parametric excitation in non-linear dynamics

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International Journal of Non-Linear Mechanics 39 (2004) 311–329

Parametric excitation in non-linear dynamics

T. Bakria ;∗, R. Nabergojb, A. Tondlc, F. Verhulsta

aMathematics Institute, Utrecht University, PO Box 80.010, Utrecht, TA 3508, The NetherlandsbDepartment of Naval Architecture, Ocean and Environmental Engineering, Via Valerio Trieste 10, I-34127, Italy

cZborovsk/a 41, Praha 5 CZ-15000, Czech Republic

Received 31 March 2002; accepted 16 September 2002

Abstract

Consider a one-mass system with two degrees of freedom, non-linearly coupled, with parametric excitation in one direction.Assuming the internal resonance 1:2 and parametric resonance 1:2 we derive conditions for stability of the trivial solution byusing both the harmonic balance method and the normal form method of averaging. If the trivial solution becomes unstable,a stable periodic solution may emerge, there are also cases where the trivial solution is stable and co-exists with a stableperiodic solution; if both the trivial solution and the periodic solution(s) are unstable, we 6nd an attracting torus with largeamplitudes by a Neimark–Sacker bifurcation. The results of the harmonic balance method and averaging are compared,as well as the results on the Neimark–Sacker bifurcation obtained by the numerical software package CONTENT and byaveraging. In all cases we have good agreement.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Dynamical systems; Neimark-Sacker bifurcation; Averaging method and tori

1. Introduction

Non-linear vibrating systems often consist of two—or even more—subsystems, where one of them isexcited, the primary system, and the other ones arecoupled through non-linear terms; they form the sec-ondary system or excited system. The primary systemis an oscillator which can be excited externally, para-metrically or by self-excitation, while the secondarysystem is excited indirectly through the non-linearcoupling.In particular, autoparametric systems [1–3] rep-

resent an important example. A typical property of

∗ Corresponding author.E-mail address: [email protected] (T. Bakri).

autoparametric systems is the existence of a semitriv-ial solution of the di>erential equations of motionin which the primary system oscillates while thesecondary system is at rest. In such problems, it isessential to study the boundary limits of the stabil-ity region for the trivial solution and to establishwhether they are determined by the stability lim-its of the primary system, i.e., whether they arechanged by the action of the secondary (excited)system [3].In the case where the non-linear coupling terms do

not allow for the existence of a semitrivial solution,the secondary system must oscillate when the primarysystem is oscillating; the system is sometimes calledhetero-parametric. In [4], a hetero-parametric exampleof an externally excited single-mass system havingtwo degrees of freedom was analysed.

0020-7462/04/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.PII: S0020 -7462(02)00190 -7

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312 T. Bakri et al. / International Journal of Non-Linear Mechanics 39 (2004) 311–329

M

x

yForce Field

−εc cos2ηtεc cos2ηt

Fig. 1. Two degrees of freedom single-mass system with simplyparametric excitation in the presence of a force 6eld parallel tothe y-direction.

Also, in the present paper, we shall consider asingle-mass system, but with parametric excitationin the primary system and a non-linear coupling ex-pressed by second degree terms in the di>erentialequations. The parametric excitation acts, for ex-ample, due to kinematic excitation of the supportsthrough non-linear springs (see Fig. 1). The systemis simply parametric when both non-linear springs inthe x-direction are identical and two kinematic ex-citations are simultaneously acting, having the sameamplitude and frequency but opposite phase.The equations of motion are

Cx + �x2x + �0x + (1 + �c cos 2�t)x + x3 + axy = 0;

Cy + �y + q2y + bx2 = 0: (1)

For the damping coeGcients we have �; �0; �¿ 0,furthermore c¿ 0; ¿ 0; � is a small positive param-eter. Apart from linear damping we have assumed thepresence of progressive damping to ensure a limitedvibration amplitude, even at parametric resonance.We could drop the assumption of a parallel force

6eld and look at the consequences of much moregeneral forces. We stress here that the non-linearcoupling terms chosen here are typical. We can add,for instance, 10 quadratic terms (xy; yy, etc.) in the

equations for x and y but most of these terms willvanish by averaging normalisation. In particular, forthe x-equation all the added potential terms will be ofno inIuence, of the other terms only xy; xy; xy willremain after averaging. In the equation for y we havea similar e>ect at this order of approximation.In the case of a single kinematic excitation, the sys-

tem is subjected to a combination of parametric andexternal excitation (see [5]). A one-mass system withtwo degrees of freedom with parametric excitation inone direction is treated in [6].Note that nowadays the parametric excitation can

be provided by modern actuators in mechatronic andsmart structures, e.g., in actively controlled magneticbearings. In this way, the elastic mounting with peri-odically varying sti>ness can practically be conceived.In the next section, we present a harmonic balance

calculation which gives us the response and resonancecurves for certain sets of parameters. In Section 3,we perform a scaling and 6rst-order averaging to theequations of motion. This leads to explicit results onthe stability of the trivial solution. Also, we 6nd fam-ilies of periodic solutions of which we determine thestability; because of the complexity of the expressionsthis is quite surprising. A comparison of quantitativeresults obtained by harmonic balance and averagingcalculations are given in Section 4. This is of inter-est as the harmonic balance method involves no errorestimate and no existence of periodic solutions result;see for a discussion [2, Section 9.6]. The averagingmethod, on the other hand, is a normal form methodwhich produces both quantitative results with error es-timates and existence properties; for an introductionand references see also [2, Section 9].A particular situation arises in Section 3 when both

the trivial solution and the periodic solution(s) areunstable. In Section 5, we 6nd that this instability istriggered o> by a Neimark–Sacker bifurcation (sec-ondary Hopf bifurcation) of the periodic solution.This results in an attracting torus with fairly largeamplitudes.The Neimark–Sacker bifurcation was 6rst pin-

pointed by using the numerical bifurcation programCONTENT. An unusual feature is that this bifurca-tion can also be identi6ed and analysed using theaveraging method.It is remarkable how di>erent this result of an at-

tracting torus is from the dynamics of a related system

T. Bakri et al. / International Journal of Non-Linear Mechanics 39 (2004) 311–329 313

studied in [7]. In that paper an internal 1:1-resonanceis assumed which leads to period-doubling bifur-cations and the presence of a strange attractor. Forrelated topics on parametric excitation (see [10,11]).

2. Harmonic balance calculations

The non-linear coupling terms play an importantrole in determining the frequency tuning so as to pro-duce signi6cant e>ects. For example, in the consid-ered system, both coupling terms are of second degreeand therefore it can be judged that the optimal tuninginto internal resonance is of 1:2 type, i.e., the strongestdeIection can be expected for q � 2.

