Quasi-periodic solutions and stability for a weakly damped nonlinear quasi-periodic Mathieu equation

Quasi-Periodic Solutions and Stability for a Weakly Damped Nonlinear Quasi-Periodic Mathieu Equation

Article in Nonlinear Dynamics · February 2002

DOI: 10.1023/A:1014496917703

CITATIONS

26

READS

58

3 authors, including:

Some of the authors of this publication are also working on these related projects:

Energy harvesting using quasi-periodic vibrationsView project

The 4th International Conference on Structural Nonlinear Dynamics and Diagnosis (CSNDD'2018), Tangier, Morocco, 25-27 June 2018View project

Mohamed Belhaq

Université Hassan II de Casablanca 146PUBLICATIONS 1,089CITATIONS

SEE PROFILE

All content following this page was uploaded by Mohamed Belhaq on 08 September 2015.

The user has requested enhancement of the downloaded file.

two successive reductions. In a first step, the multiple-scales method is applied to the original equation to de- rive a first reduced differential amplitude-phase system having periodic components. The stability of stationary solutions of this reduced system is analyzed. In a second step, the multiple-scales method is applied again to the first reduced system (RS) to obtain a second autonomous differential amplitude-phase RS. The problem for approximating QP solutions of the original system is then transformed to the study of stationary regimes of the induced autonomous second RS. Explicit analytical approximations to QP solutions are obtained and comparisons to numerical integration are provided.

Keywords:Quasi-periodic Mathieu equation, quasi-periodic solutions, stability analysis, multiple-scales tech- nique, double-reduction procedure.

1. Introduction

In the last two decades, much interest has been devoted to studying the dynamic behav- ior of quasi-periodically driven nonlinear oscillators from both the numerical and analytical points of view. Attention has been focused on developing an analytical technique to construct approximations of QP solutions and to analyze resonance phenomena and stability [1–5].

The investigation and characterization of strange attractors and their bifurcation in a quasi- periodically driven system was considered in [6, 7] and the existence of strange nonchaotic attractors on two frequency tori was discussed [8, 9]. For the characterization of different routes to chaos via the QP scenario, see, for instance, [10, 11]. The problem of suppression and control of chaos in averaged QP oscillators was also analyzed [5, 12].

In this work we will restrict our attention to the construction of explicit analytical ap- proximations of QP solutions to a weakly damped nonlinear QP Mathieu equation. Stability analysis of these solutions is also provided. Without being exhaustive, let us quote some works on the available techniques dealing with the approximation of such solutions. Ness [1] consid- ered a cubic nonlinear oscillator subjected to parametric and external excitation and analyzed resonance phenomena and stability using an averaging method. The original equation was reduced to a differential amplitude-phase system with periodic coefficients. Chua and Ushida [2, 3] applied the spectral-balance method for constructing QP solutions to a nonlinear circuit driven by multiple input frequencies. In contrast to the standard harmonic-balance technique, the solution in the spectral balance is represented by a Fourier multidimensional series. The problem for determining a QP solution is numerically transformed to the resolution of an al-

gebraic system. Quang et al. [13] analyzed the interaction phenomena between parametric and external excitations in the case of commensurate frequencies, and studied periodic solutions using multiple-scales and numerical simulations. Szabelski and Warminski [14] investigated the influence of a self-excitation on a nonlinear oscillator driven by parametric and external forcing in a commensurate case. Yagazaki and co-workers [4] analyzed the dynamics of an oscillator subjected to parametric and external excitations in an incommensurate case and with a weakly cubic nonlinear component. The Van der Pol transformation was applied to obtain a linear approximation of a periodic solution of the averaged system. In a recent work, Belhaq and Houssni [5] proposed a strategy for constructing an explicit asymptotic expansion of QP solutions to an oscillator with cubic and quadratic nonlinearities, subjected to parametric and external excitations having incommensurate frequencies. The method consists in reducing the original QP system to the so-called second reduced autonomous system performing two successive reductions realizing a nonlinear approximation of solutions.

