BIFURCATIONS IN AN AUTOPARAMETRIC SYSTEM IN
��� INTERNAL RESONANCE WITH PARAMETRIC
EXCITATION
�revised preprint nr� �����
S� Fatimah�� and M� Ruijgrok
Mathematisch Instituut� University of Utrecht� PO Box ������� ���� TA Utrecht�
The Netherlands
Abstract� We consider an autoparametric system which consists of anoscillator� coupled with a parametrically�excited subsystem� The oscil�lator and the subsystem are in � � � internal resonance� The excitedsubsystem is in � � � parametric resonance with the external forcing�The system contains the most general type of cubic nonlinearities� Us�ing the method of averaging and numerical bifurcation continuation� westudy the dynamics of this system� In particular� we consider the sta�bility of the semi�trivial solutions� where the oscillator is at rest and theexcited subsystem performs a periodic motion� We �nd various typesof bifurcations� leading to non�trivial periodic or quasi�periodic solu�tions� We also �nd numerically sequences of period�doublings� leadingto chaotic solutions�
�� Introduction
An autoparametric system is a vibrating system which consists of at leasttwo subsystems� the oscillator and the excited subsystem� This system isgoverned by dierential equations where the equations representing the oscillator are coupled to those representing the excited subsystem in a nonlinearway and such that the excited subsystem can be at rest while the oscillatoris vibrating� We call this solution the semitrivial solution� When this semitrivial solution becomes unstable� nontrivial solutions can be initiated� Formore backgrounds and references see Svoboda� Tondl� and Verhulst ��� andTondl� Ruijgrok� Verhulst� and Nabergoj ����We shall consider an autoparametric system where the oscillator is ex
cited parametrically� of the form�
x�� k�x� q��x ap���x f�x� y� � �
y�� k�y� q��y g�x� y� � �
�����
The �rst equation represents the oscillator and the second one is theexcited subsystem� An accent� as in x�� will indicate dierentiation with
�corresponding author
�
� Bifurcations in an Autoparametric System in ��� Internal Resonance
respect to time � and x� y � R� k� and k� are the damping coe�cients�q� and q� are the natural frequencies of the undamped� linearized oscillatorand excited subsystem� respectively� The functions f�x� y� and g�x� y�� thecoupling terms� are C� and g�x� �� � � for all x � R� The damping coe�cients and the amplitude of forcing a are assumed to be small positivenumbers� We will consider the situation that the oscillator and the externalparametric excitation are in primary � � � resonance and that there existsan internal � � � resonance�There exist a large number of studies of similar autoparametric systems�
The case of a � � � internal resonance has been studied by Ruijgrok ��� andOueini� Chin� and Nayfeh ���� in the case of parametric excitation� In Ruijgrok ��� the averaged system is analyzed mathematically� and an applicationto a rotor system is given� In Oueni� Chin� and Nayfeh ��� theoretical results are compared with the outcomes of a mechanical experiment� Tien�Namachchivaya� and Bajaj ��� also consider the situation that there existsa � � � internal resonance� now however with external excitation�In Tien� Namachchivaya� and Bajaj ��� and in Bajaj� Chang and John
son ��� the bifurcations of the averaged system are studied� and the authorsshow the existence of chaotic solutions� numerically in Bajaj� Chang� andJohnson ��� and by using an extension of the Melnikov method in Tien�Namachchivaya� and Bajaj ���� for the case with no damping� In Banerjee and Bajaj ���� similar methods as in Tien� Namachchivaya� and Bajaj��� are used� but now for general types of excitation� including parametricexcitation�The case of a � � � internal resonance has received less attention� In
Tien� Namachchivaya� and Malhotra ��� this resonance case is studied� incombination with external excitation� The author shows analytically thatfor certain values of the parameters� a �Silnikov bifurcation can occur� leading to chaotic solutions� In Feng and Sethna ��� parametric excitation wasconsidered� and also here a generalization of the Melnikov method was usedto show the existence of chaos in the undamped case�In this paper we study the behavior of the semitrivial solution of system