A periodic solution of Eq. (1) in the internal res-onance 1:2 can be approximated using the harmonicbalance method; when inserting the term cos 2(�t− )for cos 2�t in order to facilitate the computations, welook for a solution of the form

x = R cos �t;

y = Y0 + A cos 2�t + B sin 2�t; (2)

where is the initial phase, representing the shift ofthe time origin suitable for simplifying the analyticalsolution. Equating corresponding terms of the Fourierseries yields the following algebraic equations:

(1− �2 + 34 R2)R+ a(Y0 + 1

2 A)R=− 12 �cR cos 2 ;

− 14 ��R3 + (12 aB− �0�)R=− 1

2 �cR sin 2 ;

(q2 − 4�2)A+ 2��B+ 12 bR2 = 0;

−2��A+ (q2 − 4�2)B= 0;

q2Y0 + 12 bR2 = 0: (3)

From the third, fourth and 6fth equations of (3), thefollowing relations are obtained:

Y0 =− 12q2

bR2;

A=−12bR2 q2 − 4�2

�;

B=−bR2 ��

; (4)

where

�= (q2 − 4�2)2 + 4�2�2: (5)

0.96 0.98 1 1.02 1.04η

0.2

0.4

0.6

0.8

1

R

Fig. 2. Resonance curve R(�) corresponding to the parametersq = 2; = 0 without coupling (a = b = 0).

In particular, the vibration amplitude of the secondary(excited) system is given by

r =√

A2 + B2 = 12 bR2�−1=2: (6)

Squaring and adding the 6rst two equations of (3) andusing (4), we obtain the frequency response amplitudeof the primary system:[1− �2 +

34R2 − 1

2ab(

2q2

+q2 − 4�2

�

)R2]2

+(�0�+

14��R2 +

12ab

��

R2)2

− 14�2c2 = 0;

(7)

which can be used for determining R in dependence on�. Then, by means of Eq. (6), the vibration amplituder in dependence on � can also be determined. It isinteresting to see that R depends only on the productab, i.e. the case when a and b are both positive ornegative yields the same result, if the absolute valuesof a and b are the same.The results of this harmonic balance approach

are presented in terms of resonance curves showingthe oscillation amplitude R (in x-direction), r (iny-direction) and constant deIection Y0 in dependenceof �, the half-frequency of parametric excitation. Thefollowing parameter values are used in this section:�c= 0:2; �= 0:4; �= 0:1, and �0 = 0. The relativelyhigh progressive damping coeGcient ensures ratherlimited vibration amplitudes at parametric resonance.For reasons of comparison, in Fig. 2 ( = 0) andFig. 3 ( = 0:2), we show the resonance curve R(�)


0.96 0.98 1 1.02 1.04 1.06 1.08η

0

0.2

0.4

0.6

0.8

R

Fig. 3. Resonance curve R(�) corresponding to the parametersq = 2; = 0:2 without coupling (a = b = 0); the dashes refer tothe unstable solution.

0.94 0.96 0.98 1 1.02 1.04η

0.2

0.4

0.6

0.8

R, r

, - Y

0

Fig. 4. Resonance curve R(�); r(�) and constant deIection Y0 cor-responding to the parameters q=2; =0 with coupling (a=b=0:5);numerical results are indicated by dots.

-1

0

1

[y]

0.9 1.0 1.1η

0.9 1.0 1.1η

-1

0

1

[x]

Fig. 5. Numerically computed extreme values [x] and [y] of oscillation amplitudes along the x- and y-direction versus excitation frequency� in the case �c = 0:2; �0 = 0; � = 0:4; � = 0:1; a = b = 0:5; = 0; q = 2. The arrows indicate the direction of the jumps in vibrationamplitude when increasing or decreasing the frequency.

for the case of zero cross-coupling (a = b = 0). Thefull line denotes the stable solution, the dashed linethe unstable one.Fig. 4 shows the analytically predicted amplitudes

R; r and the constant deIection Y0 in dependenceon the frequency � for the case q = 2; = 0 andcross-coupling (a = b = 0:5), as determined by Eqs.(2). We can see that the resonance curves for the mo-tion in the x-direction exhibit two peaks, having max-ima smaller than the corresponding maxima for thecase where a= b=0 (see Fig. 2). In the same 6gure,the results of the analysis have been supplemented bydirect numerical solution of the di>erential equations(1). We can see that there is a good agreement be-tween analytical and numerical results, also due to therelatively small value of constant deIection.In a further analysis, the numerical solution was

obtained when increasing and decreasing the exci-tation frequency � by small steps in suitable timeintervals. The extreme values (maxima and min-ima) of the oscillation amplitudes along the x- andy-direction, denoted [x] and [y], were recorded andare shown in Fig. 5. The arrows indicate the di-rection of jumps in vibration amplitude due to thecontinuous change of excitation frequency. As a re-sult, one can observe a noticeable bilateral hysteresise>ect.Two sets of similar diagrams show the e>ect of

detuning from internal resonance: q = 1:8 (Figs. 6and 7) and q = 2:2 (Figs. 8 and 9). In both cases,the amplitudes R are higher than for the case q = 2,while the amplitudes r are smaller than for q = 2


(see Figs. 4 and 5). The shape of the resonance curvesdoes not exhibit the double peak form anymore: forq = 1:8, the resonance curve is bended towards thehigher values of � and towards the smaller values of �for the alternative q = 2:2. The vibration character isclose to a harmonic one because for both [x] and [y]

q = 1.8γ = 0

0.9 1.0 1.1η

0.0

0.5

1.0

Rr

- Y

0

-Y0

R

r

Fig. 6. Oscillation amplitudes R; r and constant de-Iection Y0 versus excitation frequency � in the case�c=0:2; �0 =0; �=0:4; �=0:1; a=b=0:5; =0; q=1:8. Com-parison between analytical predictions (full line, stable solution;dashed line, unstable solution) and numerical solutions.

-1

0

1

[y]

0.9 1.0 1.1η

0.9 1.0 1.1η

-1

0

1

[x]

Fig. 7. Numerically computed extreme values [x] and [y] of oscillation amplitudes along the x- and y-direction versus excitation frequency� in the case �c = 0:2; �0 = 0; �= 0:4; � = 0:1; a= b= 0:5; = 0; q= 1:8. The arrows indicate the direction of the jumps in vibrationamplitude when increasing or decreasing the frequency.

the scatter of the record points is small, although thesolution cannot be considered as a periodic solution inthe strict sense but rather as a transient vibration withslightly changing frequency. The hysteresis e>ect isreduced for q = 1:8, but becomes more pronouncedfor q= 2:2.

q = 2.2

γ = 0

0.9 1.0 1.1η

0.0

0.5

1.0

Rr

- Y

0

-Y0

R

r

Fig. 8. Oscillation amplitudes R; r and constant de-Iection Y0 versus excitation frequency � in the case�c=0:2; �0 =0; �=0:4; �=0:1; a=b=0:5; =0; q=2:2. Com-parison between analytical predictions (full line, stable solution;dashed line, unstable solution) and numerical solutions.


