Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 3, pp. 603-613, Warsaw 2013

SYNCHRONIZATION OF TWO FORCED DOUBLE-WELL DUFFING

OSCILLATORS WITH ATTACHED PENDULUMS

Piotr Brzeski, Anna Karmazyn, Przemysław Perlikowski

Lodz University of Technology, Division of Dynamics, Łódź, Poland

e-mail: przemyslaw.perlikowski@p.lodz.pl

We investigate the dynamics of two coupled Duffing oscillators with attached pendulums
forced kinematically by a common signal.Our attention is focused on different kinds of syn-
chronizationwhich can appear in the considered system.Different types of coupling (spring,
damper and spring and damper simultaneously) are taken into account.We show in a two-
-parameters space (amplitude and frequency of excitation) existence of complete and phase
synchronization and asynchronous ranges.

Key word: Duffing pendulum system, bifurcational analysis, synchronization

1. Introduction

Investigation of systems consisting of pendulums is an issue which focused attention of many
scientists. The synchronization between coupled systems with pendulums was discovered in
the XVIIth century by the Dutch researcher Christian Huygens. He showed that a couple of
mechanical clocks hanging from a common support synchronize (Huygens, 1665). There are
a lot of practical applications of such systems (Blekhman, 1998). In recent years, many new
types of synchronizations have been detected (Rulkov et al., 1995; Rosenblum et al., 1996;
Boccaletti et al., 2002; Stefanski et al., 2007; Yanchuk et al., 2008), which manifests a strong
or weak interaction between coupled nonlinear oscillators. The strongest one is the complete
synchronization (CS) (Pecora and Carroll, 1990, 1991). This idea is well-known and has been
applied inmanyfields of science (Watts andStrogatz, 1998; Strogatz, 2001; Pikovsky et al., 2001;
Tass, 2003; Stefanski and Kapitaniak, 2003; Stefanski et al., 2007; Balanov et al., 2009). The
CS has also been extended for slightly different systems in experimental investigations (Sekieta
and Kapitaniak, 1996; Argyris et al., 2005; Perlikowski et al., 2008a). Another important type
of synchronization is called the phase synchronization (PS) (Rosenblum et al., 1996). ThePS in
nonlinear systems is defined as an appearance of the reaction between the phases of subsystems
(or between the phase of the subsystems and the driving signal), while the amplitudes can still
be chaotic and uncorrelated. This type of synchronization is typical in parametrically excited
systems where different types of inner resonances can be observed (Miles, 1988; Clifford and
Bishop, 1995, 1996).
In this work, we consider two Duffing-pendulum systems coupled by a spring and/or a

damper. The dynamics of the investigated systems is complex even in the uncoupled case.
Many published papers focus on behaviour of a pendulumattached to a forced linear oscillator.
One of the first articles (Hatwal et al., 1983a,b) gives approximate solutions by the method
of harmonic balance in the primary parametric instability zone, which allows calculation of the
separate regionsof stableandunstableharmonic solutions.Furtheranalysis allowsunderstanding
of the dynamics around primary and secondary resonances (Bajaj et al., 1994; Cartmell and
Lawson, 1994; Balthazar et al., 2001; Song et al., 2003; Warminski and Kecik, 2006). A good
understanding of dynamics of systemswith a linear base gives possibility to extend investigation
to systems with a non-linear base. Non-linearity in the considered class of systems is usually



604 P. Brzeski et al.

introducedby changing the linear spring into anonlinear one (Warminski et al., 2001;Warminski
and Kecik, 2009) or a magnetorheological damper (Kecik andWarminski, 2011).
In a few papers on this topic, one can find an analytical study of the dynamics of a Duffing-

-pendulumsystem around principal and secondary resonances (Warminski et al., 2001;Warmin-
ski and Kecik, 2006, 2009; Vazquez-Gonzalez and Silva-Navarro, 2008; Macias-Cundapi et al.,
2008). Recently, we presented the complete bifurcation diagram containing oscillating and ro-
tating solutions (Brzeski et al., 2012). Our analysis also shows that in this class of two degree
of freedom systems one can expect many coexisting solutions, hence their practical application
should be preceded by careful investigation (Chudzik et al., 2011; Maistrenko et al., 1997).
In our previous papers (Perlikowski et al., 2008b; Perlikowski, 2008) we describe the syn-

chronization between two coupled single-well Duffing oscillators forced by the common signal.
We show there detailed analysis of synchronization phenomena and compare different methods
of synchronization detection.
In this paper, we consider two coupled (by the spring and/or damper) double-well Duffing

systems each with the attached pendulum.We focus investigations on synchronization between
them.Change from single-well to double-well Duffing oscillators cause an increase of complexity
– even in the periodic case systems can oscillate around opposite equlibriums (Kapitaniak, 1985,
1988; Miles, 1989; Robinson, 1989; Yamasue and Hikihara, 2004; Czolczynski et al., 2008).
In Section 2, we present model of the considered system. In Section 3, we present an over-

view of possible synchronization scenarios. Section 4 is devoted to numerical investigation of
synchronization. Finally, in Section 5 we conclude our work.