Here we will take advantage of this technique to analytically approximate QP solutions for the weakly damped nonlinear QP Mathieu equation

¨

x+αx˙ +(ω²+hcos(t)+ρcos(νt))x+ξ x³=0, (1) where damping α, nonlinear component ξ, excitation amplitudes hand ρ, and frequency ν are small. The quantities ω and are the proper and parametric frequencies. An overdot denotes differentiation with respect to timet. Equation (1) can serve as a one-mode model to mechanical systems submitted to two simultaneous additional parametric modulations having incommensurate frequencies. Rand and co-workers [15, 16] studied the linear case of Equa- tion (1) (ξ =0) and analyzed the stability chart using various strategies based on numerical integration, the harmonic-balance method, the standard perturbation technique, and the Lya- punov exponent. Comparison of these various methods was reported and excellent agreement was shown. A slight variant of Equation (1) was considered earlier by Gumowski [17], namely a QP Mathieu equation with a small cubic nonlinearity component. The quasi-periodicity effect was introduced by modulating the amplitude of the parametric excitation to the Mathieu equation. The generalized averaging technique [18] was performed in a Cartesian formulation to reduce the QP Mathieu equation into a differential system with periodic coefficients. The question concerning the approximations of periodic solutions of the first RS was pointed out but not tackled. Other papers have focused on the transition curves for QP systems. Schweitzer [19] studied the conditions for the stability or instability of solutions to a linear QP system.

Weidenhammer [20, 21] applied a perturbation method to linear and nonlinear QP Mathieu equations and determined an analytic expression for transition curves.

Note that in [5] we have combined the multiple-scales method with the Bogolioubov–

Mitropolsky method [18] to perform the double reduction. Comparisons were carried out only at the first RS level. Here we focus attention on the construction of explicit approximate analytical QP solutions of Equation (1) and examine their validity by comparing to the direct numerical integration of the original system (1). The procedure adopted here consists of ap- plying the multiple-scales technique twice to obtain the second autonomous RS. The principal advantage of this procedure is the realization of a nonlinear approximation of QP solutions.

We will restrict ourselves to the study of periodic solutions near the generating parametric resonance 1:2. Several possible parametric ‘satellite’ resonances can occur in the vicinity of this resonance. Here we analyze QP solutions close to the ‘satellite’ resonance 1:2.

The paper is divided into four sections. In Section 2, we apply the multiple-scales technique to obtain the first RS system having periodic components. In Section 3, we derive the second

¨

x+ω²x = −ε(ρ˜cos(νt)x)−ε²(µα˜˜x˙+µh˜˜cos(εt)x˜ + ˜ξ x³), (2) where the ‘fast’ timet and the ‘slow’ timeτ = εt are assumed to be independent. We will restrict our analysis to QP motions in the vicinity of the primary resonance one-half (v≈2ω).

Using the multiple-scales technique [22, 23], an approximation of QP solutions to Equa- tion (2) is sought in the form

x(t)=x0(T0, T1, T2)+εx1(T0, T1, T2)+ε²x2(T0, T1, T2)+ · · ·, (3) whereTn = εⁿT0. In terms of the variableTn, the time derivative becomes d/dt = εD1+ ε²D2+ · · ·, whereDn = ∂/∂Tn. Substituting Equation (3) into Equation (2) and equating coefficients of like powers ofε, one obtains the following systems:

D²₀x0+ω²x0 = 0, (4)

D²₀x1+ω²x1 = −2D0D1x0− ˜ρcos(νT0)x0, (5) D²₀x2+ω²x2 = −2D0D1x1−2D0D2x0−D₁²x0−µαD˜˜ 0x0

− ˜ρcos(νT0)x1−µh˜˜cos(T˜ 1)x0−ξ x₀³. (6) The general solution of Equation (4) can be written as

x0(T0, T1, T2)=A(T1, T2)e^iωT0 +cc, (7) where ‘cc’ denotes the complex conjugate of the preceding terms and A is to be deter- mined through the elimination of secular terms from the next-order equations. Substituting Equation (7) into Equation (5), we obtain