������ This is done by using the method of averaging� It is found that severalsemitrivial solutions can coexist� These semitrivial solutions come in pairs�connected by a mirrorsymmetry� However� only one of these �pairs of� semitrivial solutions is potentially stable� In section � we study the stability ofthis particular solution� the results of which are summarized in �dimensionalstability diagrams� In section � the bifurcations of the semitrivial solutionare analyzed� These bifurcations lead to nontrivial solutions� such as stableperiodic and quasiperiodic orbits� In section � we show that one of thenontrivial solutions undergoes a series of perioddoublings� leading to astrange attractor� The chaotic nature of this attractor is demonstrated bycalculating the associated Lyapunov exponents�Finally� we mention that in the averaged system we encounter a codi
mension � bifurcation� The study of this rather complicated bifurcation will
S� Fatimah and M� Ruijgrok �
be described in a separate paper� where we also use a method similar tothe one used in Tien� Namachchivaya� Malhotra ��� to show analytically theexistence of �Silnikov bifurcations in this system�
�� The averaged system
We will take f�x� y� � c�xy� �
�x�� g�x� y� � �
�y� c�x
�y� and p��� �cos �� � Let q�� � � ��� and q�� � � ���� where �� and �� are the detunings
from exact resonance� After rescaling k� � � �k�� k� � � �k�� a � ��a� x �p��x�
and y �p��y� then dropping the tildes� we have the system�
x�� x ��k�x� ��x a cos ��x
�
�x� c�y
�x� � �
y�� y ��k�y� ��y c�x
�y �
�y�� � �
�����
It is possible to start with a more general expression for f�x� y� and g�x� y��for instance including quadratic terms� We have limited ourselves to thelowestorder resonant terms� which in this case are of third order� and whichcan be put in this particular form by a suitable scaling of the x� y� and � coordinates� This is not a restriction� as a more general form for the couplingterms leads to the same averaged system and normal forms�The system ����� is invariant under �x� y�� �x��y�� �x� y�� ��x� y�� and
�x� y�� ��x��y�� In particular the �rst symmetry will be important in theanalysis of this system� We emphasize that these symmetries do not dependon our particular choice for f�x� y� and g�x� y�� but are a consequence of the��� and ��� resonances and the restriction that we have an autoparametricsystem� i�e� that g�x� �� � � for all x � R�We will use the method of averaging �see Sanders and Verhulst ���� for
appropriate theorems � to investigate the stability of solutions of system������ by introducing the transformation�
x � u� cos � v� sin � � x� � �u� sin � v� cos ������
y � u� cos � v� sin � � y� � �u� sin � v� cos ������
After substituting ����� and ����� into ������ averaging over � � and rescaling� � �
� �� � we have the following averaged system�
� Bifurcations in an Autoparametric System in ��� Internal Resonance
u�� � �k�u� ��� � �
�a�v� v��u
�� v���
�
�c�u
��v�
�
�c�v
��v�
�
�c�u�v�u�
v�� � �k�v� � ��� �
�a�u� � u��u
�� v����
�
�c�u
��u� �
�
�c�v
��u� �
�
�c�u�v�v�
u�� � �k�u� ��v� v��u�� v���
�
�c�u
��v�
�
�c�v
��v�
�
�c�u�v�u�
v�� � �k�v� � ��u� � u��u�� v����
�
�c�u
��u� �
�
�c�v
��u� �
�
�c�u�v�v�
�����
�� The semi�trivial solution
In this section we investigate the semitrivial solutions of system �����and determine their stability� From section �� the semitrivial solutions correspond to u� � v� � �� so that we have�
u�� � �k�u� ��� � �
�a�v� v��u
�� v���
v�� � �k�v� � ��� �
�a�u� � u��u
�� v���
�����
Apart from ��� ��� the �xed points of system ����� correspond with periodicsolutions of system ������ The nontrivial �xed points are
��u���v������ R�k
��
���� ��a R�
��q���� �
�a R�
��� k��
�� R�k�q���� �
�a R�
��� k��
�A
�����
where
R��� ��� �
r�
�a� � k�������
Assuming R� �� �� there are three cases� depending on the value of a and ��
�� If a � �p��� k��� there is one solution for R�
�
�� If �� � � and �k� � a � �p��� k��� there are two solutions for R
��
�� For a � �k�� there is no solution for R���
