weidenhammer [6, 7], and Zounes and Rand [8, 9]. In a recent paper, Lakrad and Belhaq [10] analysed two-period QP solutions of a two-degree-of-freedom shallow arch system subjected to a slow parametric excitation and a fast external excitation.
The main purpose of the present work is to investigate analytically three-period QP solutions of Equation (1). The procedure to construct these solutions is as follows: In a first step, we apply, as in Belhaq and co-workers [3, 4], the multiple-scales method [11, 12] two times to reduce the QP equation (1) to an autonomous amplitude–phase system describing the modulation of the slow flow dynamic. The fixed points of this autonomous system correspond to two-period QP oscillations of Equation (1) and a limit cycle of the autonomous system corresponds to a three-period QP solution of the original system (1). In a second step, we approximate these two- and three-period QP oscillations by investigating, respectively, the fixed points and the limit cycle of the reduced autonomous system. We validate our analytical approximations by comparison with numerical integrations.
2. Slow Flow Dynamic and Autonomous System
We introduce two small parameters ε and µ, such that 0< µ ε 1 and we scale Equation (1) as
¨
x + ω
2x = −ε( ˜ ρ cos(νt)x) − ε
2µ[ ˜˜ h cos(ε t)x ˜ − α ˜˜ x] ˙ − ε
2β ˜ x
2x, ˙ (2) with the scaling: ρ = ε ρ, ˜ h = µ h ˜ = µε
2h ˜˜ , α = µ α ˜ = µε
2α, ˜˜ β = ε
2β ˜ and = ε . Note that ˜ the perturbation parameters ε, µ and η, considered in Section 4, are considered as small positive non- dimensional parameters that are artificially introduced to serve as a bookkeeping devices and will be set equal to unity in the final analysis; see Nayfeh and Balachandran [13]. We perform our study of QP motions in the vicinity of the primary generating resonance which is expressed as
ν = 2ω + εσ, (3)
where σ is a detuning parameter. Using the method of multiple scales, we determine a uniform approx- imation to the solution of Equation (2) in the form
x(t) = x
0(T
0, T
1, T
2) + ε x
1(T
0, T
1, T
2) + ε
2x
2(T
0, T
1, T
2) + O ( ε
3) , (4) where T
n= ε
nt. In terms of the variable T
n, the time derivative becomes
dtd= D
0+ εD
1+ ε
2D
2+ . . ., where D
n= ∂/∂ T
n. Substituting Equation (4) into Equation (2) and equating coefficients of like powers of ε, we obtain the following hierarchy of problems:
– Order ε
0:
D
20x
0+ ω
2x
0= 0, (5)
– Order ε
1:
D
20x
1+ ω
2x
1= −2 D
0D
1x
0− ρ ˜ cos (νT
0)x
0, (6)
– Order ε
2:
D
20x
2+ ω
2x
2= −2 D
0D
2x
0− 2D
0D
1x
1− D
21x
0− ρ ˜ cos(ν T
0)x
1−µ h ˜˜ cos( ˜ T
1)x
0+ µ α(D ˜˜
0x
0) − β(D ˜
0x
0)x
02. (7)
The solution of the first-order problem can be expressed as
x
0= A(T
1, T
2)e
(iωT0)+ cc, (8)
where cc denotes the complex conjugate of the preceding terms. The complex-valued function A(T
1, T
2) is to be determined by eliminating the secular terms at the next level of approximations.
Substituting Equation (8) into Equations (6) and (7), using Equation (3) and eliminating the secular terms, we obtain the modulation equations
D
1A = i
4ω ρ ˜ Ae ¯
iσT1, (9)
D
2A = − iσ ρ ˜
8ω
2Ae ¯
iσT1+ i ρ ˜
232ω
3A + i ρ ˜
28νω(2ω + ν) A + iµ h ˜˜
2ω cos( ˜ T
1)A + µ α ˜˜
2 A − β ˜
2 A
2A. ¯ (10) Letting A =
12ae
iθwhere a and θ are real functions, combining Equations (9) and (10) into the expression
d Adt= ε D
1A + ε
2D
2A and separating real and imaginary parts, we obtain the slow flow modulation equations of amplitude and phase with a parametric forcing
da
dt = − 4ω − ν
8ω
2ρ a sin(2 γ ) + µ α ˜ 2 a − β
8 a
3, (11)
a dγ
dt = a ν − 2ω
2 − 4ω − ν
8ω
2ρ a cos(2 γ ) − 1
32ω
3+ 1
8νω(2ω + ν)
ρ
2a − µ h ˜
2ω a cos( t ) , (12) where γ =
12σ T
1− θ = εσ t /(2 − θ) with εσ = ν − 2ω. Here the parameter µ appears in this slow flow system as a new perturbation parameter.
