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Increased and Reduced Order Synchronization of 2D and 3D Dynamical Systems

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Samuel Toluwalope Ogunjo at Federal University of Technology, Akure
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This paper presents increased and reduced order generalized synchronization (GS) of two different chaotic systems with different order based on active control technique. Through this technique, suitable control functions were designed to achieve generalized synchronization between: (i) 2D Duffing oscillator as the drive and 3D Lorenz system as the response system and (ii) 3D Lorenz system as the drive and 2D Duffing oscillator as the response system. Corresponding numerical simulation results are presented to verify the effectiveness of technique. Reduced order synchronization between the two systems gives faster synchronization time than the increased order synchronization of the two systems.
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ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.16(2013) No.2,pp.105-112
Increased and Reduced Order Synchronization of 2D and 3D Dynamical Systems
Samuel Ogunjo ∗
Department of Physics, Federal University of Technology, Akure
(Received 6 January 2013 ,accepted 5 September 2013 )
Abstract: This paper presents increased and reduced order generalized synchronization (GS) of two differ-
ent chaotic systems with different order based on active control technique. Through this technique, suitable
control functions were designed to achieve generalized synchronization between: (i) 2D Duffing oscillator
as the drive and 3D Lorenz system as the response system and (ii) 3D Lorenz system as the drive and 2D
Duffing oscillator as the response system. Corresponding numerical simulation results are presented to verify
the effectiveness of technique. Reduced order synchronization between the two systems gives faster synchro-
nization time than the increased order synchronization of the two systems.
Keywords:Increased order, reduced order, synchronization, active control, Lorenz system, Duffing oscillator
1 Introduction
Pecora and Carroll [1] pioneered the synchronization of chaotic systems. Since chaotic systems are sensitive to initial
conditions, one would expect the difference of two different systems to diverge exponentially in time. However, the
addition of the right functions to one of the systems will reduce this difference to zero such that the two systems follow the
same trajectory after a period of time. This is the concept of synchronization. It is defined by Boccaleti et al [2] as a process
wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a
common behavior due to a coupling or to a forcing (periodical or noisy). From its proposition, chaos synchronization has
gained a lot of attention over the last two decades due to its potential applications in vast areas of physics and engineering
sciences. The results of chaos synchronization are utilized in biological sciences[3], economics and finance[4], chemical
reactions, secure communication[5] and cryptography and data encryption.
Over the years, many types of synchronization have been developed by many researchers including complete synchro-
nization [1], phase synchronization [6], lag synchronization [7], generalized synchronization [8], anticipated synchro-
nization [9] and projective synchronization[10]. Projective synchronization was later extended to general class of chaotic
systems and not only partially linear systems and this is called generalized projective synchronization [11, 12]. Projective
synchronization is one of the most interesting synchronization scheme because of proportional relation that exist between
the dynamical state variables of the drive and the response systems, a feature that can be used to extend binary digit to
M-ary digit for fast communication [13]. Bai and Lonnggren [14] proposed the method of identical chaos synchroniza-
tion using active control. The ubiquitous applications of active control technique has encouraged researchers to introduce
active control based on different stability criteria. It was shown recently([15]) that active control is simpler with a more
stable synchronization time and hence, it is more suitable for practical implementation.
Many researchers have attempted synchronization of similar systems using different techniques[14, 16] amongst others
while some authors have synchronized different systems [17, 18]. There are real situations where systems of different
other need to be synchronized e.g the order of the thalamic neurons can be different from the hippocampal neurons,
the synchronization between heart and lungs, the synchronization in neuron systems and certain biomechanical systems
(such as biological implants). Few researchers have done extensive work on increased order synchronization [19–21]
and reduced order synchronization[22–24]. Hence, the investigation of synchronization of different chaotic systems even
though they are of different orders is very important from the perspective of practical application and control theory.
The aim of this paper is to show numerically, the synchronization between the 2D Duffing oscillator and 3D Lorenz
system when either is the master system. Furthermore, it is the aim of this paper to determine which of the synchronization
scheme (reduced or increased) will be more effective using a new quantifier - synchronization time.