These results are summarized in Figure �� The phaseportraits of system����� in the �u�� v��plane for a speci�c value ���� a� in these regions areindicated in Figure �� Note that the �xedpoints come in pairs and aresymmetric with respect to ��� ���
S� Fatimah and M� Ruijgrok �
0σ
Region I
Region II
Region III2k1
a
1
Figure �� The parameter diagram of system ����� in the���� a�plane
u
v
1
1
u
v
1
1
u
v
1
1
Figure �� The phaseportraits of system ����� in the�u�� v��plane for speci�c values ���� a� in region I� II� andIII� respectively�
�� Stability of the semi�trivial solution
In this section we will study the stability of the semitrivial solution depending on the values of the forcing amplitude a and the detunings ��� ��in system ������ From section �� we �nd that the semitrivial solution corresponding to R�
�with the plus sign is always stable �as a solution of �������
therefore we will only study this semitrivial solution and ignore the unstablesemitrivial solutions�
� Bifurcations in an Autoparametric System in ��� Internal Resonance
Write the averaged system ����� in the form�
X� � F �X������
where X �
�BB�
u�v�u�v�
�CCA and
�F
�X�
�A�� A��
A�� A��
������
where A��� A��� A��� and A�� are � � � matrices depending on u�� v�� u�and v�� At the solution ��u���v�� �� ��� corresponding to the semitrivialsolution of system ������ we have �F
�X� AX with
A�
�BB�
�k���u�v� ����
�a��v�
��R�
�� �
�����
�a��u�
��R�
��k���u�v� � �
� � �k���
�c�u�v� ���
�
�c�u
�
�� �
�c�v
�
�
� � �����
�c�v
�
��
�
�c�v
�
��k��
�
�c�u�v�
�CCA
����
u� and v� satisfy ����� and R��satis�es ������ Let
A �
�A�� �� A��
�
To get the stability boundary of system ������ we solve detA � detA��detA�� ��� From the equation detA�� � � we �nd
��q
��a
� � k����q
��a
� � k�� � ��� � �
or
a � �k� or a � �q��� k�������
and from detA�� � �� we have�
�� � ���c�R
���r
�
��c��R
��� k�� where R�
�� �
k�
c������
Because R��is a function of �� and a� equation ����� now gives a relation
between ��� ��� and a� Graphically� this corresponds to a surface in the�dimensional parameter space ���� a� ���� This surface is shown in Figure� for �xed values of k�� k�� c�� c� � � and in Figure � for �xed values of k��k�� c� � � and c� � ��In �gure � we show the stability boundary in the ���� ���plane for a �xed
value of a � �k�� Inside the curve the semitrivial solution is unstable�outside it is stable� In these numeric calculations we �xed k� � �� k� � �and c� � �� We took c� � �� for the case c� � � and c� � �� for the casec� � ��
S� Fatimah and M� Ruijgrok
0
σ1 6k1
4k1
a 2k1
0
kc
2
2 k2c 2
-4-8 0
4k2c 2
σ2
Figure �� The parameterspace of system ����� in the�a� ��� ���space for c� � ��
σ1 6k1
4k1
a 2k1
0
kc
2
2 k2c 2
-4-8 0
4k2c 2
0
σ2
Figure �� The parameterspace of system ����� in the�a� ��� ���space for c� � ��
Bifurcations in an Autoparametric System in ��� Internal Resonance
-20
-15
-10
-5
0
5
sigma2
-10 -8 -6 -4 -2 0 2 4sigma1
σ 2
σ 1
-5
0
5
10
15
20
sigma2
-10 -8 -6 -4 -2 0 2 4sigma1 σ
σ 2
1
�a �b
Figure �� The parameter diagram of system ����� in the���� ���plane for �xed a � ������ In �gure �a� for c� � �and in �gure �b� for c� � ��
0
1
2
3
4
5
6
-20 -17.4 -14.8 -12.2 -9.6 -7 -4.4 -1.8 0.8 3.4 6sigma1
L2_NormRo
K
L
M
σ10
1
2
3
4
5
6
0 5 10 15 20 25 30 35 40 45 50 55 60 65a
L2_NormRo
N
O
P
Q
.a
�a �b
Figure �� The stability diagram of the response of system������ �a� against the detuning �� for �xed a � ���� and�� � ���� and �b� against the forcing a for �xed �� � ��and �� � ����� A solid line means that the semitrivialsolution is stable and the dashed line that it is unstable�
In Figure � �a� we show the response of R�� � u�� v�� for �xed a and ���
note that R� � R�� We �nd that between the branch points L and M thesemitrivial solution is unstable� In Figure � �b� we show the response R� for�xed �� and ��� We have indicated that the semitrivial solution is unstablebetween the branch points N and O � and between the branch points P and
S� Fatimah and M� Ruijgrok �
Q� Figure � does not depend on the sign of c�� We have similar �gures forthe case c� � �� �� � � and for the case c� � �� �� � ��
�� Bifurcations of the semi�trivial solution