Now we perform the second averaging on the slow flow system (11)–(12) to derive an autonomous amplitude–phase modulation system of the slow flow near the principal resonance 2:1 of the gen- erating resonance ν 2ω. We transform Equations (11) and (12) to the Cartesian form via the transformation
u = a cos (γ ), v = −a sin (γ ). (13)
This gives du
dt = µ α ˜
2 u + E
1v − µ h ˜
2 ω cos(t)v − β
8 (u
2+ v
2)u, (14)
d v dt = µ α ˜
2 v + E
2u + µ h ˜
2 ω cos(t )u − β
8 (u
2+ v
2)v, (15)
where
E
1= ν − 2ω
2 + 1
4ω
2 − ν 2ω
− 1
32ω
3+ 1
8νω(2ω + ν)
ρ
2, (16)
E2 = − ν − 2 ω
2 + 1
4 ω
2 − ν 2 ω
+
1
32 ω
3+ 1 8 νω (2 ω + ν )
ρ
2. (17)
Using the multiple-scale technique, we seek a second-order uniform expansion of solution of Equations (14) and (15) near the origin in the form
u(t, µ) = µu
1(T
0, T
1, T
2) + µ
2u
2(T
0, T
1, T
2) + . . . , (18) v(t, µ) = µv
1(T
0, T
1, T
2) + µ
2v
2(T
0, T
1, T
2) + . . . , (19) where T
0= t and T
n= µ
nT
0. In terms of the variable T
n, the time derivative is transformed as
d/dt = D
0+ µD
1+ µ
2D
2+ . . . , (20)
where D
nare given earlier. Substituting Equations (18) and (19) into Equations (14) and (15), and equating coefficients of the same powers of µ, we obtain equations governing the u
jand v
jas – Order O(µ
1)
D
20u
1+
20u
1= 0, (21)
v
1= E
−11D
0u
1. (22)
– Order O ( µ
2)
D
20u
2+
20u
2= − D
0D
1u
1+ α ˜
2 (D
0u
1) + E
1−D
1v
1+ α ˜ 2 v
1+ h ˜
2ω cos(T
0)u
1− h ˜
2ω cos( T
0)(D
0v
1) + h ˜
2ω sin( T
0) v
1, (23)
v
2= E
−11D
0u
2+ D
1u
1− α ˜ 2 u
1+ h ˜
2ω v
1cos ( T
0)
. (24)
– Order O(µ
3)
D
20u
3+
20u
3= − D
0D
1u
2− D
0D
2u
1+ α ˜
2 D
0u
2− E
1(D
1v
2) − E
1( D
2v
1) + E
1α ˜ 2 v
2+ E
1h ˜
2ω cos(T
0)u
2− E
1β 8 v
1u
21+ v
21− h ˜
2ω cos(T
0)D
0v
2+ h ˜
2ω sin(T
0)v
2− β 8 D
0u
1u
21+ v
12− β
4 u
1(u
1D
0u
1+ v
1( D
0v
1)), (25) v
3= E
−11D
0u
3+ D
1u
2+ D
2u
1− α ˜ 2 u
2+ h ˜
2ω cos( T
0) v
2+ β 8 u
1u
21+ v
12. (26) Here
20= −E
1E
2is the proper frequency of system (14)–(15). Consequently, the product E
1E
2must take negative values. We examine the response of system (14)–(15) in the vicinity of the satellite resonance 2
0of the generating one ν 2ω. This is expressed as
= 2
0+ µ
2σ
1, (27)
where σ
1is a detuning parameter. The solution of Equation (21) can be expressed as
u
1= A
1(T
1, T
2)e
i0T0+ cc, (28)
where A
1(T
1, T
2) is slowly varying complex amplitude of the slow flow dynamic (14)–(15), to be determined from the next order of expansion. Substituting Equation (28) into Equation (22), we obtain
v
1= i
0E
1A
1(T
1, T
2) e
i0T0+ cc. (29)
Substituting Equations (28) and (29) into Equation (23) and eliminating secular terms leads to D
1A
1= α ˜
2 A
1− i h E ˜
18ω
0q
00A ¯
1e
iσ1T2. (30)
The solution of Equations (23) and (24) is given by
u
2(T
0, T
1) = Q
1A
1(T
1, T
2)e
i(+0)T0, (31) v
2(T
0, T
1) = i Q
2A
1(T
1, T
2)e
i(+0)T0+ i Q
3A ¯
1(T
1, T
2)e
i(−0)T0. (32) Substituting Equations (28)–(29), (31)–(32) into Equation (25) and eliminating secular term, we obtain
D
2A
1= − β 4
1 +
20E
12A
21A ¯
1− i h ˜
2q
01A
1. (33)
Letting A
1=
12a
1e
iθ1where a
1and θ
1are real functions, substituting into expression
d Ad T1= µ D
