Revista de Ciencias Tecnológicas (RECIT). Volumen 3 (1): 10-22
Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925
Volumen 5 (1): e214. Enero-Marzo, 2022. https://doi.org/10.37636/recit.v5n1e214.
ISSN: 2594-1925
1
Research article
Master-slave synchronization in the Rayleigh and Duffing
oscillators via elastic and dissipative couplings
Sincronización maestro-esclavo en los osciladores Rayleigh y
Duffing mediante acoplamientos elásticos y disipativos
Ulises Uriostegui-Legorreta , Eduardo Salvador Tututi-Hernández
Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Av. Francisco J. Mújica S/N,
C.P. 58060, Morelia, Michoacán, México
Corresponding author: Ulises Uriostegui Legorreta, Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana
de San Nicolás de Hidalgo, Av. Francisco J. Mújica S/N, C.P. 58060, Morelia, Michoacán, México. E-mail:
uli_mat@hotmail.com. ORCID: 0000-0001-9905-6060
Received: November 17, 2021 Accepted: February 4, 2022 Published: February 16, 2022
Abstract. - In this work a master-slave configuration to obtain synchronization between the Rayleigh and
the Duffing oscillators is studied. For this configuration, we analyze the system when the dissipative
coupling and one that combines the elastic and dissipative couplings are used. We analyzed the coupling
parameters to find the range where synchronization between the oscillators is achieved. We found
synchronization in the oscillators for large values of the coupling parameter. Our numerical findings show
that for the dissipative coupling, there exists partial synchronization while for the others there is complete
synchronization.
Keywords: Nonlinear dynamics; Control of chaos; Synchronization.
Resumen. - En este trabajo se estudia una configuración maestro-esclavo para obtener sincronización
entre los osciladores Rayleigh y Duffing. Para esta configuración, analizamos el sistema cuando se utiliza
el acoplamiento disipativo y uno que combina los acoplamientos elástico y disipativo. Analizamos los
parámetros de acoplamiento para encontrar el rango donde se logra la sincronización entre los osciladores.
Encontramos sincronización en los osciladores para valores grandes del parámetro de acoplamiento.
Nuestros hallazgos numéricos muestran que para el acoplamiento disipativo existe una sincronización
parcial mientras que para los demás existe una sincronización completa.
Palabras clave: Dinámica no lineal; Control del caos; Sincronización.
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1. Introduction
Since the seminal work of Pecora and Carroll on
synchronization [1], numerous works on chaos
that comprises diverse areas such as lasers,
chemical reactions, electronic circuits, biological
systems, among others, have been studied. In
particular, low-dimensional systems have been
of interest in order to understand the
synchronization and chaotic behavior in nature.
The most studied and representative systems are
the Lorenz, Chua, Rössler, van der Pol, Rayleigh,
Duffing and their variations [2-7].
The Rayleigh oscillator is much like the van der
Pol oscillator. The Rayleigh and Duffing
oscillators are the paradigmatic circuits to study
chaos in systems of low-dimensional. The first
gives a limit cycle and the last provides the
prototype of a strange attractor. It is well known,
that a limit cycle is a closed trajectory in phase
space having the property that at least one other
trajectory spirals into it, when . In other
words, the limit cycle is an isolated trajectory; it
spirals either towards or away from the limit
cycle. An attractor is called strange if it has a
fractal structure. This is often the case when the
dynamics on the attractor is chaotic. If a strange
attractor is chaotic, it exhibits sensitive
dependence on the initial conditions. Studies
focused on the Rayleigh oscillator reveal that it
possesses a rich dynamical structure, especially
when the oscillator is forced. This system
exhibits complex bifurcation structures with an
important number of periodic states, a chaotic
region and islands of periodic states, showing, in
addition, transitions from chaos to stable states.
The dynamics based on identical or distinct linear
oscillators presenting the same kind of attractors
is still under study [8,9]. Nevertheless, the
dynamics of these systems in states of different
attractors is of current interest and it could give
rise to important information. The control of
chaos is concerned with using some designed
control to modify the characteristics of a
nonlinear system. A number of methods such as
active control, adaptive control, optimal control
and sliding mode control exist for the control of
chaos in systems [10-13]. Several kinds of
synchronization play an important role in the
study of chaos such as phase synchronization,
anticipated synchronization, generalized
synchronization, projective synchronization, that
have been studied and applied to a chaotic and
unified system by J. Yan et. al [14]. A. Razminia
et. al [15] have obtained complete
synchronization in chaotic systems of fractional
order through sliding mode control. A. Ouannas
et. al [16] present new approaches to study
coexistence of some kinds of synchronization
between hyperchaotic systems such as hybrid
synchronization and anti-synchronization. E.