D²₀x1+ω²x1= −2iωD1Ae^iωT0− 1

2ρA˜ e^i(ω+ν)T0−1

2ρ˜A¯e^i(ν−ω)T0+cc. (8) Introducing the detuning parameter σ according to v = 2ω+εσ, substituting into Equa- tion (8), and eliminating the secular terms leads to

D1A= i

4ωρ˜A¯e^{iσ T1}, (9)

where the overbar indicates the complex conjugate. The solution of Equation (8) is then given by

x1(T0, T1, T2)= ρ˜

2ν(2ω+ν)A(T1, T2)e^i(ω+ν)T0 +cc. (10) Substituting Equations (7) and (10) into Equation (6) and eliminating the secular terms, one obtains

−2iωD2A−D₁²A−iωµαA˜˜ −µhA˜˜ cos(T˜ 1)−3˜ξ A²A¯− ˜ρ² A

4ν(2ω+ν) =0. (11) Combining Equations (9) and (11) with the expressionA˙ =D0A+εD1A+ε²D2A, substitut- ingA=(1/2)ae^iβ, whereaandβare real functions, separating the real and imaginary parts, we obtain the first reduced-modulation equations of amplitude and phase system in the polar form

da

dt = −µ1

2αa˜ − 1 4ω

2− ν

2ω

ρasin(2γ ), adγ

dt = ν−2ω

2 a− 1

4ω

2− ν 2ω

ρacos(2γ )− 1

32ω³ρ²a− 3 8ωξ a³

− 1

8ωυ(2ω+v)ρ²a−µ 1

2ωha˜ cos(T1), (12)

whereγ =(1/2)σ T1−β. Periodic solutions of this system (12) correspond to QP solutions of Equation (1) near the generating resonance 1:2. Here the parameterµ appears in system (12) as a new perturbation parameter. In the unperturbed autonomous case,µ=0, this system takes the form

da

dt = − 1 4ω

2− ν

2ω

ρasin(2γ ), adγ

dt = ν−2ω

2 a− 1

4ω

2− ν 2ω

ρacos(2γ )

− 1

32ω³ρ²a− 3

8ωξ a³− 1

8ων(2ω+ν)ρ²a, (13)

whose stationary nontrivial solutions (a = 0), corresponding to da/dt = dγ /dt = 0, are given by

R²+(Q2−Q1)R−Q1Q2=0, (14)

where a²= 8ω

3ξR, Q1= −ρ² 1

32ω³+ 1

8ων(2ω+ν)

+ρ4ω−ν

8ω² +ν−2ω 2 and

Q2=ρ² 1

32ω³ + 1

8ων(2ω+ν)

+ρ4ω−ν

8ω² −ν−2ω

2 .

Figure 1. Bifurcation curves of quasi-periodic solutions in the(ρ , ν)plane forω=1.1.

Equation (14) gives a real and positive solution when the following conditions are satisfied:

! >0, Q1>0; Q2<0, (15)

where! = (Q2+Q1)² is the discriminant of Equation (14). Figure 1 illustrates the corre- sponding regions in the parameter space limited by the bifurcation curves of the QP solutions of Equation (1) for the parameter valuesω = 1.1. In the hatched regions, there exists three fixed points, two centers, and one saddle denoted, respectively, byC0,Cⁱ₂, andS₂ⁱ;i=1,2 and the subscript indicates the order of the cycle. Only one center exists in the unhatched region.

3. Second Reduced System and Quasi-Periodic Solutions

In this section, we derive the second RS and determine an explicit approximation of QP solutions to Equation (1) corresponding to periodic solutions of system (12). This will be done by constructing an asymptotic expansion of the periodic solutions of (12) close to a stationary solution of the unperturbed system (13). Note that a linear approach was proposed in [4] to approximate periodic solutions to a first RS system. Here we follow the strategy proposed in [5] to obtain relevant nonlinear approximate solutions. To appreciate the accuracy of the nonlinear approach, see [5] where comparisons to the linear method and to numerical integration are shown.