-1

0

1

[y]

0.9 1.0 1.1η

0.9 1.0 1.1η

-1

0

1

[x]

Fig. 9. Numerically computed extreme values [x] and [y] of oscillation amplitudes along the x- and y-direction versus excitation frequency� in the case �c = 0:2; �0 = 0; �= 0:4; � = 0:1; a= b= 0:5; = 0; q= 2:2. The arrows indicate the direction of the jumps in vibrationamplitude when increasing or decreasing the frequency.

3. Scaling and �rst-order averaging

We shall now use a normal form method (aver-aging) which enables us to obtain more detailed in-formation about the solutions and for which preciseerror estimates are known. As before we considersystem (1) in the vicinity of the origin. To makethis more explicit we introduce the following generalscaling:

x = ��x Nx; y = ��y Ny; a= ��a Na; b= ��b Nb;

� = �� N�; �= �� N�;

= �� N; and �0 = ��0 N�0;

where, as before, � is a small, positive parameter. Bal-ancing the terms in the 6rst equation of system (1)yields

�� + 2�x = � + 2�x = �a + �y = ��0 = 1:

Balancing the terms in the second equation of system(1) yields

�� = �b + 2�x − �y = 1:

We have here six equations with eight unknowns,hence the scaling is not unique. Solving these equa-tions yields

��0 = �� = 1; � = �� = 1− 2�x; �a = 1− �y;

�b = 1 + �y − 2�x:

The third and fourth equations yield

06 �x6 1=2; (8)

06 �y6 1: (9)

We can freely choose �x as well as �y. However, oneshould 6rst examine the magnitude of the physicalparameters involved in system (1) before 6xing thevalues of �x and �y. Here, we base our choice onthe parameters of Section 2. We decided to take theparameter � to be O(1). This immediately means�x =1=2. The parameter a was taken to be equal to b.This implies �y=�x=�a=�b=1=2 and thus, x=�1=2 Nx,y= �1=2 Ny, a= �1=2 Na, b= �1=2 Nb, �= � N�, �= N�, = N and�0 = � N�0.Introducing this scaling into system (1) and omit-

ting the bars yields

Cx + (1 + �c cos 2�t)x + ��0x

+ �axy + �(�x2x + x3) = 0;

Cy + q2y + ��y + �bx2 = 0: (10)

The next steps are the usual ones in averaging approx-imations; see for instance [8,12, Chapter 11].We introduce the following transformation:

x(t) = x1(t) cos t + x2(t) sin t;

x(t) =−x1(t) sin t + x2(t) cos t;


y(t) = y1(t) cos qt +1qy2(t) sin qt;

y(t) =−qy1(t) sin qt + y2(t) cos qt:

Averaging the resulting system of di>erential equa-tions for x1;2; y1;2 yields

x1a = �

{ (−a

q�(q)y2a − 1

2�0

)xa1

+ (a�(q)y1a + c�(2�))x2a + O(|X |3)}

;

x2a =−�

{(−a�(q)y1a − c�(2�)) x1a

+(−a

q�(q)y2a +

12�0

)x2a + O(|X |3)

};

y 1a =�q(−2b�(q)x1ax2a − 1

2 �qy1a);

y 2a =−�(−b�(q)x21a + b�(q)x22a +12 �y2a); (11)

where the subscript a indicates ‘approximation’ and

�(q) =

{− 14 if q=±2;

0 otherwise:

3.1. Stability of the trivial solution (the generalcase)

Linearising system (11) around the origin yields

ddt

(X

Y

)= A

(X

Y

);

where

A=

− 12 �0� c�(2�)� 0 0

c�(2�)� − 12 �0� 0 0

0 0 − 12 �� 0

0 0 0 − 12 ��

:

The eigenvalues of the matrix A are

�1 = (−c�(2�)− 12 �0)�; �2 = (c�(2�)− 1

2 �0)�;

�3 = �4 =− 12 ��:

Conclusion. According to this linear analysis we candistinguish between the following cases:

• � �= ±1: In this case, all the eigenvalues have anegative real part. The averaged trivial solution isasymptotically stable. This result holds also for theoriginal system (10).

• � = ±1 and �0 ¿c=2: This case yields asymptoticstability as well for the averaged system (11). Theresult also holds for the original system (10). This isremarkable as we have resonance and energy trans-fer to the system.

• � = ±1 and �0 ¡c=2: One eigenvalue has, in thiscase, a positive real part. This implies the instabilityof the averaged trivial solution. The result also holdsfor the original system (10).

• � = ±1 and �0 = c=2: In this case, one of thefour eigenvalues of the matrix A is zero, whereasthe other three are eigenvalues with negative realparts. Linear analysis is, in this case, not conclu-sive regarding the stability properties of the aver-aged trivial solution. We must therefore examinemore closely the Iow in the centre manifold. SeeAppendix A [9,13,16].

3.2. The near resonance case q2 =4+ ��, �=1+ ��

Let us assume that the half-frequency of paramet-ric excitation � is not exactly equal to ±1. We shallinstead allow a margin of detuning of magnitude ��with � being a constant not dependent on �. We alsoallow a margin of detuning in q2 of magnitude ��. Inthis way, system (10) becomes

Cx + (1 + �c cos 2(1 + ��)t)x + ��0x + �axy

+ �(�x2x + x3) = 0;

Cy + 4y + ��y + ��y + �bx2 = 0: (12)

Transforming = (1 + ��)t and di>erentiating withrespect to yields the following:

Cx + (1 + �c cos 2 )x − 2��x + ��0x + �axy

+ �(�x2x + x3) + O(�2) = 0;


Cy + 4(1− 2�� + ��=4)y + ��y + �bx2

+O(�2) = 0: (13)

Applying the method of averaging as before produces

x1a = �

{ (a8y2a − �0

2− �

8(x21a + x22a)

)x1a

+(−a4y1a − c

4− � +

38(x21a + x22a)

)x2a

};

x2a =−�

{ (a4y1a +

c4− � +

38(x21a + x22a)

)x1a

+(a8y2a +

�02

+�8(x21a + x22a)

)x2a

};

y 1a =�2

(12bx1ax2a − �y1a +

�4y2a − 2�y2a

);

y 2a =−�

(b4x21a −

b4x22a +

�2y1a

+12�y2a − 4�y1a

): (14)

Linearising system (14) around the origin yields

ddt

(X

Y

)= A

(X

Y

);

where

A=

− �02 � −(c4 +�)� 0 0

−(c4 −�)� −�02 � 0 0

0 0 −12 �� (�8 −�)�

0 0 (− 12 �+4�)� −1

2 ��

The eigenvalues of the matrix A are

�1 =−2�0 +√

c2 − 16�2

4�;

�2 =−2�0 +

√c2 − 16�2

4�;

�3 =−��2

+ i(� − 8�)�

4;

�4 =−��2

− i(� − 8�)�

4:

Conclusion. We introduce the quantity z� = �=cto analyse the eigenvalues of the matrix A. z� willplay in the sequel an important role in the studyof the periodic solutions. This will be the mainsubject of the next subsection. According to linearanalysis, we can distinguish between the followingcases:

• |z�|¿ 14 , c

2−16�2 ¡ 0: this case encloses two sub-cases, namely:1. �0 ¿ 0: This case yields asymptotic stability of

the averaged trivial solution as all the eigenval-ues have negative real part. The result holds alsofor the original system (13).