On the stability boundary shown in Figure �� the semitrivial solutionundergoes a pitchfork bifurcation� We have used the bifurcation continuation program CONTENT �Kuznetsov ����� to study the nontrivial solutionsbranching from these bifurcation points� We �nd that the results dependon the values of c� and c�� For positive values of c� and c�� the results aresummarized in Figure ��
-20
-15
-10
-5
0
5
sigma2
-10 -8 -6 -4 -2 0 2 4sigma1
I
II
III
IV
A
B
C
σ2
σ 1
Figure � The parameter diagram of system ����� in the���� ��� plane for c� � ��
We have �xed a � �k� for values ���� ��� in Region I� where the semitrivial solution is stable� Crossing the boundary from Region I into Region IIit becomes unstable and an attracting nontrivial solution is born� Crossingthe boundary from Region II into III the semitrivial solution becomes stableand a small� unstable nontrivial solution appears� Crossing the boundaryfrom Region III into region IV the stable and unstable nontrivial solutionscollide and disappear in a saddlenode bifurcation�
�� Bifurcations in an Autoparametric System in ��� Internal Resonance
-4
-2.4
-0.8
0.8
2.4
4
-15 -13.3 -11.6 -9.9 -8.2 -6.5 -4.8 -3.1 -1.4 0.3 2sigma2
u2
σ2
u2AB
C
C
Figure � The bifurcation diagram of the semitrivial solution in the ���� u��plane for c� � ��
-20
-15
-10
-5
0
5
sigma2
0 2 4 6 8 10 12 14a
σ2
A
B
C
I
II
III
IV
Figure �� The parameter diagram of system ����� in the�a� ���plane� for c� � ��
In Figure � we �x a � ����� and �� � �� �see Figure �� and show theu� component of the nontrivial solution as �� is varied� We have indicatedthe bifurcation points A� B� and C corresponding to Figure �� There is aninterval for �� where two stable solutions coexist and on the boundaries ofthis interval hysteresis jumps occur�
S� Fatimah and M� Ruijgrok ��
It is possible to make similar diagrams in the �a� ���plane� keeping ���xed� Again we �nd similar bifurcations� see Figure �� Note that the pointsA� B correspond to the branching points in Figure �� and C corresponds toa saddlenode bifurcation�
0
1
2
3
4
5
6
-20 -17.4 -14.8 -12.2 -9.6 -7 -4.4 -1.8 0.8 3.4 6sigma1
L2_Norm
K
L
M
L’
Ro
R
σ10
1
2
3
4
5
6
0 5 10 15 20 25 30 35 40 45 50 55 60 65a
L2_NormR
a
Ro
N
O
P
Q
P’
.
�a �b
Figure ��� The stability diagram of the nontrivial solution R together with the response R� of the semitrivialsolution for c� � �� �a� against the detuning �� for �xeda � ���� and �� � ���� and �b� against the forcing a for�xed �� � �� and �� � �����
In Figure �� �a� we show the amplitude of the nontrivial solution R �pu�� v�� u�� v��� together with the amplitude of the semitrivial solution
for �xed a and �� �see Figure ��a��� We �nd two nontrivial solutions incertain interval of ��� One is stable and another is unstable� We also showthat between points L� and M the nontrivial solution is stable and thereexists an unstable nontrivial solution between points L and L�� In Figure���b� we show the amplitude R� together with the amplitude of the semitrivial solution R� for �xed �� and �� �see Figure � �b��� The nontrivialsolution is stable between points P� and Q and there exists an unstablenontrivial solution between points P and P��For negative values of c� we �nd dierent phenomena in the behavior
of solutions of system ������ On the stability boundary shown in Figure �� the semitrivial solution undergoes a pitchfork bifurcation but then thenontrivial solutions which branches from this bifurcation point undergoes aHopf bifurcation� Again we have used CONTENT to study the nontrivialsolution branching from these bifurcation points� The results are illustratedin Figure ���
�� Bifurcations in an Autoparametric System in ��� Internal Resonance
-20
-10
0
10
20
sigma2
-10 -8 -6 -4 -2 0 2 4sigma1
A
BC
D
E
I
II
III
IV
VVI
σ2
σ1
VI
Figure ��� The parameter diagram of system ����� in the���� ���plane for c� � � and �xed a � ������
We have �xed a � �k� for values ���� ��� in region I� In this region thesemitrivial solution is stable� Crossing the boundary from Region I intoRegion II it becomes unstable and a stable nontrivial solution appears�Crossing the boundary from Region II into III the nontrivial solution becomes unstable� A supercritical Hopf bifurcation occurs at the boundarybetween Region II and III� Crossing the boundary from Region III into IVthe semitrivial solution becomes stable and another small unstable nontrivial solution appears� Crossing the boundary from Region IV into V theunstable nontrivial solution changes its stability and again it undergoes asupercritical Hopf bifurcation� Finally� crossing the boundary V into VI thestable and unstable nontrivial solution collide and disappear in a saddlenode bifurcation�In Figure �� we �x a � ���� and �� � �� �see Figure ��� and show the