2. Model of the system

Themodel of the system taken under consideration is shown inFig. 1. It consists of twomutually
coupled subsystemswith common driving. Each subsystem is composed of a double-well Duffing
oscillatormoving in thevertical direction and thependulumsuspendedto themass of theDuffing
oscillator. Motion of each subsystem is described by two generalized coordinates: the vertical
position of theDuffingoscillator is describedby the coordinate yi, and the angular displacement
of the pendulum is given by the angle ϕi, where i=1,2. The systems are coupled thought the
spring or/and damper.

Fig. 1. Model of the system

2.1. Equations of motion

The evolution of the systemcanbedescribedby the set ofODEsderivedusingLagrange equ-
ations of the second kind.Thekinetic energy T , potential energy V , andRayleigh dissipation D
are given by the following formulas



Synchronization of two forced double-well Duffing oscillators ... 605

T =
2
∑

i=1

[1

2
(M+mi+µi)ẏ

2
i +
(

mi+
1

2
µi
)

liẏiϕ̇i sinϕi+
1

2

(

mi+
1

3
µi
)

l2i ϕ̇
2
i

]

V =
1

2
kS
(

y1−y2
)2

+
2
∑

i=1

[1

2
k1y
2
i +
1

4
k2y
4
i +
1

2
k3
(

yi−y3
)2

+
(

mi+
1

2
µi
)

gli(1− cosϕi)
]

(2.1)

D=
1

2
cS(ẏ1− ẏ2)

2+
2
∑

i=1

1

2
cDy

2
i

where M is mass of the Duffing oscillators, mi is mass at the end of each pendulum, µi and
li correspond to mass and length of the rod of the i-th pendulum, k1 and k2 are linear and
nonlinear parts of the spring stiffness and k3 is stiffness of the forcing spring. The viscous
damping coefficient of the Duffing oscillators is given by cD. The coupling between subsystems
is characterized by two parameters: stiffness kS and viscous damping cS.

The generalized forces are given by the following formula

Q=
2
∑

i=1

Mi(ϕ̇i) (2.2)

where Mi(ϕ̇i) = ciϕ̇i is damping momentum of the i-th pendulum with the damping coeffi-
cient ci. The dampers of the pendulums are located in pivot points (not shown in Fig. 1). Using
formulas (2.1) and (2.2), one can derive four coupled second order ODEs

(M+m1+µ1)ÿ1+ cDẏ1+(−k1+k3)y1+k2y
3
1 +
(

m1+
1

2
µ1
)

l1(ϕ̈1 sinϕ1+ ϕ̇
2
1cosϕ1)

+ cS(ẏ1− ẏ2)+kS(y1−y2)= k3y3

(M+m2+µ2)ÿ2+ cDẏ2+(−k1+k3)y2+k2y
3
2 +
(

m2+
1

2
µ2
)

l2(ϕ̈2 sinϕ2+ ϕ̇
2
2cosϕ2)

+ cS(ẏ2− ẏ1)+kS(y2−y1)= k3y3
(

m1+
1

3
µ1
)

l21ϕ̈1+ c1ϕ̇1+
(

m1+
1

2
µ1
)

l1(ÿ1+g)sinϕ1 =0

(

m2+
1

3
µ2
)

l22ϕ̈2+ c2ϕ̇2+
(

m2+
1

2
µ2
)

l2(ÿ2+g)sinϕ2 =0

(2.3)

In numerical calculations, we assumed that the Duffing oscillators and pendulums are
identical. We used the following values of their parameters: M = 2.0kg, k1 = 900N/m

3,
k2 =200000N/m, cD =3.9Ns/m.We have distinguished parameters of the pendulums suspen-
ded to each subsystemand assumed the following values of pendulumsparameters: m1 =0.1kg,
µ1 = 0.1kg, l1 = 0.05m, c1 = 5.94 · 10

−5Nms, m2 = 0.1kg, µ2 = 0.1kg, l2 = 0.05m,
c2 = 5.94 · 10

−5Nms. The Duffing oscillators are kinematically forced thought linear springs
k3 =200N/m. Motion of the base is described by a harmonic function: y3 = e(t) =Acos(ωt),
where A=0.04m.We neglect the static deflation of Duffing systems.