∗E-mail address:stogunjo@futa.edu.ng
Copyright c
⃝World Academic Press, World Academic Union
IJNS.2013.10.15/750
106 International Journal of Nonlinear Science, Vol.16(2013), No.2, pp. 105-112
2 System description
The Duffing oscillator is given by the expression
¨x+b˙x+dV (x)
dx =g(f, ω, t)(1)
where xdenotes the displacement from the equilibrium position. fis the forcing strength, bis the damping parameter
and g(f, ω, t)is the periodic driving force given by T=2π
ω.ωis the angular frequency of the driving force. V(x)is the
potential giving by Taylor series.
V(x) = 1
2αx2+1
4βx4(2)
where αand βare potential parameters. The Duffing oscillator can be single-well (α > 0,β > 0) or double-well
(α < 0,β > 0). The chaotic behaviour of the single-well Duffing oscillator is shown in Figure 1.
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Figure 1: Phase space portrait of the single-well Duffing oscillator with parameters α=−0.5, β = 0.5, f = 1, b =
0.1ω= 0.5.
The Lorenz system is perhaps the first chaotic system in existence, proposed by Lorenz in 1963. It is a 3D autonomous
dynamical system given by the expression
˙y1=−σy1−σy2
˙y2=−y1y3+ry1−y2
˙y3=y1y2−b1y3
(3)
The system is chaotic with three equilibria when r > 24.74 and Lyapunov exponent 1.497,0,−22.46. The Lorenz
system has found application in thermal convection and laser dynamics. Phase space of the Lorenz system commonly
referred to as ”butterfly effect” is shown in Figure 2.
3 Methodology
If two systems synchronized due to the addition of a functional relation between the states of the systems, a new form of
synchronization called generalized synchronization is realized [23]. It was shown in [23] that generalized synchronization
exist between a drive system xand yif there exists a function φsuch that y=φ(x). The approach used in this work is
based on the proof given in [23] and active control method in ref. [14].
IJNS email for contribution: editor@nonlinearscience.org.uk
Samuel Ogunjo: Increased and Reduced Order Synchronization of 2D and 3D Dynamical Systems 107
−20 −10 0 10 20
0
10
20
30
40
50
x1
x3
Figure 2: The butterfly effect as seen in the Lorenz system with parameters set to σ= 10, r = 28, b =8
3.
3.1 Increased order synchronization
Equation (1) can be written in the form
˙x1=x2
˙x2=−bx2−αx1−βx3
1+fcos ωt (4)
Taking the Duffing oscillator in the form of equation 1 as the drive system and the 3D Lorenz system equation 3 as the
response system. Adding the control, the response system becomes
˙y1=−σy1−σy2+u1
˙y2=−y1y3+ry1−y2+u2
˙y3=y1y2−b1y3+u3
(5)
The goal is to determine the control functions u1(t), u2(t)u3(t)such that lim
t→∞ ∥y−φ(x)∥= 0.
The error state of the system is obtained as
e1=y1−φ1
e2=y2−φ2
e3=y3−φ3
(6)
The case of φ(x)=(gx1−x2, j x1, x2)Tis considered. gand jare positive integers. Using equation 6, the error
dynamical system is obtain the error dynamical system
˙e= ˙y−˙φ= ˙y−Dφ(x)f(x)(7)
where Dφ(x)is the Jacobian matrix of the map φ(x)
Dφ(x)
∂φ1(x)
∂x1
∂φ1(x)
∂x2
∂φ1(x)
∂x3
∂φ2(x)
∂x1
∂φ2(x)
∂x2
∂φ2(x)
∂x3
∂φ3(x)
∂x1
∂φ3(x)
∂x2
∂φ3(x)
∂x3
(8)
In the following, an active control of the form
u(t) = v(t) + Dφ(x)f(x)
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108 International Journal of Nonlinear Science, Vol.16(2013), No.2, pp. 105-112
is adopted to design the controllers. Where v(t) = (v1(t), v2(t), v3(t))T.