To implement the multiple-scales method to Equation (12), it is convenient to introduce the variable change

u=acos(γ ), v= −asin(γ ) (16)

to transform system (12) to the equivalent Cartesian form du

dt = −µα˜

2u+Q1v− 3ξ

8ω(u²+v²)v−µ h˜

2ωcos(t)v, dv

dt = −µα˜

2v+Q2u+ 3ξ

8ω(u²+v²)u+µ h˜

2ωcos(t)u. (17)

We seek a second-order uniform approximate periodic solution of system (17) in the vicinity of a stationary regime in the form

u(t;µ) = u0+µu1(T0, T1, T2)+µ²u2(T0, T1, T2)+ · · ·,

v(t;µ) = v0+µu1(T0, T1, T2)+µ²v2(T0, T1, T2)+ · · ·, (18) whereTn=µⁿtand(u0, v0)denote the coordinates of a stationary solution of the unperturbed system (13). In terms of the variablesTn, the time derivatives are written as before, d/dt = µD1+µ²D2+ · · ·, whereDn=∂/∂Tn. Substituting Equations (18) into Equations (17) and equating coefficients of the same powers ofµ, one obtains up to third-order the following systems

D²₀u1+²₁u1= −α˜

2S⁻¹v0+ h˜

2ωS⁻¹u0cos(t)+ h˜

2ωv0sin(t), (19)

v1=S

D0u1+α˜

2u0+ h˜

2ωv0cos(t)

, (20)

D²₀u2+²₁u2 = −D0D1u1−S⁻¹D1v1− α˜

2 + 3ξ

4ω(u0v1+v0u1)

D0u1

− h˜

2ωcos(t)+ 3ξ

4ω(u0u1+3v0v1)

D0v1

+ h˜

2ωsin(t)−α˜ 2S⁻¹

v1

+ 3ξ

8ωS⁻¹(3u1u²₁+2v0u1v1+u0v²₁)+ h˜

2ωS⁻¹cos(t)u1, (21) v2=S

Dou2+D1u1+ α˜

2u1+ 3ξ

8ω(2u0u1v1+3v0v₁²+v0u²₁)+ h˜

2ωv1cos(t)

(22) and

D²₀u3+²₁u3 = D²₁u1−2S⁻¹D1v2−2S⁻¹D2v1+

˜ α+ 3ξ

2ω(u0v1+v0u1)

D1u1

+ h˜

ωcos(T0)+ 3ξ

2ω(u0u1+3v0v1)

D1v1

+ α˜²

4 − h˜²

4ω²cos²(T0)

u1+h˜α˜

2ωcos(T0)v1

+ ˜hR1cos(T0)u2+ α˜

2R2+ h˜

2ωsin(T0)

v2

Figure 2. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂² in the phase portrait, forh = 0.015, = 0.175,α = 0.001, ρ = 0.08,ω = 1.1,ξ = 0.5 andν = 2.415.

+++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

+R3u1u2+R4v1u2+R5u1v2+R6v1v2

+ α˜

2R7+ ˜hR8cos(T0)

u1v1+ α˜

2R9+ ˜hR10cos(T0)

v₁² +

α˜

2R11+ ˜hR12cos(T0)

u²₁+R13u1v²₁+R14u³₁. (23) Here

1=

−Q1+ 3ξ

8ω(u²₀+3v₀²) Q2+ 3ξ

8ω(v₀²+3u²₀) 1/2

is the proper frequency of system (17) and the quantities S, S and Ri (i = 1, . . . ,14) are given in Appendix A.

The first step in the present study consisted in averaging the original equation (1) on the rapidly dynamic near its generating resonance 1:2. In this second stage, we analyze the

‘satellite’ resonance 1:2, say 21. Then we introduce the new detuning parameterσ1

according to=21+σ, and setσ1=µ²σ˜1andT0=21T0+ ˜σ1T2. Thus, the solution of Equation (19) may be written in the form

u1=A1(T1, T2)e^i1T0+ ˜h(F2−iF3)e^iT0 +α˜

2F1+cc, (24)

whereA1is the slowly varying complex amplitude determined from the higher-order expan- sion. Substituting Equation (24) into Equation (20), one obtains

v1=iG2A1(T1, T2)e^i1T0 + ˜h(G3(F3+iF2)+G4)e^iT0+α˜

2G1+cc. (25)