2. �0 = 0: In this case, two of the four eigenval-ues are purely imaginary, whereas the othertwo are eigenvalues with negative real parts.Linear analysis is, in this case, not conclusiveregarding the stability of the averaged triv-ial solution. We therefore must examine moreclosely the Iow in the centre manifold. SeeAppendix A.

• |z�|6 14 , c2 − 16�2¿ 0: this case encloses three

subcases, namely:1. �0 ¿ 1=2

√c2 − 16�2: This case yields asymp-

totic stability of the averaged trivial solu-tion as all the eigenvalues have negative realpart. This result holds also for the originalsystem (13).

2. �0 ¡ 1=2√

c2 − 16�2: In this case, �2 ¿ 0which immediately implies the instability of theaveraged trivial solution. This result holds alsofor the original system (13).

3. �0 = 1=2√

c2 − 16�2: According to whether|z�| = 1=4 or not, we have (�1 = �2 = 0) or(�2 = 0 and �1 ¡ 0). The rest of the eigen-values has in both cases negative real parts.Linear analysis is, in this case, not conclusiveregarding the stability of the averaged triv-ial solution. We therefore must examine moreclosely the Iow in the centre manifold. SeeAppendix A.


Remark. The case (|z�|¿ 1=4 and �0 = 0) corre-sponds to a Hopf-bifurcation, with respect to theparameter �0, of the averaged trivial solution as twoof the four eigenvalues cross the imaginary axis withnon-zero speed. The introduction of the detuningparameter � has a clear inIuence on the stabilityproperties of the trivial solution.

3.3. The periodic solutions

We look in this section for periodic solutions ofsystem (13). We introduce, for this purpose, thephase-amplitude transformation

x(t) = R1(t) cos(t + "(t));

x(t) =−R1(t) sin(t + "(t));

y(t) = R2(t) cos(2t + (t));

y(t) =−2R2(t) sin(2t + (t)):

Averaging the resulting system yields

R1a = �R1a

{a4R2a sin(2"− ) +

c4sin 2"

−�8R21a −

�02

};

"a = �

{a4R2a cos(2"− ) +

c4cos 2"

+38R2

1a − �

};

R2a =�2R2a

{−bR21a

4R2asin(2"− )− �

};

a =�2

{bR2

1a

4R2acos(2"− ) +

12(� − 8�)

}: (15)

Equating the right-hand side of system (15) to zeroyields the non-trivial critical points of these equa-tions which correspond with 2#-periodic solutions ofthe original system (13). Without loss of general-ity we assume R2a ¿ 0. For notational simplicity, we

introduce the following quantities:

�=−3(4�2 + �2

�) + ab��

|b|c√4�2 + �2

�

;

$ =�(4�2 + �2

�) + 2ab�

|b|c√4�2 + �2

�

;

z =�0c; z� =

�c; and �� = � − 8�:

The results for the non-trivial critical points corre-sponding with periodic solutions are summarised asfollows:

R21a =

2√4k2 + �2

�

|b| R2a; (16)

cos(2"− ) =−sgn(b)��√4�2 + �2

�

; (17)

sin(2"− ) =−2 sgn(b)�√

4�2 + �2�

; (18)

cos 2"= $R2a + 2z; (19)

sin 2"= �R2a + 4z�; (20)

R+2a =

−(2$z + 4�z�) +√

�2 + $2 − 4(z�− 2z�$)2

�2 + $2

¿ 0; (21)

R−2a =

−(2$z + 4�z�)−√

�2 + $2 − 4(z�− 2z�$)2

�2 + $2

¿ 0: (22)

3.4. Stability of the periodic solutions in the case ofexact resonance, i.e. � = 0, � = 0

Linearising system (15) around the non-trivial crit-ical points yields

ddt

(X

Y

)= �A

(X

Y

);


where

A=

− �R21a4

−3R31a

4 − aR1a sgn(b)4 0

3R1a4 −�0 − �R2

1a4 0 − aR2a sgn(b)

4

|b|R1a

4 0 − k2 0

0 |b|R21a

4R2a0 − |b|R2

1a8R2a

:

The eigenvalues of this matrix are unfortunately toocomplicated to write down explicitly. In Appendix Bwe prove the following results for the stability of theperiodic solutions:

1. Suppose ab¿ 0; z ¡ 1=2, then the periodic solu-tion with amplitude along the y-direction equal toR+2a will be stable no matter what the other param-

eters are.2. The periodic solution with amplitude along the

y-direction equal to R−2a will, if it exists, always

be unstable.3. Given the parameters c; �0; �; ; b and �, suppose

that ab¡ 0 and z¡ 12 , then there exists as ¿ 0 such

that the periodic solution with amplitude R+2a along

the y-direction will be stable provided |a|¡as.4. Given the parameters c; �; ; b and �, suppose

that $¡ 0, then there exist as1, as2 both positiveand zs ¿ 1

2 such that the periodic solution with am-plitude R+

2a along the y-direction will be stable forall 1=26 z6min(1=2

√1 + ($=�)2; zs), provided

as1 ¡ |a|¡as2.5. Given the parameters c; �0; �; ; b and �, suppose

that ab¡ 0, then there exists 0¡au ¡∞ suchthat the periodic solution with amplitude R+

2a alongthe y-direction will become unstable provided|a|¿ au.

3.5. Stability of the periodic solutions in the case�0 = 0, = 0

Linearising system (15) around the non-trivial crit-ical points yields

ddt

(X

Y

)= �A

(X

Y

);

where

A=

− �R21a4

2�R1a4

−akR1a sgn(b)2D

aR1aR2a�� sgn(b)4D

0 − �R21a4

−a�� sgn(b)4D

−a�R2a sgn(b)2D

�R1a|b|2D

R21a��|b|4D − k

2−R2

1a��|b|8D

−R1a��|b|4DR2a

�R21a|b|

2DR2a

R21a��|b|8D R2

2a

−�R21a|b|

4DR2a

;

D =√4�2 + �2

�:

The analysis in Appendix C of the Routh–Hurwitzsystem of conditions (see [14,15]) for the stability ofthe periodic solutions yields:

1. Suppose ab¿ 0 and |�|¡ 4√2�, then the periodic

solution with amplitude along the y-direction equalto R+

2a will, if it exists, be stable no matter what theother parameters are.

2. Suppose ab¿ 0 and |�|¡ 1=2�, then the periodicsolution with amplitude along the y-direction equalto R+

2a will, if it exists, be stable no matter what theother parameters are.

3. The periodic solution with amplitude along they-direction equal to R−

2a will, if it exists, alwaysbe unstable.