1A
1+ µ
2D
2A
1+ . . . , and separating real and imaginary parts, we obtain the autonomous equation of the amplitude and phase describing the modulation of the slow flow dynamic (11)–(12)
˙
a
1= F
1a
1+ F
2a
13+ F
4a
1sin(2γ
1), (34)
γ ˙
1= F
3+ F
4cos(2γ
1), (35)
where γ
1=
−220t − θ
1and the coefficients Q
i(i = 1 , 2 , 3), q
00, q
01and F
i(i = 1 , . . . , 4) are given in the appendix. The stationary regims (fixed points) of the system (34)–(35) correspond to periodic solution of the slow flow system (11)–(12) and, consequently, to two-period QP motion of the original Equation (1). Similarly, a periodic solution (limit cycle) of Equations (34) and (35) corresponds to two-period QP solution of Equations (11)–(12) and to three-period QP solution of Equation (1).
3. Two-Period Quasi-Periodic Solution
The stationary regimes of system (34)–(35) are obtained by setting the time derivatives equal to zero,
˙
a
1= γ ˙
1= 0. Eliminating γ
1from the resulting algebraic system, we obtain
F
22a
41+ 2F
1F
2a
12+ C
1= 0, (36)
which is a quadratic equation in a
21. Here C
1= F
12+ (F
3− F4)( F
3+ F4) and we let
1=
−F
22(F
32− F
42) the discriminant of Equation (36). For
1> 0, Equation (36) has real positive so- lutions: one solution when C
1< 0 and two solutions when C
1> 0 . These are given by
a
1 1,22= − F
1±
F
42− F
32. (37)
The stability of these solutions is determined by calculating the eigenvalues of the Jacobian matrix of system (34)–(35) which are given by
λ
1= 2
F
1+ F
2a
12, λ
2= 2F
2a
12. (38)
Since F
2is a negative number, the nature of the fixed points depends on the sign of λ
1.
In Figure 1, we draw the conditions corresponding to the threshold for the existence of periodic solutions near the satellite resonance 2:1 of the system (14)–(15). We can distinct three regions. In region I, where
1> 0 and C
1< 0, there are two possible solutions: an unstable trivial solution and a larger stable one. Within region II, where
1> 0 and C
1> 0, there are three possible solutions: one unstable, one larger stable and the trivial unstable solution. Within region III, i.e. when
1< 0, only an unstable focus exists. We will see in the next section that in this region, a stable limit cycle may also exist.
Figure 2 shows a typical family of the frequency-response curves. The solid and broken lines represent stable and unstable fixed points, respectively. At the turning point separating an unstable branch and a stable one, a saddle-node bifurcation takes place.
In Figure 3, we show the amplitude–response curves. The effect of varying the excitation frequency on the response of the solution is illustrated.
Figure 1. Bifurcation curves of quasi-periodic solutions of the original system (1) near the resonance 2:2:1 forα = 0.01, β=0.01,ω=1.1,ν=2.415 andρ=0.08.
Figure 2. Frequency-response curves forα=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08. Solide lines stand for stable and dashed lines for unstable solutions.
Figure 3. Effect of the slow frequency on the parametric excitation-response curve forα=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08.