Figure 3. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂² forh=0.015,=0.175,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

Substitution of Equations (24) and (25) into Equation (21) and elimination of the secular terms leads to

D1A1= αA˜ 1

2 (E1−iE2)+ ˜hA¯1(E3+iE4)e^iσ˜1T2, (26) where expressions ofFi (i =1,2,3),Gi andEi (i =1, . . . ,4) are given in Appendix B. The solution of Equation (21) is then given by

u2 = α˜h˜

2 (M1+iM2)e^iT0 + ˜h²(M3+iM4)e^2iT0 + ˜hA1(M5+iM6)e^i(1+)T0 +A²₁(M7+iM8)e^2i1T0 +α˜²

4 M9+ ˜h²(M10+iM11)+M12A1A¯1+cc. (27) Substituting Equation (27) into Equation (22) yields

v2 =

SD1A1+A1

˜ α

2(N1+iN2)

e^i1T0 +α˜h˜

2 (N3+iN4)e^iT0

Figure 4. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂¹ forh=0.009,=0.181,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

+A²₁(N5+iN6)e^2i1T0+ ˜h²(N7+iN8)e^2iT0+ ˜hA¯1(N9+iN10)e^i(−1)T0 + ˜hA1(N11+iN12)e^i(1+)T0+ α˜²

4 N13+N14A1A¯1+ ˜h²(N15+iN16)+cc. (28) With the help of Equations (24) and (25), the elimination of the secular terms from Equa- tion (23) gives

D2A1 = ˜α²A1(E5+iE6)+ ˜h²A1(E7+iE8)+A²₁A(E¯ 9+iE10)

+ ˜αh˜A¯1(E11+iE12)e^iσ˜1T2. (29) Here Mi (i = 1, . . . ,12) and Ni (i = 1, . . . ,16) are given in Appendix C, and Ei (i = 5, . . . ,12) are given in Appendix D. Equations (26) and (29) can be combined to describe the modulation of the complex amplitude to third-order with respect to the original time. Indeed, substituting these equations into expressionA˙1=µD1A1+µ²D2A1+ · · ·, yields

A˙1 = αA1

2 (E1−iE2)+hA¯1(E3+iE4)e^iσ˜1T2 +α²A1(E5+iE6)+h²A1(E7+iE8) +µ²A²₁A¯1(E9+iE10)+αhA¯1(E11+iE12)e^iσ˜1T2. (30) LettingA1in the polar formA1 = (1/2)ase^iβs, whereas andβs are real, substituting Equa- tion (29) into Equation (30), and separating the real and imaginary parts, we obtain the second modulated autonomous RS

1 2

das

dt = −as

4 E1+has

2 (E3cos(2γ_s)−E4sin(2γ_s))+α²

2 E5as +h² 2 E7as

+1

8E9µa³_s +αh

2 as(E11cos(2γs)−E12sin(2γs)),

Figure 5. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂¹ forh=0.009,=0.181,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

1 2

dγs

dt = −as

4(−21)−αE2

4 as+has

2 (E3sin(2γs)+E4cos(2γs))+α² 2 E6as

+h²

2 E8as +1

8E10µa_s³+αh

2 as(E11sin(2γs)+E12cos(2γs)), (31) whereγs =(σ1/2)t−βs,E9µ=µ²E9andE10µ=µ²E10.

Stationary solutions of this second RS (31), given by setting das/dt =0 and dγs/dt =0, correspond to periodic solutions of Equation (17). It is worth noting [24, 25] that by construct- ing QP solutions of weakly nonlinear oscillators using perturbation methods, one can come to a conclusion on the stability of the QP solutions by studying the stability of the corresponding fixed points of the second RS (Equations (31)).

The steady-state solutions are given either byas =0, which is a possible solution, or by combining Equations (31) to obtain

Asa⁴_s +Bsa_s²+Cs =0. (32)

Figure 6. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂¹ forh=0.0465,=0.184,α=0.001,ρ=0.08,ω=1.1,ξ =0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

Here

As = E_9µ² +E_10µ²

16 , Bs = E9µZ1+E10µZ4

2 and Cs =Z₁²+Z₄²−Z₂²−Z₃², whereZ1,Z2,Z3andZ4are given in Appendix E.