Hence,
˙e=−σe1−gσx1+σx2−σe2−σj x1+v(t)
˙e2=−y1y3+re1+rgx1−rx2−e2−jx1+v2(t)
˙e3=y1y2−be3−b1x2+v3(t)
(9)
We re-define the control functions to eliminate all terms which are not linear in e1, e2, e3as follows:
v1(t) = gσx1−σx2+σjx1+w1(t)
v2(t) = y1y3−rgx1+rx2+jx1+w2(t)
v3(t) = −y1y2+b1x2+w3(t)
(10)
substituting (10) into (9) the error is obtained as
˙e1=−σe1−σe2+w1(t)
˙e2=re1−e2+w2(t)
˙e3=−be3+w3(t)
(11)
The error system in (11) to be controlled is a linear system with control input w1(t), w2(t), w3(t)as a function of
e1, e2, e3. As long as these feedbacks stabilize the system, e1, e2, e3will converge to zero as time tends to infinity.
Using the active control method, we choose
w1(t)
w2(t)
w3(t)
=A
e1
e2
e3
(12)
There are many possible choices for the elements of the matrix Aprovided that all eigenvalues of Ahave negative real
part. As a result, the system given by (11) will be stable. For simplicity, the following is chosen as suitable value for
A[23]
A=
(λ1+σ)σ0
−r(λ2+ 1) 0
0 0 (λ3+b)
(13)
3.2 Reduced order synchronization
The Duffing oscillator (4) is used as the response while the Lorenz system given in (3) is taken as the drive system to
demonstrate reduced order synchronization. With the controls added, the response system becomes
˙x1=x2+u1
˙x2=−bx2−αx1−βx3
1+fcos ωt +u2
(14)
The goal is to design appropriate active controllers ui(t), i = 1,2such that the trajectory of the slave system (14) asymp-
totically approach the drive system (3) for a given functional relation and finally undergoes generalized synchronization.
The error states are defined as
e1=x1−φ1
e2=x2−φ2
(15)
As in section 3.1, a simple linear generalized synchronization of the φ(x) = (φ1, φ2) = (y3, y2−y1)Tis utilized. Using
the notation in (15), the error dynamical system is given as in (7).
An active control of the form u(t) = v(t) + Dφ(x)f(x)is adopted to design the controllers. Where v(t) = (v1(t), v2(t)).
Hence,
˙e1=e2+y2−y1+v1(t)
˙e2=−be2−by2+by1−αe1−αy3−βx3
1+fcos(ωt) + v2(t)(16)
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Samuel Ogunjo: Increased and Reduced Order Synchronization of 2D and 3D Dynamical Systems 109
The control functions are re-defined to eliminate all terms which cannot be shown in the form e1, e2as follows:
v1(t) = −y2+y1+w1(t)
v2(t) = by2−by1+αy3+βx3
1−fcos(ωt) + w2(t)(17)
substituting (17) in (16), we get
˙e1=e2+w2(t)
˙e2=−be2−αe1+w1(t)(18)
The error system (18) to be controlled is a linear system with a control input w1(t), w2(t)as functions of e1, e2. As long
as these feedbacks stabilize the system, e1, e2will converge to zero as time tends to infinity. This implies that systems (3)
and (4) are synchronized in a generalized differentiable transformation y=φ(x). Using the active control method, we
choose
(w1(t)
w2(t))=A(e1
e2)(19)
where A is a constant matrix. There are many possible choices for the elements of the matrix A provided that all eigen-
values of A have a negative real part. As a result, the system 18 will be stable[23].
4 Results and discussion
The parameters of the Duffing oscillator are chosen as b= 0.1, α = 0.5, β = 0.5, f = 0.95, ω = 0.75 and the
parameters of the Lorenz systems are chosen as σ= 10, r = 28, b =8
3so that both exhibit chaotic behaviour without
the controller. In the case of increased-order synchronization, the initial conditions are taken arbitrarily as (0,0.1) and
(0,3,12) for the drive and response system respectively while for the reduced-order synchronization scheme the initial
conditions for the drive system is (0,1) and the response system (1,1,0). The simulation results for the errors in the
increased order synchronization scheme are shown in figure (3) while that of the reduced order scheme is shown in figure
(4) when controllers are activated at time t= 0. The average error propagation on the state system variable is defined in
ref [26, 27] as |e|=√(e2
1+e2
2)and |e|=√(e2
1+e2
2+e2
3)for both reduced order and increased order synchronization
respectively. For both cases, each of the eigenvalues are taken to be −1. Furthermore, the synchronization in both cases is
seen to be effective for both single well and double well Duffing oscillators as the error vector become zero as time tends
to infinity in both cases shown in figures (3) and (4).