4. Given the parameters c; �; �; �; b and �, sup-pose that ab¡ 0 and |z�|¡ 1

4 , then there existsas ¿ 0 such that the periodic solution with ampli-tude R+

2a along the y-direction will be stable pro-vided |a|¡as.

5. Given the parameters c; �; �; �; b and �, supposethat ab¿ 0; |z�|¡ 1

4 and |�|¿ 1=2�, then there ex-ist �1, �2 and au ¿ 0 such that the periodic solutionwith amplitude R+

2a along the y-direction will be-come unstable provided |a|¿ au and �1 ¡�¡�2.

6. Given the parameters c; �; �; �; b and �, supposethat ab¡ 0; |z�|¡ 1

4 and |�|¡ 1=2�, then thereexists au ¿ 0 such that the periodic solution withamplitude R+

2a along the y-direction will becomeunstable provided |a|¿ au.

7. Given the parameters c; �; �; �; b and �, supposethat ab¡ 0; |z�|¡ 1

4 and |�|¡ 4√2�, then there

exists au ¿ 0 such that the periodic solution withamplitude R+



4. Harmonic balance versus averaging method

The harmonic balance method, which is essentiallya Fourier projection, and the averaging method areclassical methods but they are seldom compared; seealso the discussion in [2, Chapter 9].In Figs. 10 and 11, we have a superposition of

the results obtained by both methods in terms ofresonance curves showing the oscillation amplitudesin x- and y-direction. There is good agreement be-tween the results of both methods. Making � smaller

0.94 0.96 0.98 1 1.02 1.04 1.06η

0

0.2

0.4

0.6

0.8

R

Fig. 10. Oscillation amplitude R versus excitation frequency �by superposition of the results from the harmonic balance (boldline) and the 6rst-order averaging method corresponding to theparameters �=0:1; c=2; a=b=0:5=

√�, �0=0, �=0:4; �=1; =0,

� = 0 (q = 2). The dashes refer to the unstable solutions.

0.94 0.96 0.98 1 1.02 1.04 1.06η

0.1

0.2

0.3

0.4

r

Fig. 11. Oscillation amplitude r versus excitation frequency �by superposition of the results from the harmonic balance (boldline) and the 6rst-order averaging method corresponding to theparameters �=0:1; c=2; a=b=0:5=

√�, �0=0; �=0:4; �=1; =0,

� = 0 (q = 2). The dashes refer to the unstable solutions.

0.985 0.99 0.995 1 1.005 1.01 1.015η

0

0.1

0.2

0.3

0.4

R

Fig. 12. Oscillation amplitude R versus excitation frequency� by superposition of the results from the harmonic balance(bold line) and the 6rst-order averaging method correspond-ing to the parameters � = 0:025; c = 2; a = b = 0:25=

√�,

�0 = 0; �= 0:4; �= 1; = 0; �= 0 (q= 2). The dashes refer tothe unstable solutions.

0.985 0.99 0.995 1 1.005 1.01 1.015η

0.05

0.1

0.15

0.2

r

Fig. 13. Oscillation amplitude R versus excitation frequency� by superposition of the results from the harmonic balance(bold line) and the 6rst-order averaging method correspond-ing to the parameters � = 0:025; c = 2; a = b = 0:25=

√�,

�0 = 0; �= 0:4; �= 1; = 0; �= 0 (q= 2). The dashes refer tothe unstable solutions.

and keeping the parameters a and b constant in theaveraged system does improve this agreement, as ex-pected. See Figs. 12 and 13.

5. The Neimark–Sacker bifurcation

We know from the propositions above that the pe-riodic solution with amplitude along the y-directionequal to R+

2a becomes at some point unstable. This


is an interesting situation as also the trivial solutionis unstable. The solutions show then a very di>erentbehaviour as they are attracted to a torus on whichwe have quasi-periodic dynamics. One of the possiblecauses for this behaviour can be a Neimark–Sackerbifurcation, also known as a secondary Hopf bifurca-tion. We have made use of the special software CON-TENT to pinpoint this bifurcation; see [9], AppendixC. We 6rst found a periodic solution when ab¡ 0with b¡ 0 and small. When running CONTENT, weused the method of continuation with, this time, b as acontrol parameter and monitored the multipliers of theperiodic solution. CONTENT’s results are presentedbelow.

5.1. Numerical data generated by CONTENT

The following parameters, with respect to system(10), have been used in our numerical analysis. � =0:1; c = 1; a = 0:5=

√�; � = 1; �0 = 0; � = 1; � =

0:4; =0:2, and �=0:8 (q=2:02). System (10) has tobe made autonomous to be able to spot the bifurcationwith CONTENT. Its dimension becomes therefore of

NS

x [0]

x [2

]

3

2

1

0

-1

-2

-3

Fig. 14. Figure generated by CONTENT; for the case under consideration see Section 5.1. The cycles (prior to the bifurcation) areprojected on the (x=

√�; y=

√�) plane. The outer cycle with dots is the one that bifurcates. NS stands for Neimark–Sacker bifurcation.

the sixth order:

x1 = x2;

x2 =−(1 + �cz1)x1 − �ax1y1 − �(�x21x2 + x31);

y 1 = y2;

y 2 =−(4 + ��)y1 − ��y2 − �bx21 ;

z1 = z1 − 2z2 − z1(z21 + z22);

z2 = z2 + 2z1 − z2(z21 + z22): (23)

A Neimark–Sacker bifurcation has been found at thecritical value bc=−0:179576=

√�. Note that, because

of the scaling introduced in Section 3, to obtain theoriginal values of b corresponding to system (1), wehave to multiply with

√�.

The corresponding multipliers computed by CON-TENT, presented in the modulus-argument form, areas follows:

)1 = 1; "1 = 0:207607;

)2 = 1; "2 =−0:207607;


Fig. 15. Case b¿bc with vibration records showing attraction to a periodic solution of x(t) (left) and y(t) (right) corresponding to theparameter values � = 0:1; c = 1, a = 0:5=

√�; b =−0:1=

√�; �0 = 0, � = 1; � = 0:4; = 0:2; � = 1; � = 0:8 (q = 2:02).

250 500 750 1000 1250 1500 1750 2000t

-6

-4

-2

2

4

6

x

250 500 750 1000 1250 1500 1750 2000t

-1.5

-1

-0.5

0.5

1

1.5

y

Fig. 16. Case where b is just above bc with vibration records showing ‘slow’ attraction to a periodic solution (just before bifur-cation) of x(t) (left) and y(t) (right) corresponding to the parameter values � = 0:1; c = 1, a = 0:5=

√�; b = −0:15=

√�; �0 = 0,

� = 1; � = 0:4; = 0:2; � = 1; � = 0:8 (q = 2:02).