v� component of the nontrivial solution as �� is varied� We have indicatedthe bifurcation points A� B� C� D� and E corresponding to Figure ��� Thepoints A and C indicate the branching points of the semitrivial solution�The points B and D indicate Hopf bifurcation points and E a saddlenodebifurcation point� Again there is an interval for �� where two stable solutionscoexist and on the boundaries of this interval hysteresis jumps occur�
S� Fatimah and M� Ruijgrok ��
As we discussed in the case c� � �� for c� � � we �nd similar bifurcationsin the �a� ���plane �see Figure ���� keeping �� �xed� The points A� B� C�D� and E on the curves correspond to the bifurcation points in Figure ���
-4
-2.4
-0.8
0.8
2.4
4
-20 -16.5 -13 -9.5 -6 -2.5 1 4.5 8 11.5 15sigma2
v2
A
B
B
C
D
D
E
E
σ2
v2
Figure ��� The stability diagram of system ����� in the���� v��plane for c� � ��
-20
-15
-10
-5
0
5
10
sigma2
0 2 4 6 8 10 12 14a
σ 2
A
B
C
D
E
I
II
III
IV
V
VI
Figure ��� The parameter diagram of system ����� in the�a� ���plane for c� � �
�� Bifurcations in an Autoparametric System in ��� Internal Resonance
We note that the amplitude of the nontrivial solution of system ����� is
R �pu�� v�� u�� v�� � In Figure � �a� we have depicted the amplitude of
the semitrivial solution R� against �� for �xed a and ��� In Figure �� �a��we show both the amplitude of the nontrivial solution R and the amplitudeof semitrivial solution R�� When �� is varied� we �nd there is an intervalof �� consisting of the unstable semitrivial solution and the stable nontrivial solution� There is also interval of �� where the semitrivial solutionis stable together with the stable nontrivial solution and the unstable nontrivial solution� In Figure �� �b� we show the amplitude R� together withthe amplitude R� against a for �xed �� and ��� Again the same behavior ofsolutions of system ����� occurs� when a is varied�
0
1
2
3
4
5
6
-25 -21.9 -18.8 -15.7 -12.6 -9.5 -6.4 -3.3 -0.2 2.9 6sigma1
L2_Norm
.
R
K
L
M
Ro
L *
σ10
1.6
3.2
4.8
6.4
8
0 10 20 30 40 50 60 70 80 90 100a
L2_Norm
.a
Ro
N
O
P
Q
Q*
R
�a �b
Figure ��� The stability diagram of the nontrivial solution R together with the semitrivial solution for c� � �� �a�against the detuning �� for �xed a � ���� and �� � ��� and�b� against the forcing a for �xed �� � �� and �� � ����
�� Period Doubling Bifurcations and Chaotic Solutions
We now consider the case that c� � � and c� � �� Not only does thesystem exhibit Hopf bifurcations� but we also observe a sequence of perioddoublings� leading to a strange attractor�In previous sections we have choosen a �xed value of a not too close to the
stability boundary given by a � �k�� This was done because when a is in theneighbourhood of �k�� complications can arise� since then also det A�� � ��and we can have doublezero eigenvalues� This problem can be studiedanalytically by considering a codimension � bifurcation� this will be carriedout in a separate paper� As a �rst result from this bifurcation analysiswe mention the occurence of global bifurcations� involving heteroclinic andhomoclinic loops� We also �nd a homoclinic solution of �Silnikov type� Itis wellknown �see Kuznetsov ���� and Wiggins ����� that the existence ofsuch a homoclinic loop is conected with chaotic solutions� We therefore
S� Fatimah and M� Ruijgrok ��
conjecture that the chaotic solutions we �nd numerically are the result ofthe �Silnikov phenomenon�In the numeric calculations� presented in this section interesting behavior
of solutions of system ����� occurs near the stability boundary�
A
B
C
D
E
I
σ2
II
III
IV
V
VI
Figure ��� The parameter diagram of system ����� in the�a� ���plane for c� � � close to the stability boundary�
-4
-2.4
-0.8
0.8
2.4
4
-10 -8.5 -7 -5.5 -4 -2.5 -1 0.5 2 3.5 5 6.5 8sigma2
v2
σ2
v2A
B
B
C
D
E
ED