−0.1
0
0.1
−5
0
5
0 10 20 30 40 50 60 70 80 90 100
0
10
20
Time
Figure 3: Error in the synchronization scheme between drive (Duffing) and response (Lorenz) in Increased-order synchro-
nization.
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110 International Journal of Nonlinear Science, Vol.16(2013), No.2, pp. 105-112
−0.04
−0.02
0
0.02
0 10 20 30 40 50 60 70 80 90 100
−1
−0.5
0
0.5
Time
Figure 4: Error values in the synchronization scheme between drive(Lorenz) and response (Duffing) in Reduced-order
synchronization.
4.1 Synchronization time
Synchronization between two systems occur at a time tafter the activation of the control function. The time taken for the
response system x(t)to attain the dynamics of the drive system y(t)after the activation of the control function is regarded
as the synchronization time. In Figure 5, this is represented by b. The region ahas significant error associated with
the transmitted signal, hence recovered signal cannot be trusted. The synchronization time can be used to quantify the
0 5 10 15 20 25
−1.5
−1
−0.5
0
0.5
1
1.5
Time
a
b
Figure 5: Synchronization between the drive and response system of an arbitrary synchronized system. Point bis the
synchronization time. In the region a, received signal will ’noisy’ due to the error.
effectiveness of a synchronization scheme as a short time interval means an efficient synchronization. Hence, information
can be received faster without much error. According to Zhang et al., [25], perturbation by noise from measurement and
environment cannot be ignored in synchronization as is time. At the port of the response system, the input signal is not at
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Samuel Ogunjo: Increased and Reduced Order Synchronization of 2D and 3D Dynamical Systems 111
the same time instant as that of the drive system such that the drive and response system can be written as:
dx(¯
t)
d¯
t=f(¯
t, x(¯
t)),¯
t=t−τ
dˆy(t)
dt =ˆ
f(t, ˆy, x(¯
t))
(20)
The time taken for the magnitude of error in each of the synchronization scheme (increased and reduced order) to approach
zero is used to determine their efficiency. The graph is shown in figure (6). Synchronization time is seen to decrease
with increasing coupling strength. This implies that in synchronizing a Duffing oscillator to Lorenz system, the time
for synchronization decreases with time and better time is obtained for reduced-order active control synchronization.
Practically, encoded message between the two chaotic system will be received in shorter time using reduced-order active
control and as the coupling strength is reduced better results are obtained.
Figure 6: Synchronization time for both increased order and reduced order synchronization between the Duffing oscillator
and Lorenz chaotic system.
5 Conclusion
In this paper, the generalized synchronization of 2D Duffing oscillator and 3D Lorenz systems for both increased order
and reduced order using active control technique. Suitable control functions which achieve both increased and reduced
synchronization were designed. Simulation results were presented to illustrate the effectiveness of the proposed method.
Furthermore, the efficiency of the two schemes were compared using a quantity - synchronization time. The result can
be extended to both single-well Duffing oscillator. From the results obtained, it can be concluded that reduced order
synchronization offers a faster time to synchronization than increased order. The Duffing oscillator has been extended to
the triple-well, Duffing-van-der Pol Oscillator and ϕ6Duffing oscillator. Therefore, experimental research can be done to
see which system offers the best synchronization time and practical application to secure communication.
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In this paper, we tried to achieve synchronization between two chaotic systems of different order in master-slave configuration. We mainly focused on reduced order synchronization case. Achieving synchronization of two non-identical non-linear systems is often a very challenging problem. This paper proposes an adaptive law while utilizing to attain linear matrix inequality (LMI) framework so as to identify the conditions to attain synchronization goal. As all natural systems involve uncertainty and are exposed to external disturbances in real practice. Here a detailed analysis to tackle such dynamical systems which contain such effects, is performed. Using Lyapunov stability theory, the asymptotic convergent behavior of the error dynamics between master and slave systems of proposed class under the control effect is proved. The efficacy of the proposed control scheme is tested on different chaotic systems, provided as example. Finally, detailed simulation results are provided in the end of the article to illustrate the effectiveness of the proposed work.