250 500 750 1000 1250 1500 1750 2000t

-6

-4

-2

2

4

6

x

250 500 750 1000 1250 1500 1750 2000t

-2

-1

1

2

3

y

Fig. 17. Case b¡bc with vibration records of attraction to a torus of x(t) (left) and y(t) (right) corresponding to the parameter values� = 0:2; c = 1; a = 0:5=

√�, b =−0:2=

√�; �0 = 0; � = 1; � = 0:4, = 0:2; � = 1; � = 0:8 (q = 2:02).

)3 = 1; "3 = 0;

)4 = 0:623983; "4 =−0:163293;

)5 = 0:623983; "5 = 0:163293;

)6 = 3:4873× 10−6; "6 = 0:

We conclude that when the control parameter bgoes below the threshold value bc = −0:179576=

√�

a Neimark–Sacker bifurcation takes place (see Fig.14). Numerical results shown in Figs. 15–17 con6rm

this very clearly. We can see from these 6gures thatas the parameter b decreases, it takes longer for theperiodic solution to stabilise. When b drops belowbc, the periodic solution looses its stability in thebifurcation.

5.2. Averaging method results

Remarkably enough, one can also track down theNeimark–Sacker bifurcation by looking for a Hopfbifurcation of the non-trivial critical points of the


averaged system (15). This way, we will be able tolocate this bifurcation without use of sophisticatedsoftware like CONTENT. We have done this in thecase �0=0 and compared the averaged results with themore accurate data obtained by CONTENT. Startingat b=−0:1=

√�, the averaged system has a non-trivial

critical point which undergoes a Hopf bifurcation atthe critical value bc � −0:171=

√�. This is a rather

good 6rst-order approximation of the more accuratevalue bc = −0:179576=

√� computed by CONTENT.

The relative error in b is as follows:

∣∣∣∣bc − bc

bc

∣∣∣∣ � 5%:

The averaging method predicts with satisfactory pre-cision the bifurcation point bc. This precision will im-prove if we take � to be smaller.

6. Conclusions

1. The instability intervals in parameter space ofthe trivial solution of the parametrically excitedprimary system are not changed much by the pres-ence of the non-linear coupling to the secondaryoscillator.

2. In the examined system we chose the prominentresonance 1:2 both for the internal and the para-metric resonance. For other resonances, the e>ectof non-linear coupling to the stability of the trivialsolution is expected to be even weaker.

3. Non-trivial solutions become important whenthe trivial solution is unstable. In particular, theattracting torus which we found and which ischaracterised by fairly large amplitudes can beundesirable from an engineering point of view.

4. There are not many comparisons of the harmonicbalance method and other analytic approximationmethods in the literature. We have found goodagreement between the periodic solution results ofharmonic balance and averaging.

5. In most cases Neimark–Sacker bifurcations arestudied by numerical means. Interestingly, we canalso analyse this bifurcation in our problem byusing the normal form method of averaging. Theresults are in good agreement with the numerics.

Appendix A. Study of the centre manifold

When the trivial solution is non-hyperbolic, theproblem of stability becomes much more complicatedas the stability of the trivial solution in the centremanifold of the averaged system does not have toimply the stability in the original system. There isnot much known on the relation between centre man-ifolds in (averaging) normal forms and the originalsystem. In spite of this, we shall study the Iow inthe averaged centre manifold and check numericallywhether this suggests that the result holds for the orig-inal system. In the following, we present the resultsof Section 3 with details for one interesting case. Forthe background see [9] and the references therein:

• The case �=±1; �0 = c=2: In this case, one of thefour eigenvalues was equal to zero. Study of theaveraged centre manifold reveals that the Iow isgoverned by the following di>erential equation:

u=− �ab�(q)2

�qu3 + O(u4):

The averaged trivial solution is stable providedq = ±2 and ab¿ 0. Using numerical data, weconjecture that these results hold for the originalsystem as well.

• The case |z�|¡ 1=4; �0 = 1=2√

c2 − 16�2: Inthis case, one of the four eigenvalues equals zero.Study of the averaged centre manifold reveals thatthe Iow is governed by the following di>erentialequation:

u=− �abc(�0� + �(� − 8�))2�0(4�2 + (� − 8�)2)

u3 + O(u5):

We see from this equation that the trivial solutioncan be both stable and unstable depending on the pa-rameters involved in the system. Numerical analysissuggests this holds for the original system as well.

• The case |z�| = 1=4; �0 = 0: In this case, we aredealing with a two-dimensional centre manifold.This case is of such complexity that even the re-striction of the Iow to the averaged centre manifolddoes not yield a di>erential equation which can bestudied easily. We shall therefore draw our con-clusion from numerical analysis only. We 6nd thetrivial solution in this case to be unstable.


• The case |z�|¡ 1=4; �0 = 1=2√

c2 − 16�2: Thiscase will be studied in detail below.

A.1. Study of the centre manifold |z�|¿ 14 , �0 = 0

We 6rst apply the following transformation (see[13]:

x1a =1

2(c − 4�)x2a;

x2a =−1

2√16�2 − c2

x1a:

After which, we restrict the Iow to the centre manifoldand introduce the complex coordinates

y = x1a + ix2a; Ny = x1a − ix2a:

The resulting system is then normalised using the com-plex transformation

y = z + h2(z);

where h2(z) denotes a polynomial in z of degreem¿ 2. This normalisation eliminates all third-orderterms except z2 Nz. The system can then be transformedback into Cartesian coordinates. Omitting the tildesyields

x1a =Re(�1)x1a − Im(�1)x2a

+(�(�0)x1a − $(�0)x2a)(x21a + x22a)

+O(|xa1|5; |x2a|5);

x2a = Im(�1)x1a + Re(�1)x2a

+(�(�0)x2a + $(�0)x1a)(x21a + x22a)

+O(|xa1|5; |x2a|5): (24)

The parameters �(�0) and $(�0) will be speci6ed later.We 6nd it more convenient to work in polar coordi-nates. The system transforms to

r = Re(�1)r + �(�0)r3 + O(r5);

,= Im(�1) + $(�0)r2 + O(r4): (25)

We are especially interested in the dynamics of theIow near �0=0. It is therefore natural to Taylor expand

the coeGcient around �0 = 0 so that:

r =−�02

r + �(0)r3 + O(�0r3; r5);

,=

√16�2 − c2

4+ $(0)r2 + O(�0r2; r4): (26)

Taking �0 = 0 and neglecting the higher-order termsin the system produces

r = �(0)r3;

,=

√16�2 − c2

4+ $(0)r2: (27)

where

�(0) =

2ab�(c−4�)(c2(�−16�)+2�(4�2+(�−16�)2))(4c2−(4�2−�(�−16�)))2+16�2(�−8�)2

;

$(0) =−ab(c−4�)-

4√16�2−c2(16c4−8c2(4�2−�(�−16�))

+−ab(c−4�)-

(4�2+�2)(4�2+(�−16�)2))(28)

with

-= (32c6(8� − �)− 4c4(1536�3 + 4�2(56� − 5�)

+ 64�2� − 56��2 + 3�3)

−32�2�(16�4 + (128�2 − 24�� + �2)2

+ 8�2(160�2 − 24�� + �2))

+ c2(131072�5 + 16�4(24� − �)− 16384�4�

−512�3�2 − 64�2�3 + 24��4 − �5

+ 8�2(3328�3 − 416�2� + 24��2 − �3)))

=(4�2 + (� − 16�)2): (29)

The sign of �(0) is decisive for the stability of thetrivial solution. If �(0)¡ 0, the trivial solution will bestable, if �(0)¿ 0, the trivial solution will be unsta-ble. After a detailed study of the numerator we cameto the following conclusion. The sign of �(0) dependsstrongly on all the parameters involved in this system.Some cases will yield stability and others will give in-stability. This result holds of course for the averagedsystem. Numerical analysis suggests the stability re-sults hold for the original system as well.