Campos et. al [17] analyzed the multimodal
synchronization on the master-slave
configuration. J.S. González et. al [18] studied
the synchronization between two different
coupled chaotic oscillators with an external
force. The itinerary synchronization between
piecewise linear systems with different number
of attractors was studied by A. Anzo-Hernández
et. al [19]. The hybrid function projective
synchronization of chaotic systems has been
developed and used on systems where the
parameters of the system are unknown by
applying adaptive control, A. Khan et. al [20]. A.
Karimov et. al [21] have studied the adaptive
generalized synchronization between an analog
circuit and a computer model by comparing the
numerical methods used on the computer
simulation of chaotic systems.
Some applications of the Rayleigh and Duffing
oscillators go from physics to biology,
electronics, chemistry and many other fields. For
instance, a possible application of
synchronization in chaotic signals is to
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implement secure communication systems, since
chaotic signals are usually broadband, noise like,
and difficult to predict the behavior. They can
also be used for masking information bearing
waveforms [22,23,24]. In robotics, the oscillators
have been included to control joint hips and
knees of human-like robots to ensure the
mechanical system follows the right path. The
generated signals can be used as reference
trajectories for the feedback control [25,26].
Other application is in artificial intelligence. In
fact, the oscillators have shown usefulness to
training neuronal network and recognition of
chaotic systems [27,28].
As far as the coupling between the Rayleigh and
Duffing oscillators is referred, we can mention
three different couplings, namely: gyroscopic,
dissipative and elastic [29-34]. Among the
diverse way of coupling, the most used are the
elastic and dissipative ones [34,35,36]. In a
previous work [34], it is analyzed a different
approach of synchronizing two distinct
oscillators of low-dimensional, using the
aforementioned couplings. Uriostegui et. al [37]
studied synchronization between the van der Pol
and Duffing oscillators by using the elastic,
dissipative and a combination of both couplings.
It was found that the elastic coupling leads to no
synchronization, whilst with the dissipative one
it is reached partial synchronization. For the
combination of both couplings, it is reached
complete synchronization.
In this work, we study and compare two types of
couplings by using the Rayleigh and Duffing
systems: the dissipative and the used previously
by Uriostegui et. al [34]. It is important to
remark that the studies in the literature on this
kind of synchronization is based only on one
coupling. An outline of this work is as follows.
In Sec. 2, it is briefly studied the main features of
the Rayleigh and Duffing oscillators. In Sec. 3,
we study and compare two types of couplings
using the Rayleigh and Duffing systems upon the
master-slave configuration. In Sec. 4, some
conclusions and an outlook are presented.
2. Dynamics of the oscillators
The dynamics of the forced Rayleigh oscillator is
described by the following nonlinear differential
equation:
(1)
The Rayleigh oscillator is characterized by
nonlinear damping. The variable denotes the
position, the time, and is a parameter that
governs the nonlinearity and damping. The
external forcing is given by the harmonic
function, with amplitude and frequency .
We have defined the function:
(2)
as the Rayleigh potential, which represents a
single-well (see Fig. 1 (a)). The potential has a
minimum located at . In order to express
Eq. (1) as a dynamical system and to analyze the
fixed points, we set and drop the forcing
to obtain
(3)
We can observe from Eq. (3) that the only fixed
point is located at . For the case
when , the Rayleigh oscillator satisfies the
Rayleigh-Liénard theorem, giving a limit cycle in
the phase space, around the origin.
On the other hand, the Duffing oscillator is a
nonlinear dynamical system governed by
(4)
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where
(5)
and is positive and it denotes a dissipative
parameter, is a positive constant that controls
the nonlinearity of the system, and is the
amplitude of the external forcing, being its
frequency. The potential in Eq. (5) represents a
double-well shown in Fig. 1 (b). The local
minima of this potential are located in
and the local maximum is located at . As a
dynamical system the Duffing equation in (4) (no
forcing) can be cast as
(6)
where we set . The fixed points for this
system are located in the phase space at
and
. The first
one at is a saddle point, while the
others, depending on the parameter α, they can be
stable or unstable points. For the points
result stables, for the case, the resulting
dynamics is of type center and for case,
the points result unstable. In particular, when the
damping is positive , the trajectory of the
system is spiral stable, conversely, for a damping
negative , the trajectory is spiral unstable
at the fixed points
in both
cases.
(a) (b)
Figure 1. The potentials and . (a) The
potential corresponds to the Rayleigh oscillator. (b) The
Duffing oscillator (ε = 1).