Equation (32) governing the steady-state response gives real and positive solutions in the form

a²_s

1s₂ = −Bs ±√

!s

2As

if one of the following conditions are satisfied (see Figure 10)

!s >0,

Cs

As >0 and_A^Bs

s <0, or !s >0,

Cs

As <0. (33)

Here!s =B_s²−4A_sCsis the discriminant of Equation (32).

Finally, the approximation up to the second order of periodic solutions of system (12) close to the ‘satellite’ resonance of order 1:2 takes the form

u(t) = u0+ascos t

2 −γs

+2h(F2cos(t)+F3sin(t))+α 2F1

+αh(M1cos(t)−M2sin(t))+2h²(M3cos(2t)−M4sin(2t)) +has

M5cos

3

2t−γs

−M6sin 3

2t −γs

+a_s²

2(M7cos(t−2γ_s)

−M8sin(t−2γs))+α²

4 M9+2h²M10+a_s²

4 M12, (34)

Figure 7. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂¹ forh=0.0465,=0.184,α=0.001,ρ=0.08,ω=1.1,ξ =0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

v(t) = v0−asG2sin t

2 −γs

+2h((G3F3+G4)cos(t)−G3F2sin(t)) +α

2G1+αS

2 asE1cos 1

2t−γs

+αS

2 asE2sin 1

2t−γs

+hSasE3cos 1

2t +γs

−hSasE4sin 1

2t+γs

+αas

2

N1cos 1

2t −γs

−N2sin 1

2t −γs

+αh(N3cos(t)−N4sin(t))+a_s²

2(N5cos(t −2γs)−N6sin(t −2γs)) +2h²(N7cos(2t)−N8sin(2t))

Figure 8. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂² forh=0.117,=0.175,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

+has

N9cos

1

2t+γs

−N10sin 1

2t+γs

+has

N11cos

3

2t−γs

−N12sin 3

2t −γs

+α²

4 N13+a²_s

4 N14+2h²N15. (35)

The response is exactly tuned to the frequency of the excitation with a phase shifted backward by the value

γs = 1 2tan⁻¹



 −Z1Z3+

a_s² 4E10Z4

Z2

Z1Z2+

a²_s

4E10+Z4

Z3



 (36)

resulting from the vanishing of Equations (31).

Combining Equations (7), (10), (16), and (34–36), we obtain a second-order approximation of the QP solutions to Equation (1) in the form

x(t) = u(t)cos νt

2

−v(t)sin νt

2

+ ρ

2ν(2ω+ν)

u(t)cos 3νt

2

−v(t)sin 3νt

2

, (37)

whereu(t)andv(t)are given by Equations (34) and (35).

Figure 9. Comparison between numerical and analytical periodic solutions of the reduced system (17) nearC₂² forh=0.117,=0.175,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (17); —: analytical approximation (Equations (34) and (35)).

In Figures 2, 4, 6, and 8, we present comparisons between the analytical approximation of periodic solutions (Equations (34) and (35), solid line), near the center C₂ⁱ (i = 1,2), and a direct numerical integration of the first RS (17) using a Runge–Kutta method (crossed line). In Figures 3, 5, 7, and 9, we show the same comparisons in the time history plane. The subharmonic character of the solution is illustrated in Figures 6–9.

In Figure 10, we show the conditions (33) corresponding to the threshold for the existence of subharmonic oscillations of order 1:2 of the second RS (31). The stability of these solutions is determined by calculating the eigenvalues of the Jacobian matrix of (31)

J =

Z₁+³₄E_9µa_s²

0+Z₂cos(2γs0)−Z₃sin(2γs0) −2Z₂as0sin(2γs0)−2Z₃as0cos(2γs0)

−¹₂E_10µa_s0 −2Z₂cos(2γ_s0)+2Z₃sin(2γ_s0)

. (38)

These are given by

λ²−tr(J )λ+det(J )=0, (39)

Figure 10. Bifurcation curves of a subharmonic periodic solution of order 1:2 nearC₂¹in the(h, )plane for α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415.