2.2. Dimensionless equations of motion:

Introducing dimensionless time τ = tp1, where p1 =
√

[g(m1+µ1/2)]/[l1(m1+µ1/3)] is the
natural frequency of first pendulumwe reach dimensionless equations



606 P. Brzeski et al.

(1+m1B +µ1B)ẍ1+ cBẋ1+(−k1B +k3B)x1+k2Bx
3
1+
(

m1B +
1

2
µ1B
)

· l1B(ϕ̈1 sinϕ1+ ϕ̇
2
1cosϕ1)+ cSB(ẋ1− ẋ2)+kSB(x1−x2)= k3BAB cos(µτ)

(1+m2B +µ2B)ẍ2+ cBẋ2+(−k1B +k3B)x2+k2Bx
3
2+
(

m2B +
1

2
µ2B
)

· l2B(ϕ̈2 sinϕ2+ ϕ̇
2
2cosϕ2)+ cSB(ẋ2− ẋ1)+kSB(x2−x1)= k3BAB cos(µτ)

(

m1B +
1

3
µ1B
)

l21Bϕ̈1+ c1Bϕ̇1+
(

m1B +
1

2
µ1B
)

l1Bẍ1 sinϕ1

+
(

m1B +
1

3
µ1B
)

l21B sinϕ1 =0

(

m2B +
1

3
µ2B
)

l22Bϕ̈2+ c2Bϕ̇2+
(

m2B +
1

2
µ2B
)

l2Bẍ2 sinϕ2

+
(

m2B +
1

2
µ2B
)

ol1Bl2B sinϕ2 =0

(2.4)

where: yst =
M+m1+µ1
|k1+k3|

, o=
m1B+µ1B/3
m1B+µ1B/2

, AB =
A
yst
, µ= ω

p1
, cB =

cD
Mp1
, cSB =

cS
Mp1
, kSB =

kS
Mp2
1

,

k1B =
k1
Mp2
1

, k2B =
k2y
2

st

Mp2
1

, k3B =
k3
Mp2
1

, and liB =
li
yst
, miB =

mi
M
, µi =

µi
M
, ciB =

ci
Mp1y

2

st

, xi =
yi
yst
,

ẋi =
ẏi
ystp1
, ϕ̇i =

ϕ̇i
p1
, ẍi =

yi
ystp

2

1

, ϕ̈i =
ϕ̈i
p2
1

for i=1,2.

The dimensionless parameters have the following values: o = 0.8889, AB = 1.297,
l1B = 1.622, m1B = 0.05, µ1 = 0.05, l2B = 1.622, m2B = 0.05, µ2 = 0.05, cB = 0.1321,
c1B = 0.002104, c2B = 0.002104, k1B = 2.039, k2B = 0.4307, k3B = 0.4531. The dimensionless
parameters of coupling: kSB, cSB and the excitation frequency µ are control parameters. Pen-
dulums have a linear resonance at µ=1.0, while for theDuffing oscillators the linear resonance
occurs at µ=1.2. Such diversificationmeans that during the increasing frequency of excitation
the resonance of pendulums occurs faster than the resonance of the Duffing oscilators, and we
do not observe an overlapping of those two resonances.

3. Types of synchronization

Generally, between coupled oscillators, one can observe different types of synchronization. The
strongest relation between coupled oscillators is the CS (Pecora and Carroll, 1990, 1991) which
takesplacewhenthe identical subsystemsexhibitmotionwith the sameamplitudeand frequency.
In the analyzed system, the sufficient condition for this type of synchronization is

lim
t→∞
‖z(t)−w(t)‖=0 (3.1)

where z(t) and w(t) are state vectors of the two coupled systems.

WhenDuffingoscillators are in theCSstate, thependulumsarealso completely synchronized.
In our system, there is no transfer of vertical forces between Duffing oscillators and pendulums,
hence there is no qualitative difference between in-phase and anti-phase synchronization. In-
phase or anti-phase motion depends only on initial conditions. Therefore, both in-phase and
anti-phase synchronization of the pendulums is described as complete synchronization.