Remark. From the study above, we can state thatwhen �0 is near zero and �(0) is positive, the averagedsystem has an unstable cycle. The ‘averaged’ trivialsolution is in this case asymptotically stable. If, forsome values of the parameters, the Iow in the centremanifold is stable or unstable, then reversing the signof the parameter a or bwill make the Iow respectivelyunstable or stable. Numerical results suggest this holdsalso for the original system.

Appendix B. Stability of the periodic solutions inthe case of exact resonance, i.e. � = 0; � = 0

The results of Section 3.4 can be obtained usingthe Routh–Hurwitz criterion to study the stability ofthe periodic solutions. A necessary and suGcient con-dition for the stability of the periodic solution trans-lates in our case, after some simpli6cations, to thefollowing:

�0 + � +2��R2a

|b| ¿ 0;

8�(�2+92)k2R32a+4k(9�02+�2(3�0+4k))R2

2a|b|+2�R2a|b|2(2(�20+4�0k+2k2)+akR2a sgn(b))

+ |b|3((2�0+k)2+a(�0+k)R2a sgn(b))¿ 0;

8�(�2+92)k3R32a+4k2R2

2a|b|(9�02+�2(3�0 + 2k)

+ a�2R2a sgn(b))+2�kR2a|b|2(2�20+4�0k

+ k2 + 2a(�0+k)R2a sgn(b))

+ |b|3(�0k(2�0+k)+a(�0+k)2R2a sgn(b))

¿ 0;

4(�2 + 92)k2R2a + 4�k|b|(�0 + aR2a sgn(b))

+ a|b|2 sgn(b)(2�0 + aR2a sgn(b))¿ 0: (30)

Proposition 1. Suppose ab¿ 0; z ¡ 1=2, then theperiodic solution with amplitude along the y-directionequal to R+

2a will be stable no matter what the otherparameters are.

Proof. This can easily be seen from the Routh–Hurwitz equations. The condition z¡ 1

2 guarantees

the existence of the periodic solution as $ is posi-tive.

Proposition 2. The periodic solution with amplitudealong the y-direction equal to R−

2a will, if it exists,always be unstable.

Proof. The fourth equation of the Routh–Hurwitz sys-tem can be rewritten as follows:

(�2 + $2)b2c2R2a + 4��|b|�0 + 2a|b|b�0 ¿ 0: (31)

Having done this, it is now easy to see that R−2a will

always yield a negative number.

Remark. From Eq. (31) we see that R+2a will always

satisfy the fourth inequality of the Routh–Hurwitz sys-tem.

Proposition 3. Given the parameters c; �0; �; ; band �, suppose that ab¡ 0 and z¡ 1

2 , then there ex-ists as ¿ 0 such that the periodic solution with ampli-tude R+

2a along the y-direction will be stable provided|a|¡as.

Proof. When a tends to zero, the parameter $ be-comes positive. One can easily see that the periodicsolution with amplitude R+

2a still exists because z¡ 12 .

Taking a = 0 in the Routh–Hurwitz system of con-ditions, we can see that the positivity condition ismet. As R+

2a and the Routh–Hurwitz system of condi-tions depend continuously on the parameter a, we canuse the continuity principle to prove the existence ofas ¿ 0 such that the positivity conditions will still bemet provided |a|¡as.

Proposition 4. Given the parameters c; �; ; b and�, suppose that $¡ 0, then there exist as1, as2 bothpositive and zs ¿ 1

2 such that the periodic solutionwith amplitude R+

2a along the y-direction will be stablefor all 1=26 z6min(1=2

√1 + ($=�)2; zs), provided

as1 ¡ |a|¡as2.

Proof. De6ne as1 as follows:

lim|a|→as1

$ = 0:


Taking z = 12 , we 6nd

lim|a|→as1

R2a = 0:

Putting |a|= as1 and z= 12 in the Routh–Hurwitz sys-

tem, we can see that the positivity condition is met.Using the continuity principle, we can prove the ex-istence of an open region U around ( 12 ; |as1|) in theza-plane such that the positivity condition will still besatis6ed provided the point (z; a) lies in this region U .De6ning the parameters as2 and zs such that[12 ; zs]× (as1; as2) ⊂ U

concludes the proof. The condition z6min(1=2

√1 + ($=�)2; zs) guarantees the existence of the

periodic solution.

Proposition 5. Given the parameters c; �0; �; ; band �, suppose that ab¡ 0, then there exists0¡au ¡∞ such that the periodic solution withamplitude R+


Proof. When |a| tends to in6nity (keeping the otherparameters constant), the parameter $ will tend to−∞. The parameter � is not a>ected by a (exact reso-nance). The threshold value z=1=2

√1 + ($=�)2 will

hence never be exceeded. The periodic solution withamplitude R+

2a will consequently still exist. Its magni-tude is as follows:

lim|a|→∞

aR+2a = (2�0 + c) sgn(a):

In other words R+2a becomes insigni6cant when |a|

increases. Bearing this in mind and looking at the thirdcondition of the Routh–Hurwitz system, we 6nd thatas |a| increases, the last term of this equation willbecome decisive for its sign. As

lim|a|→∞

|b|3(�0k(2�0 + k) + a(�0 + k)2R2a sgn(b))

=− |b|3(c(�0 + �)2 + �0(2�20 + 2�0� + �2))

¡ 0;

we can prove, using the continuity principle, the ex-istence of 0¡au ¡∞ such that the third expressionof the Routh–Hurwitz system will remain negative orbecome zero as |a|¿ au.

Appendix C. Stability of the periodic solutions inthe case �0 = 0, � = 0

After some simpli6cations, the Routh–Hurwitz sys-tem of conditions based on Section 3.5 becomes inthis case

� +�DR2a

|b| ¿ 0;

4�3D2R32a + 16�2D�R2

2a|b|+ �|b|3

×(D + 2aR2a sgn(b))

+ 2�R2a|b|2(8�2 + aDR2a sgn(b))¿ 0;

16�5D4�R42a + 2a�2(D2 − 8��)|b|5 sgn(b)

+ 8�4D3R32a|b|(8�2 + aDR2a sgn(b))

+ 8�3D2�R22a|b|2(D2 + 8�2 + 4aDR2a sgn(b))

+ �D�|b|4(D3 + 4aR2a(D2 + 4�2 − 8��)sgn(b))

+ 2�2D2R2a|b|3(8D�2 + aR2a(D2 + 20�2

−8��)sgn(b))¿ 0;

�2R2aD3 + 4ab�kR2aD + a|b|2 sgn(b)×(4�� + aR2aD sgn(b))¿ 0: (32)

Proposition 6. Suppose ab¿ 0 and |�|¡ 4√2�,

then the periodic solution with amplitude along they-direction equal to R+

2a will, if it exists, be stable nomatter what the other parameters are.