3. Master-slave synchronization
In this section, two different couplings for the
Rayleigh and Duffing systems are studied and
compared among themselves, namely: the
dissipative and the one that combines elastic and
dissipative couplings employed by Uriostegui et.
al [34]. Let us stress that most of the research on
synchronization is based on autonomous systems
of three-dimensional or higher [38,39,40]. Three
of the most studied nonautonomous systems of
low-dimensional with forcing are the Duffing,
van der Pol, Rayleigh and their variations, since
much of the dynamical features embedded in the
physical systems can be realized on these
systems [41,42,43]. One important implication is
that a two-dimensional continuous dynamical
system cannot give rise to strange attractors. In
particular, chaotic behavior arises only in
continuous three-dimensional dynamical
systems or higher. Most of the research on
synchronization is based on autonomous systems
that satisfy the Poincaré-Bendixson theorem.
Nevertheless, let us stress that the Rayleigh and
Duffing oscillators being two-dimensional, need
an external forcing to present chaos.
The dynamics for each oscillator under study is
described by the equations in (1) and (4). The
values of the parameters we use are as follows:
=1.2, α=0.3, ε=1, A1 =2.8, ω1 =0.2, A2 =0.5 and
ω2 =1.3. In Figs. 2 and 3 it is displayed the
respective trajectories with the initial conditions
x (0) = 1, y (0) = 2, u (0) = 1 and v (0) = -1. Let
us mention that the very same values of the
parameters and the initial conditions will be used
in the subsequent numerical simulations. The
numerical simulations were performed using the
fourth order of the Runge–Kutta method.
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Figure 2. Rayleigh oscillator described by Eq. (1).
Figure 3. Duffing oscillator described by Eq. (4).
In the configuration master-slave, the Duffing
oscillator acts as master and the Rayleigh
oscillator as slave. For this case we have
(7)
(8)
In this instance, the coupling used corresponds to
a dissipative coupling and it is represented by
, being a coupling parameter to be
varied. For the case, the system
decouples. The coupling is linear feedback to the
slave oscillator and it can be seen as a
perturbation for each oscillator in the system,
proportional to the difference of the velocity,
named in literature a dissipative coupling. We are
interested in studying how the dynamics of the
system evolves as the constant coupling
changes.
In general, the synchronization problem reduces
to finding a suitable value of the coupling
strength , (denoted by ) being in the range
, such that the master and slave
systems synchronize. Thus, for a coupling
strength , when the complete synchronization
is reached, the error function goes to zero:
(9)
When the system is in practical synchronization,
for a certain value of , the error functions
satisfy
(10)
(11)
for given positive values , and arbitrary
initial conditions. This definition is used,
because, sometimes, the errors do not exactly
converge to zero, but in practice we can still
speak of synchronized systems. In some cases, it
can be reached complete synchronization in a
single state of the system while in the other, it can
be only obtained practical or null
synchronization. Although, in practice, such as in
analog circuits, we have no total control on the
parameters used (e.g., resistors, capacitors,
transistors), which makes not possible
reproducing the required conditions in the
numerical simulations. This could give no
complete synchronization. The partial
synchronization is the phenomenon when, in a
dynamical system, only part of the state variables
synchronizes and the others do not.
Let us consider the error functions
and by taking as a control
parameter to be varied in small steps from 0 to
200. For our case, the error functions allow us to
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find the range of values for in which the
synchronization is reached in the projections
onto the and planes, as it can be
shown in Figs. 4 and 5. Notice that in the
projection onto the plane no complete
synchronization exists since the error function
do not exactly converge to zero;
the function goes to zero for large
values of . For the projection onto the
plane, the complete synchronization could be
reached for rather large values of . In order to
see this, let us observe that the errors
and can be calculated from Eqs. (7)
and (8) as:
(12)
Figure 4. The error function varying the
parameter .
Figure 5. The error function varying the
parameter .
The plots of and as a function of for a
value of , are depicted in Fig. 6.
Figure 6. Error functions and , for .
In Figure 7, it can be appreciated from a time-
series plot of and that the signals are
not in complete synchronization. On the
contrary, in Figure 8 the time-series plot of
and , show that the signals are in complete
synchronization.
Figure 7. Time-series plot and .
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Figure 8. Time-series plot and .
Let us analyze the projections onto the and
planes for a specific value of . In
this case the master system is in a chaotic regime.
In Fig. 9 (a) it is shown the behavior of the
Duffing oscillator (master) and in Fig. 9 (b) the
Rayleigh oscillator (slave). In Fig. 9 (c) we can
appreciate the fact that in the projection onto the
plane there is no complete synchronization
while in the projection onto the plane there
is only complete synchronization (Fig. 9 (d)).
(a)
(b)
(c)
(d)
Figure 9. Dissipative coupling case, for . In (a) The Duffing oscillator (master) and in (b) the Rayleigh oscillator
(slave). In (c) and (d) projections onto the and planes, respectively.