Another type of synchronization appearswhen the difference between the position ofDuffing
oscillators after every period remains the same.This can be observed in three cases. Thefirst one
corresponds to the situation whenmasses oscillate around opposite equilibria. The second case
appears when one pendulum is locked with an excitation frequency with a different ratio than
the other one (mostly it is 2 : 1 and 1 : 1). The third case can be observed when one pendulum



Synchronization of two forced double-well Duffing oscillators ... 607

is in the stable equilibrium position and the other one is oscillating. All specified cases fulfil the
following condition

∀t |[z(t)−w(t)]− [z(t+T)−w(t+T)]|=0 (3.2)

where T is the period of motion, and can be classified as the PS (Pikovsky et al., 2001).
Nevertheless, when Duffing oscillate periodically around the same or opposite equlibria one

can observe the CS between pendulumswhen the following condition is fulfilled

lim
t→∞

∥

∥

∥ |Φ(t)|− |Θ(t)|
∥

∥

∥=0 (3.3)

where Φ(t) and Θ(t) are state vectors of the pendulums.
Phase synchronization between pendulums occurs when one pendulum is locked with an

excitation with a different ratio than the second one. Hence, pendulums oscillate with different
amplitude. In our system, the sufficient condition for the PS of the pendulums is

∀t |[Φ(t)−Θ(t)]− [Φ(t+T)−Θ(t+T)]|=0 (3.4)

4. Results of numerical simulations

In our numerical simulations, we investigate synchronization between the two considered sub-
systems.We take into account three different types of coupling: (1) only by a damper, (2) only
by a spring and (3) combinated by the spring and damper. To obtain more general overview
on synchronization properties, for each case of coupling, we calculate three diagrams in a two-
-dimensional space: couping coefficient (damping coefficient and/or spring stiffness) and frequ-
ency of the external excitation. The first one shows synchronization between Duffing oscillators
with fixed pendulums (the mass of each oscillator is equal to (M+mi)). The second and third
panels show synchronization between the Duffing oscillators and pendulums, respectively (pen-
dulums oscillate freely). Such comparison enables us to present the influence of the attached
pendulums on the synchronization. To distinguish different types of synchronization, we use
symbols presented in Table 1.

Table 1. List of symbols used to distinguish different states of the system

Symbols for Duffing oscillators

For periodic orbits (PO) For chaotic attractors

⋆ complete synchronization (CS) • complete synchronization (CS)

H phase synchronization (PS) ◦ no synchronization

× no synchronization

Symbols for pendulums

no
both pendulums are in their equilibrium position (stable one)

symb.

+ one pendulum is equilibrium position (stable one) and the another one is oscillating

For periodic orbits (PO) For chaotic attractors

⋆ complete synchronization (CS) • complete synchronization (CS)
– phase or antiphase

H phase synchronization (PS) ◦ no synchronization

× no synchronization

We always start integration for the lowest value of frequency excitation (µ=0.6) with the
same initial conditions for all values of coupling: Duffing oscillators are in an unstable steady



608 P. Brzeski et al.

state (x1,2 = ẋ1,2 = 0.0) and the pendulums are slightly perturbed from the hanging-down
position (ϕ1,2 =0.0001, ϕ̇1,2 =0.0).

In Fig. 2, we show case (1) where the coupling is realized only by the damper. The damping
coefficient cSB ∈ (0,0.35) and the excitation frequency µ∈ (0.6,2.4). This range of parameters
includes linear resonances of the Duffing system and pendulum and a 2 : 1 resonance of the
pendulum. In Fig. 2a, we consider the system with fixed pendulums.When Duffing oscillators
are in the periodic regime, the CS occurs in a large range of cs, and in a few cases we observe
the PS (which coexists with the CS). For chaotic motion of Duffing oscillators, the value of
the coupling coefficient is crucial and strongly influences the synchronization properties. In the
chaotic regime theCS is possible only for larger values of cs.Worth to notice is the fact that for
a higher frequency of excitation µ, velocities of the Duffing oscillators are larger. This implies
that forces generated in the coupling damper are also higher. This explains why the minimum
value of cs for which the CS occurs decreases with an increasing excitation frequency µ.

Fig. 2. Synchronization of Duffing oscillators and pendulums, subsystems coupled by the linear damper,
(a) synchronization between Duffing oscillators with fixed pendulums, (b) synchronization between

Duffing oscillators and (c) synchronization between pendulums

Figures 2b and 2c correspond to the system with unfixed pendulums. Comparing Fig. 2a
and Fig. 2b (synchronization between Duffing oscillators) one can notice that for systems with
oscillating pendulums, the area where theDuffing oscillators have chaotic motion is muchwider
and synchronization in that range does not occur for all considered values of cs. Therefore, for
the systemwith pendulums, the coupling strength is not so crucial. It is also important that the
area where the PS occurs is larger than in the case with fixed pendulums. By analyzing Fig. 2c
(synchronization of pendulums) one can notice that motion of the pendulums is observed only
near parametric resonances (1 : 1, 2 : 1) and in areas where the Duffing oscillators have chaotic
behaviour. Outside the mentioned ranges, the pendulums reach hanging-down positions. Syn-