Figure 11. Variation of the steady-state amplitude as versus the excitation amplitude near C¹₂. —: stable;

- - -: unstable; forα=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415.

where

tr(J )=2Z1+2ρ1, ρ1= 1

2E9µa₀², ρ2= 1

2E10µa₀² and det(J )=ρ1(ρ1+2Z1)+ρ2(ρ2+2Z4).

Figure 12. Effect of the frequency on the response curves nearC₂¹. —: stable, - - -: unstable; forα = 0.001, ρ=0.08,ω=1.1,ξ=0.5 andν=2.415.

Figure 13. Effect of the damping on the response curves nearC₂¹. —: stable, - - -: unstable; forα = 0.001, ρ=0.08,ω=1.1,ξ=0.5 andν=2.415.

When det(J ) <0, the roots of Equation (39) are real and have different signs, and then the fixed point is a saddle. For det(J ) = 0, one of the eigenvalues is zero and hence the fixed point is nonhyperbolic. In the case where det(J ) >0, one has

λ= 1 2

tr(J )±

tr(J )²−4det(J )

. (40)

Hence, the fixed point is a stable node if 4ρ2(2Z4+ρ2)−4Z²₁ <0 andZ1 <0, and it is an unstable node if 4ρ2(2Z4+ρ2)−4Z₁²<0 andZ1 >0. On the other hand, the fixed point is a focus if 4ρ2(2Z4+ρ2)−4Z₁²>0, which is stable ifZ1<0 and unstable ifZ1>0.

Figure 14. Trancated analytical (a) and numerical (b) quasi-periodic solutions of the original system (1) nearC₂² with the amplitudeas₁in the time history plane, forh=0.01,=0.17,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415.

Figure 11 illustrates the effect of varying the excitation amplitude hon the steady-state response amplitudeas of the subharmonic solution close to the centerC₂¹. It follows that three bifurcation values ofh divide theh-axis into distincte intervals I, II, III and IV. In regions I and IV (h < hc₁ orhc₁ > h < hc₃), wherehc₁ =0.0056 andhc₃ =0.172, there is only one solution, namely the trivial solution which is stable. In region II (hc1 < h < hc2), wherehc2 = 0.0712, three possible solutions exist: the trivial stable solution and two nontrivial solutions, one of which is stable. In region III (h_c2 < h < hc3), there are two possible solutions: the trivial solution which is unstable, and one nontrivial solution, which is stable.

In Figure 12, we illustrate a family of excitation-response curves. The solid and broken lines represent stable and unstable fixed points, respectively. For =0.17 and= 0.175, the response curve has one turning point in which an unstable branch and a stable one meet in a saddle-node bifurcation point. For the other values of, we show that the domain of excitation amplitude is decreasing with . The effect of the damping α on the response is reported in Figure 13.

Figure 15. Direct comparison between numerical and analytical quasi-periodic solutions of Equation (1) nearC₂² with the amplitudea_s1in the time history plane, forh=0.01,=0.17,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (1); —: analytical approximation (Equation (37)).

In Figure 14 we show a comparison between the analytical (Figure 14a) QP solution, Equation (37), and a direct numerical integration of (1) (Figure 14b). A direct comparison of these results is given in Figure 15. The results are given for the amplitudeas₁ in the vicinity of the centerC₂¹.

Finally, Figures 16–18 report similar comparisons for a fixedhand different values of for the amplitudeas2. In Figures 16 and 17, the results are given in the vicinity of the center C₂¹and they are given near the centerC₂²in Figure 18.

4. Conclusions

Explicit analytical approximations of QP solutions have been constructed for a weakly nonlin- ear QP Mathieu oscillator near a parametric primary resonance (1:2, 1:2). A double multiple- scales method is applied to carry out two successive reductions of the original system. The first multiple-scales method reduces the original QP equation to a differential amplitude- phase system having periodic coefficients with forcing frequency related to the smaller of the two original forcing frequencies. Then a second multiple-scales scheme is used to study the reduced system and construct approximations of QP solutions of the original system.