Proof. The condition |�|¡ 4√2� implies D2 −

8�� ¿ 0. It is now easy to see that under this condi-tion, the Routh–Hurwitz system of conditions will besatis6ed.

Proposition 7. Suppose ab¿ 0 and |�|¡ 1=2�,then the periodic solution with amplitude along they-direction equal to R+

2a will, if it exists, be stable nomatter what the other parameters are.

Proof. The condition |�|¡ 1=2� implies D2 −8�� ¿ 0. It is now easy to see that under this condi-tion, the Routh–Hurwitz system of conditions will besatis6ed.


Proposition 8. The periodic solution with amplitudealong the y-direction equal to R−

2a will, if it exists,always be unstable.

Proof. The fourth equation of the Routh–Hurwitz sys-tem can be rewritten as follows:

(�2 + $2)b2c2DR2a + 4��ab2 sgn(b): (33)

Having done this, it is now easy to see that R−2a will

always yield a negative number.

Remark. From Eq. (33) we easily see that R+2a will al-

ways satisfy the fourth equation of the Routh–Hurwitzsystem of conditions.

Proposition 9. Given the parameters c; �; �; �; band �, suppose that ab¡ 0 and |z�|¡ 1=4, then thereexists as ¿ 0 such that the periodic solution with am-plitude R+

2a along the y-direction will be stable pro-vided |a|¡as.

Proof. When a tends to zero, the parameter � becomeszero. One can easily see that the periodic solutionwith amplitude R+

2a still exists because |z�|¡ 14 . Tak-

ing a=0 in the Routh–Hurwitz system of conditions,we can see that the positivity condition is met. As R+

2aand the Routh–Hurwitz system of conditions dependcontinuously on the parameter a, we can use the con-tinuity principle to prove the existence of as ¿ 0 suchthat the positivity conditions will still be met provided|a|¡as.

Proposition 10. Given the parameters c; �; �; �; band �, suppose that ab¿ 0; |z�|¡ 1

4 and |�|¿ 1=2�,then there exist �1, �2 and au ¿ 0 such that theperiodic solution with amplitude R+

2a along they-direction will become unstable provided |a|¿ au

and �1 ¡�¡�2.

Proof. When |�|¿ 1=2� then there are �1 and �2 suchthat

D2 − 8�� ¡ 0 ∀�∈ (�1; �2):

When |a| tends to in6nity, the periodic solution withamplitude R+

2a will still exist as |z�|¡ 14 . For its am-

plitude we have

lim|a|→∞

|a|R+2a = positive constant:

R+2a becomes consequently insigni6cant when |a| in-

creases. Bearing this in mind, we can see that the thirdcondition of the Routh–Hurwitz system becomes neg-ative as |a| tends to in6nity. We can now prove, usingthe continuity principle, the existence of 0¡au ¡∞such that the third equation of system (25) will be-come negative or zero provided |a|¿ au.

Proposition 11. Given the parameters c; �; �; �; band �, suppose that ab¡ 0; |z�|¡ 1

4 and |�|¡ 1=2�,then there exists au ¿ 0 such that the periodic so-lution with amplitude R+

2a along the y-direction willbecome unstable provided |a|¿ au.

Proof. When |�|¡ 1=2�, then D2−8�� will alwaysbe positive. When |a| tends to in6nity, the periodicsolution with amplitude R+

2a will still exist as |z�|¡ 14 .

For its amplitude we have

lim|a|→∞

|a|R+2a = positive constant:

R+2a becomes consequently insigni6cant when |a| in-

creases. Bearing this in mind, we can easily see thatthe left-hand side of the third condition of the Routh–Hurwitz system becomes negative as |a| tends to in-6nity and ab¡ 0. We can now prove, using the conti-nuity principle, the existence of 0¡au ¡∞ such thatthe third equation of system (25) will become nega-tive or zero provided |a|¿ au.

Proposition 12. Given the parameters c; �; �; �; band �, suppose that ab¡ 0; |z�|¡ 1

4 and |�|¡ 4√2�,

then there exists au ¿ 0 such that the periodic so-lution with amplitude R+

2a along the y-direction willbecome unstable provided |a|¿ au.

Proof. The proof runs similar to the previousone.

References

[1] M. Cartmell, Introduction to Linear, Parametric and NonlinearVibrations, Chapman & Hall, London, 1990.

[2] A. Tondl, M. Ruijgrok, F. Verhulst, R. Nabergoj, Auto-parametric Resonance in Mechanical Systems, CambridgeUniversity Press, New York, 2000.

[3] A. Tondl, To the analysis of autoparametric systems, Z.Angew. Math. Mech. 77 (1997) 407–418.

[4] A. Tondl, Nonlinearly coupled systems: parametric andself-excitation, Acta Tech. XCSAV 43 (1998) 493–505.


[5] A. Tondl, Nonlinearly coupled systems, InXz. Mech. (Eng.Mech.) 6 (2) (1999) 87–96 (in Czech).

[6] G. Schmidt, A. Tondl, Nonlinear Vibrations, CambridgeUniversity Press, Cambridge, 1986.

[7] A. Fatimah, M. Ruijgrok, Bifurcations in an autoparametricsystem in 1:1 internal resonance with parametric excitation,Int. J. Non-Linear Mech. 37 (2002) 297–308.

[8] F. Verhulst, Nonlinear Di>erential Equations and DynamicalSystems, Springer, Berlin, Heidelberg, 1996.

[9] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory,2nd Edition, Springer, New York, 1995.

[10] A. Tondl, R. Nabergoj, A nonlinearly coupled system withparametric excitation, StrojnZ[cky XCasopis 50 (1999) 398–411.

[11] A. Tondl, R. Nabergoj, A parametrically excited system withnonlinear coupling, Manuscript, 2000.

[12] J.A. Sanders, F. Verhulst, Averaging Methods in NonlinearDynamical Systems, Springer, New York, 1985.

[13] S. Wiggins, Introduction to Applied Nonlinear DynamicalSystems and Chaos, Springer, New York, 1990.

[14] A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics,Wiley, New York, 1995.

[15] L. Meirovitch, Methods of Analytical Dynamics,McGraw-Hill, New York, 1970.

[16] D. Ruelle, Elements of Di>erentiable Dynamics andBifurcation Theory, Academic Press, New York, 1989.

parametric excitation in non-linear dynamics

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