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A modified master-slave scheme leading to
synchronization even in the cases where the
classical master-slave scheme fails, was
considered in Ref. [34]. The system analyzed in
this reference can be separated in two parts (see
Eqs. (13) and (14)). On one side, the system
combines a non-conventional coupling, where
linear feedback is made. The elastic coupling is
proportional to the difference of the position,
, which is introduced in the velocity of
the slave system. The other part uses also another
linear feedback proportional to the difference of
the velocity (dissipative coupling), ,
introduced in the acceleration in the slave system.
For the Rayleigh and Duffing oscillators, the
equations that govern the evolution are
(13)
(14)
The errors and , are
determined by subtracting Eqs. (13) and (14),
given
(15)
The constant corresponds to the elastic
coupling and to the dissipative coupling.
Hence, and
which manifest the dependence
of on the derivative of error and the coupling
, giving more information about the dynamical
evolution of the system. Let us introduce the
vector
(16)
which is the called control vector, and it contains
the coupling we propose. Notice that the control
depends on the error and its derivative. As before,
for the case , the system decouples.
In order to study the dynamics of the system, we
vary the couplings and keeping one of
them constant, while the other is varied. Let us
consider the and
error functions. We calculate
keeping and varying in small steps
from 0 to 10. In a similar way, we obtain the error
function with and varying
in small steps from 0 to 200. As it can be
appreciated from Figs. 10 and 11, we obtain
complete synchronization, since the error
functions go to zero as the value of and are
increased. The plots of and as a function
of , for the values of and , are
depicted in Fig. 12.
Figure 10. The error function varying the
parameter .
Figure 11. The error function varying the
parameter .
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Figure 12. Error functions and with respective
values of and .
As it can be observed in Figs. 13 and 14, the time-
series plots of , and shown
that the signals are in complete synchronization.
Figure 13. Time-series plot and .
Figure 14. Time-series plot and .
Let us now analyze the projections onto the
and planes for values of and
. For this case, the Duffing oscillator is
in a chaotic regime; the Rayleigh oscillator is
maintained as the slave system. In Figs. 15 (a)
and (b) the behavior of the Duffing and Rayleigh
oscillators is shown, respectively, while in (c)
and (d), it can be observed that complete
synchronization is reached for these systems.
For certain systems, it is not possible to reach
synchronization when the classical master-slave
scheme is used. In some cases, the systems reach
complete synchronization in a single state of the
slave system as it occurs for the dynamics
contained in Eqs. (7) and (8), depending on ,
obtaining partial synchronization by using
dissipative coupling for Rayleigh and duffing
oscillators. Variations to the master-slave
scheme for some systems have been proposed to
solve certain kind of problems [44-47]. In
particular, in Ref. [34] a modified master-slave
scheme is considered that leads to
synchronization even in the cases where the
classical master-slave scheme fails.
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(a)
(b)
(c)
(d)
Figure 15. Elastic and dissipative couplings for and . In (a) The Duffing oscillator (master) and in (b) the
Rayleigh oscillator (slave). In (c) and (d) projections onto the and planes respectively.
4. Conclusions
The Rayleigh and Duffing are low-dimensional
nonautonomous systems that present chaos and
have been well studied. In this work, we have
studied the master-slave configuration in the
Rayleigh and Duffing oscillators, when
dissipative coupling is used, only complete
synchronization in the projection onto the
plane can be reached. As a matter of fact,
according to the classical master-slave coupling,
in the best of cases, it is obtained only complete
synchronization in a single state of the slave
system studied. On the other hand, the possibility
of using two coupling (elastic and dissipative, in
this case), blending up as one, allows the system
a more interesting dynamics and a broad range
for the control parameters. In this paper, we have
analyzed the synchronization in the Rayleigh and
Duffing oscillators using the combination of the
elastic and dissipative couplings. We observed
that, in a difference with other approaches, with
this new coupling, we were able of obtaining
complete synchronization in the projections onto
the and planes. In order to apply
synchronization in communication systems, it is
necessary to have a large range of the control
parameter, which is obtained in the Rayleigh and
Duffing oscillators, by employing our approach
of coupling. This kind of coupling will be applied
in others systems that do not present
synchronization through the usual methods.
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5. Acknowledgements
This work has been partially supported by
UMNSH. U. Uriostegui-Legorreta thanks to
CONACYT for financial support.
6. Authorship acknowledgements
Ulises Uriostegui Legorreta: Conceptualización;
Recursos; Ideas; Metodología; Análisis formal;
Investigación; Análisis de datos; Borrador
original; Administración de proyecto. Eduardo
Salvador Tututi Hernández: Conceptualización;
Ideas; Metodología; Análisis formal; Análisis de
datos; Escritura; Revisión y edición.
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