Figure ��� The stability diagram of system ����� in the���� v��plane for c� � � close to the stability boundary�
�� Bifurcations in an Autoparametric System in ��� Internal Resonance
u1
v1
u2
v2
�a
u1
v1
u2
v2
�b
u1
v1
u2
v2
�c
u1
v1
u2
v2
�d
Figure �� The sequence of period doubling bifurcations�The phaseportraits in the �u�� v��plane and �u�� v��planefor �a� �� � ����� �b� �� � ���� �c� �� � ������ and �d��� � �����
S� Fatimah and M� Ruijgrok �
In Figure �� we �xed �� � �k�c�� a close to �k� and c� � � �a � ����
�� � ��� and c� � ���� The bifurcations of the semitrivial solution aresimilar to the case where a is taken far away from the stability boundary�compare Figure �� and Figure ����In Figure ��� the semitrivial solution branches at point A� and then
at point C� When the semitrivial solution branches at point A� a stablenontrivial solution bifurcates and then this nontrivial solution undergoes aHopf bifurcation at point B� We point out that a �xed point and a periodicsolution in the averaged system correspond to a periodic and quasiperiodicsolution� respectively� in the original� time dependent system� A supercritical Hopf bifurcation occurs at point B for �� � ������ and at point Dfor �� � ������� Again� a saddlenode bifurcation occurs at point E for�� � ��������We �nd a stable periodic orbit for all values of �� in the interval ������ � �� � ������� As �� is decreased� period doubling of the stableperiodic solution is observed� see Figure��� There is an in�nite number ofsuch period doubling bifurcations� until the value ��� � ������ is reached�The values of �� with ��� � �� � ������ produce a strange attractor�
v2
u2
u1
Figure �� The strange attractor of the averaged system������ The phaseportraits in the �u�� v�� u��space for c� � �at the value �� � ����
To know whether the strange attractor is chaotic or not� we have calculated the Lyapunov exponents of system ������ Any system containingat least one positive Lyapunov exponent is de�ned to be chaotic� with the
� Bifurcations in an Autoparametric System in ��� Internal Resonance
magnitude of the exponent re�ecting the time scale on which system becomeunpredictable �Wolf� Swift� Swinney� and Vastano ������We �nd that the Lyapunov spectra of system ����� corresponding to pa
rameter values above are � � ������� � � �������� � � �������� and� � �������� so that the orbits displayed in Figure �� are chaotic� We havefound that for other values of c� � �� c� � �� k� and k� the same type of scenario occurs i�e� periodic solutions which after a series of periodicdoublingslead to a strange attractor with one positive and three negative Lyapunovexponents�The Lyapunov spectrum is closely related to the fractal dimension of the
associated strange attractor� We �nd that the KaplanYorke dimension ofthe strange attractor for �� � ��� is �����
�� Conclusion
An autoparametric system of the form ������ with the conditions stated inequation ������ has at most �ve semitrivial solutions� which come in pairsand are symmetric with respect to ��� ��� We have studied one semitrivialsolution� which is stable as a solution of ������ and considered its stabilityin the full system� The dependence of the stability of this solution on theforcing and the detunings is pictured in Figure � and �� We �nd that therecan exist at most one stable nontrivial periodic solution� By studying thebifurcations from the semitrivial solution� we also �nd in some cases Hopfbifurcations� leading to quasiperiodic solutions� Also� we have observedcascades of perioddoublings� leading to chaotic solutions� The fact thatthese chaotic solutions arise in the averaged system implies that chaoticdynamics is a prominent feature in the original system�
�� Acknowledgements
The authors wish to thank Prof� A� Tondl for formulating the problem� Thanks also to Prof� F� Verhulst for many suggestions and discussionsduring the execution of the research and for reading the manuscript� Wethank L� van Veen for numerically calculating Lyapunov exponents and J�M�Tuwankotta for assistance in using the program� The research was conductedin the department of Mathematics of the University of Utrecht�
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