Combining Equations (28)–(29), and (31)–(32), the second-order approximation of periodic solutions of system (14)–(15) takes the form
u(t) = a
1cos
2 t − γ
1+ Q
1a
1cos 3
2 t − γ
1, (39)
v (t) = −
0E
1a
1sin
2 t − γ
1− Q
2a
1sin 3
2 t − γ
1− Q
3a
1sin
2 t + γ
1. (40)
Figure 4. Comparison between analytical approximation of periodic solution (Equations (39) and (40)) and numerical periodic solution of system (14)–(15) forh=0.1,=0.213,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08.
Combining Equations (4), (8), (13), (39) and (40), we obtain an explicit second-order approximation of two-period QP solutions of Equation (1) in the form
x(t) = u(t ) cos ν
2 t
− v(t) sin ν
2 t
+ ρ
2 ν (2 ω + ν ) u(t ) cos 3 ν
2 t
− ρ
2 ν (2 ω + ν ) v(t ) sin 3ν
2 t
. (41)
In Figure 4, we present a comparison between the analytical approximation (system (39)–(40)) of a periodic solution of the slow flow system (14)–(15) and a direct numerical integration of the system (14)–(15) using a Runge–Kutta method.
In Figure 5, we show a comparison between the analytical approximation of two-period QP solutions (Equation (41)) and a direct numerical integration of the original Equation (1).
4. Three-Period Quasi-Periodic Solution
A three-period QP solution of the original system corresponds to a limit cycle of the system (34)–(35). To approximate the limit cycle of this system, we consider the equivalent cartesian form via the equations U = a
1cos( γ
1), V = a
1sin( γ
1)
U ˙ = F
2U + F
1(U
2+ V
2)U + (F
3− F
4)V , (42)
V ˙ = F
2V + F
1(U
2+ V
2)V − (F
3+ F
4)U. (43)
We introduce a perturbation parameter η which is small comparing to the previous parameters ε and
µ and we scale as F
1= η F ˜
1, F
2= η F ˜
2. A periodic solution of the system (42)–(43) is sought
Figure 5. Comparison between the analytical approximation of the two-period QP solution (Equation (41)), (a), and the direct numerical integration of Equation (1), (b), forh=0.1,=0.213,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08.
as
U (T
0, T
1) = U
0(T
0, T
1) + ηU
1(T
0, T
1) + η
2U
2(T
0, T
1) + . . . , (44) V (T
0, T
1) = V
0(T
0, T
1) + ηV
1(T
0, T
1) + η
2V
2(T
0, T
1) + . . . , (45) where T
n= η
nT . Substituting Equations (44) and (45) into Equations (42) and (43) and equating coefficients of like powers of η, we obtain
– Order η
0:
D
20U
0+
2pU
0= 0, (46)
V
0= 1
F
3− F
4D
0U
0(47)
– Order η
1:
D
20U
1+
2pU
1= (F
3− F
4)
− D
1V
0+ F ˜
1V
0+ F ˜
2V
0U
02+ V
02−D
0D
1U
0+ F ˜
1D
0U
0+ F ˜
2D
0U
0U
02+ V
02, (48)
V
1= 1 F
3− F
4D
0U
1+ D
1U
0− F ˜
1U
0− F ˜
2U
0U
02+ V
02, (49)
Here
2p= ( F
3− F
4)(F
3+ F
4) is the proper frequency of system (42)–(43), which corresponds to the frequency of the limit cycle. Note that the term (F
3− F
4)(F
3+ F
4) takes positive values only in the region III of Figure 1. This means that the limit cycle exists in this regions in the presence of an unstable focus.
The solutions of the first-order problem is
U
0= A
pe
ipT0+ A ¯
pe
−ipT0, (50)
V
0= i
pF
3− F
4A
pe
ipT0+ A ¯
pe
−ipT0, (51)
where A
pis the amplitude of the limit cycle that will be defined by the elimination of secu- lar terms. Introducing Equations (50) and (51) into Equation (48) and eliminating secular terms yields
D
1A
p= F ˜
1A
p+ 2 ˜ F
21 +
2p(F
3− F
4)
2A
2pA ¯
p. (52)
Let A
p=
12a
pe
iθp, where a
pand θ
pare real functions. Substituting into Equation (52), multiplying by η and separating real and imaginary parts, we obtain
˙
a
p= F
1a
p+ F
22
1 +
2p(F
3− F
4)
2a
3p, (53)
θ ˙
p= 0 . (54)
The stationary solutions of this system are a
p= 0 (trivial solution) and a
2p= − 2F
1F
21 +
(F3−2pF4)2, (55)
which corresponds to the amplitude of the limit cycle of the system (42) and (43). The stability of these solutions is determined by calculating the eigenvalues of the Jacobian matrix of the system (42)–(43).