Examination of these approximations given by the double-reduction procedure shows a good agreement with numerical integration from both the qualitative and quantitative points of view. Good agreement is obtained at the first reduction level by comparison between the first RS and the original one, and at the second reduction stage, by comparison between the second RS and the original system.

The double-reduction strategy presented in the present work has the advantage of realizing a nonlinear approximation to QP solutions. It provides an efficient asymptotic tool to construct such solutions for weakly damped nonlinear QP oscillators.

Figure 16. Trancated analytical (a) and numerical (b) quasi-periodic solutions of the original system (1) nearC₂¹ with the amplitudeas₂in the time history plane, forh=0.01,=0.16,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415.

Appendix A

S =

Q1− 3ξ

8ω(u²₀+3v²₀) ₋1

; S=

Q2+ 3ξ

8ω(3u²₀+3v₀²) ₋1

; R1 = 1

2ω

S⁻¹−S⁻¹+ 3ξ

4ω(v₀²−u²₀)

; R2=2S⁻¹+ 3ξ

4ω(u²₀−3v₀²);

R3 = 3ξ 2ωu0

3

2S⁻¹−S⁻¹

; R4= −9ξ

4ωS⁻¹v0; R5= −9ξ

4ωS⁻¹v0; R6 = −3ξ

4ωS⁻¹u0; R7= 3ξ

ωu0; R8= − 3ξ

4ω²v0; R0= 9ξ 2ωv0; R10 = − 3ξ

8ω²u0; R11= 3ξ

2ωv0; R12 = − 9ξ 8ω²u0;

Figure 17. Direct comparison between numerical and analytical quasi-periodic solutions of Equation (1) nearC₁² with the amplitudea_s2in the time history plane, forh=0.01,=0.16,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (1); —: analytical approximation (Equation (37)).

Figure 18. Direct comparison between numerical and analytical quasi-periodic solutions of Equation (1) nearC₂² with the amplitudeas₂in the time history plane, forh=0.01,=0.16,α=0.001,ρ=0.08,ω=1.1,ξ=0.5 andν=2.415. +++: Numerical simulation of (1); —: analytical approximation (Equation (37)).

R13 = 3ξ 8ω

S⁻¹−S⁻¹+ 3ξ

4ω(u²₀−3v₀²)

; R14 = 3ξ

8ω

S⁻¹−S⁻¹+ 3ξ

4ω(v₀²−3u²₀)

.

where

K1 = Q1− 3ξ

4ω(u²₀+3v₀²); K2=Q1−Q2− 3ξ

4ω(3u²₀+v₀²);

K3 = −3ξ 8ωu0

Q1− 3ξ 8ωu²₀

; K4 = 9ξ 8ωu0

Q1−2

3Q2− 9ξ 8ωu²₀

; K5 = −9ξ

4ωv0

Q2+ 3ξ 8ωv₀²

.

Appendix C

M1 = L1(−3²₁)⁻¹; M2=L2(−3²₁)⁻¹; M3=L3(−15²₁)⁻¹; M4 = L4(−15²₁)⁻¹; M5= −L5(8²₁)⁻¹; M6= −L6(8²₁)⁻¹; M7 = −L7(3²₁)⁻¹; M8= −L8(3²₁)⁻¹; M9=L9(²₁)⁻¹;

N1 = S

1+ 3ξ

4ωu0G1+ 3ξ 4ωv0F1

; N2=S 3ξ

4ωu0F1G2+ 9ξ

4ωv0G1G2

; N3 = S

−21M2+F2+ 1

4ωG1+ 3ξ

4ωu0F2G1+ 3ξ

4ωu0F1(G3F3+G4) + 9ξ

4ωv0G1(G3F3+G4)+ 3ξ

4ωv0F1F2

; N4 = S

21M1−F3− 3ξ

4ωu0F3G1+ 3ξ

4ωu0F1G3F2

+ 9ξ

4ωv0G1G3F2− 3ξ

4ωv0F1F3

,