Synchronization of two forced double-well Duffing oscillators ... 609

chronization between the pendulums occur only for periodic motion while for chaotic solutions
they stay unsynchronized. For this type of coupling, mostly the CS occurs, and regions where
the PS can be observed are small. One can see that for µ ∈ (1.85,1.97) and for µ > 2.15 the
Duffing systems synchronize their phaseswhile the pendulumsare in theCS state. This situation
is observed because the Duffing components oscillate around opposite wells.
Figure 3 corresponds to the system coupled only by the spring (cSB = 0) – case (2). The

stiffness kSB varies from 0 to 1.6, and similarly to Fig. 2, we check synchronization of the
Duffing oscillators and pendulums for different values of the excitation frequency µ. Similarly
to the situation shown in Fig. 2a, one can notice that when motion of the Duffing oscillators
is periodic, synchronization always occurs (independent on the coupling kSB value). Therefore,
the strength of the coupling changes only the type of synchronization (from the PS to the CS
and vice-versa) modifying the region of periodic motion (systems oscillate around the same or
opposite wells). When the Duffing subsystems oscillate chaotically, the synchronization occurs
for kSB > 0.72 regardless of the value of µ.

Fig. 3. Synchronization of Duffing oscillators and pendulums, subsystems are coupled by the linear
spring, (a) synchronization between Duffing oscillators with fixed pendulums, (b) synchronization

between Duffing oscillators and (c) synchronization between pendulums

Analyzing system with pendulums (Figs. 3b,c), one can notice that contrary to the system
coupledby thedamper,wedonot observeperiodicwindows in the chaotic range of the excitation
frequency µ. Chaotic attractors that exist for µ < 1.29 (around the principal resonance of
pendulum) cause asynchronous motion of the systems. The regions where the PS of Duffing
oscillators occur are larger in the case of fixed pendulums and this is a contrary situation to the
phenomenon observed in the system coupled with the damper. In Fig. 3c one can notice ranges
where only one pendulum is oscillating and the second one is in a stable equilibrium position
(marked as plus). This phenomenon is observed only in this coupling scheme, but theoretically
is possible for all types of coupling.



610 P. Brzeski et al.

Results of simulations of the system with both types of coupling (cSB 6= 0, kSB 6= 0) are
presented inFig. 4 (case (3)).Wedecide to hold the same ranges of couplingparameters as in the
previous cases, and the coupling is given by the proportional sum of cSB and kSB. The values
of coupling by the damper and spring is shown in the right and left axis of Fig. 4, respectively.
Systems that are coupledbybotha springanddamper exhibit dynamics that is characteristic for
both types of previous cases and can be regarded as their generalization. Nevertheless, there are
also some uniqueness. In Fig. 4a, the CS in the chaotic state occurs for lower values of coupling
than in separate coupling cases. The range of thePS is smaller than in the casewhen the systems
are coupled only by the spring and larger than in the case when the coupling is realized only by
the damper. In Fig. 4b, one can observe a noticeable increase in the area of periodic window,
which also occurs in Fig. 3b (for which both pendulums are in the stable equilibrium position).
The area of the PS between the pendulums can be observed in a much wider range than in
Fig. 2c and Fig. 3c, and it is typical for sufficiently large values of cSB and kSB.

Fig. 4. Synchronization of Duffing oscillators and pendulums subsystems coupled by the linear spring
and damper, (a) synchronization between Duffing oscillators with fixed pendulums, (b) synchronization

between Duffing oscillators and (c) synchronization between pendulums

5. Conclusion

In this paper, we analyse dynamical behaviour of two coupled Duffing oscillators with attached
pendulums. We show how different types of couplings affect synchronization between the sub-
systems. From the practical point of view, periodic motion of the coupled system is the best,
and the coupling which causes periodization is prominent. One can see that for different types
of couplings, the dynamics is generally similar because in most cases motion of pendulums is
observed around parametric resonances, where a locking phenomenon occurs. For the coupling
realized by a spring and damper, chaotic ranges are smallest especially for the 2 : 1 resonance.



Synchronization of two forced double-well Duffing oscillators ... 611

The coexistence of different types of synchronization is visible by the rapid changes in behaviour
of the subsystems.

Acknowledgment

This work has been supported by the Foundation for Polish Science, TeamProgrammeunder project

TEAM/2010/5/5.

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Manuscript received June 22, 2012; accepted for print October 28, 2012