These are given by the characteristic equation
λ
2− 2 F
1λ + F
22+ (F
3− F
4)(F
3+ F
4) = 0, (56) which has two real solutions if the discriminant
2= 4
F122
> 0, where
1is given in Section 2, and two complex conjugate solutions if
2< 0. This means that the trivial solution in region III of Figure 1 is an unstable focus.
Substituting Equations (50) and (51) into Equation (48) and (49) gives
U
1= −i Z A
3pe
3ipT0+ cc, (57) V
1=
pZ
4( F
3− F
4)
A
3pe
3ipT0− 2 A
2pA ¯
pe
ipT0+ cc . (58)
where Z is given in the appendix.
Hence, the first-order approximation of the periodic solution of the system (41) and (43) is written as
U (t) = a
pcos(
pt + θ
p), (59)
V (t) = −
pF
6a
psin(
pt + θ
p), (60)
where a
pis given by Equation (55). Consequently, the two-period QP solution of the system (14)–(15) is approximated by
u(t) = U (t) cos
2 t
+ V (t ) sin
2 t
+ Q
1U(t ) cos 3
2 t
+ Q
1V (t) sin 3
2 t
, (61)
v(t) = −
0E
1+ Q
3U (t) sin
2 t
+
0
E
1− Q
3V (t ) cos
2 t
−Q
2U(t ) sin 3
2 t
+ Q
2V (t ) cos 3
2 t
. (62)
Finally, the three-period QP solution of Equation (1) is approximated by Equation (41) in which the slow flow amplitudes u(t ) and v(t) of the slow flow dynamic are now given by Equations (61) and (62).
In Figure 6 we show a comparison between the approximate periodic solution (59)–(60) and the numerical integration of the autonomous system (42)–(43).
In Figure 7 we present a comparison between the approximate two-period QP solution (61), (62) and the numerical solution of the slow flow system (14)–(15).
Figure 6. Comparison between the approximate periodic (limite cycle) solution (59)–(60) and the numerical integration of the autonomous system (42)–(43) forh=0.04,=0.24,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08.
Figure 7. Comparison between the approximate two-period QP solution (61), (62), (a), and the numerical solution of the slow flow system (14)–(15), (b), forh=0.04,=0.24,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08.
Figure 8. Comparison between the analytical three-period QP solution (41), (61) and (62), (a), and the numerical solution of the original Equation (1), (b), forh=0.04,=0.24,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.08.
Figure 9. Comparison between the approximate periodic solution (59)–(60) and the numerical integration of the autonomous system (42)–(43) forh=0.02,=0.24,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.04.
Figure 8 shows a comparison between the analytical three-period QP solution (41), (61) and (62) and the numerical solution of the original Equation (1).
Finally, in Figures 9–11, same comparisons (as in Figures 6–8) are given for different parameter values of the excitation amplitudes ρ and h .
5. Conclusion
In this paper, we have shown that the double reduction method, applied successfully to approximate
two-period QP solutions in systems driven by two-period QP forcing, can also be implemented to
Figure 10. Comparison between the approximate two-period QP solution (61), (62), (a), and the numerical solution of the slow flow system (14)–(15), (b), forh=0.02,=0.24,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.04.
Figure 11. Comparison between the analytical three-period QP solution (41), (61) and (62), (a), and the numerical solution of the original Equation (1), (b), forh=0.02,=0.24,α=0.01,β=0.01,ω=1.1,ν=2.415 andρ=0.04.
construct three-period QP solutions. To this end, three successive multiple-scales methods were applied to construct explicit analytical three-period QP solution for the self-excited quasi-periodic Mathieu equation in the vicinity of the parametric primary resonance 2:2:1.
In a first step, we have used the multiple-scales method to reduce the original QP oscillator to a slow flow amplitude and phase system having a parametric periodic term.
In a second step, a multiple-scales technique is applied on this slow flow system to perform the second averaging leading to an autonomous amplitude–phase system describing the modulation of the slow flow dynamic. Steady-state solutions of this induced autonomous system corresponding to two-period QP regimes of the original system are investigated to construct explicit analytical two-period QP solutions.
In a last step, we perform a multiple-scales scheme on the autonomous system to obtain analytical
approximation of a limit cycle. This limit cycle corresponds to a three-period QP solution of the original
system.
The comparisons of the analytical approximations with direct numerical integrations, at each level of averaging, show the validity of the analytical approximations and illustrate the efficiency of the proposed approch to investigate three-period QP solutions in nonlinear oscillators exhibiting such motions.
Appendix
q
00= 1 +
20E
12−
0E
12q
01= − 1
64ω
20
2E
121 +
20+E20 1( + 2
0) + E
12q
001 +
320−E2 0 120
+ 2(2 +
0)( +
0)
1 +
20+E20 1( + 2
0) − 2
0(2 +
0) E
12−
1 + 3
20−
0E
12
Q
1= − h E
11 +
20+E20 14ω( + 2
0) Q
2= E
1−10
h
4ωE
1+ ( +
0)Q
1Q
3= − h 8ω
01 + 3
20−
0E
12F
1= α 2 F
2= − β
16
1 +
20E
12F
3= − 2
02 + h
2q
01F
4= h E
1q
004ω
0Z = F
22
p1 −
2p(F
3− F
4)
2Acknowledgements
This work was supported by the “Centre National de la Recherche Scientifique et Technique” under the Grant: PARS-Physique 04. F.L. would like to thank the Alexander von Humboldt Foundation for the financial support.
References
1. Schmidt, G., ‘Application of the theory of nonlinear oscillations: Interaction of self-excited forced and parametrically excited vibrations’, inThe 9th International Conference on Non-linear Oscillations, 3 Kiev, Naukowa Dumka, 1984.
2. Szabelski, K. and Warminski, J., ‘Self-excited system vibrating with parametric and external excitations’,Journal of Sound and Vibration187(4), 1995, 595–607.
3. Belhaq, M. and Houssni, M., ‘Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations’,Nonlinear Dynamics18, 1999, 1–24.
4. Belhaq, M., Guennoun, K., and Houssni, M., ‘Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation’, International Journal of Non-Linear Mechanics37, 2002, 445–460.
5. Rand, R., Guennoun, K., and Belhaq, M., ‘2:2:1 Resonance in the quasi-periodic Mathieu equation’,Nonlinear Dynamics 31, 2003, 367–374.
6. Weidenhammer, F., ‘Nicht-linear Schwingungen mit fast-periodischer Parametererregten’, Zeitschrift für Angewandte Mathematik und Mechanik61, 1961, 633–638.
7. Weidenhammer, F., ‘Intabilitäten eines ged¨ampften Schwingers mit fast-periodischer Parametererregung’,Ingeniour-Archiv 49, 1980, 187–193.
8. Zounes, R. and Rand, R., Transition curves for the quasi-periodic Mathieu equation’,SIAM Journal on Applied Mathematics 58, 1998, 1094–1115.
9. Zounes, R. and Rand, R., ‘Global behavior of a nonlinear quasi-periodic Mathieu equation’,Nonlinear Dynamics27, 2002, 87–105.
10. Lakrad, F. and Belhaq, M., ‘Solutions of a shallow arch under fast and slow excitations’, inIUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, 8–13 June 2003, Rome, Italy.
11. Nayfeh, A. H.,Perturbation Methods, Wiley, New York, 1973.
12. Nayfeh, A. H. and Mook, D. T.,Nonlinear Oscillations, Wiley, New York, 1979.
13. Nayfeh, A.H. and Balachandran, B.,Applied Nonlinear Dynamics, Wiley, New York, 1995.
14. Guennoun, K., Houssni, M., and Belhaq, M., ‘Quasi-periodic solutions and stability for a weakly damped nonlinear quasi- periodic Mathieu equation’,Nonlinear Dynamics27, 2002, 211–236.