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... Different synchronization methods and techniques have been used to study synchronization between two similar integer order systems (Ojo and Ogunjo 2012), two dissimilar integer order systems of same dimension (Motallebzadeh et al. 2012;Femat and Solís-Perales 2002), two similar or dissimilar systems with different dimensions Ogunjo 2013), three or more integer order system (compound, combination-combination synchronization Ojo et al. 2014b, a;Mahmoud et al. 2016, discrete systems (Liu 2008Kloeden 2004;Ma et al. 2007), fractional order system of similar dimension (Lu 2005), fractional order synchronization of different dimension (Bhalekar 2014;Khan and Bhat 2017), circuit implementation of synchronization (Adelakun et al. 2017), impulsive anti-synchronization of fractional order (Meng et al. 2018), discrete fractional order systems (Liu 2016), multiswitching (Ogunjo et al. 2018), and synchronization between integer order and fractional order systems (Chen et al. 2012). ...
Modified hybrid combination synchronization of chaotic fractional order systems
Article
Full-text available
  • Mar 2022
  • SOFT COMPUT
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization and anti-synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
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... Recently, the topic of synchronization between different dimensional chaotic and hyperchaotic systems attract more and more attentions. Until now, many effective control schemes have been introduced to achieve chaos synchronization between dynamical systems with different dimensions such as full state hybrid projective synchronization [7], inverse matrix projective synchronization [8], generalized synchronization [9], inverse generalized synchronization [10], hybrid synchronization [11], − synchronization [12], Q-S synchronization [13], reduced order synchronization [14] and increased order generalized synchronization [15]. Recently, a new type of synchronization, called − φ generalized synchronization, has been proposed to synchronize chaotic and hyperchaotic systems with different dimensions. ...
Λ − φ generalized synchronization: application to fractional hyperchaotic systems with arbitrary dimensions and orders
Article
Full-text available
  • Aug 2020
This paper investigates the − φ generalized synchronization between non-identical fractionalorder systems characterized by different dimensions and different orders. The − φ generalized synchronization combines the inverse matrix projective synchronization with the generalized synchronization. In particular, the proposed approach enables − φ generalized synchronization to be achieved between n-dimensional master system and m-dimensional slave system in different dimensions. The technique, which exploits nonlinear controllers, stability property of integer-order linear systems and Lyapunov stability theory, proves to be effective in achieving the − φ generalized synchronization. Finally, the approach is applied between 4-D and 5-D fractional hyperchaotic systems with the aim to illustrate the capabilities of the novel scheme proposed herein.
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... Different synchronization methods and techniques have been used to study synchronization between two similar integer order systems [26], two dissimilar integer order systems of same dimension [20,8], two similar or dissimilar systems with different dimensions [27,28,22], three or more integer order system (compound, combination-combination synchronization) [25,24,19], discrete systems [14,12,18], fractional order system of similar dimension [16], fractional order synchronization of different dimension [2,11], circuit implementation of synchronization [1] and synchronization between integer order and fractional order systems [3]. ...
Modified hybrid combination synchronization of chaotic fractional order systems
Preprint
Full-text available
  • Sep 2018
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization, anti-synchronization and project synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
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... The synchronization of chaotic systems with different dimensions has been studied for integer order systems. Some of the studies include increased order synchronization of two systems (Ogunjo, 2013;Ojo et al., 2014c), hybrid function projective combination synchronization (Ojo et al., 2014b), reduced order function projective combination synchronization (Ojo et al., 2014a), backstepping fuzzy adaptive control (Wang and Fan, 2015), and experimental designs (Adelakun et al., 2017) (Fig. 15.1). ...
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... As a consequence of increasing interest in discussing the synchronization phenomena of chaotic nonlinear dynamical systems numerous synchronization types and schemes have been put forward for consideration for instance generalized synchronization [34], projective synchronization [35], modified projective synchronization [36], function projective synchronization [37], modified-function projective synchronization [38], hybrid synchronization [39] and hybrid function projective synchronization [40]. Few years back another kind of synchronization was developed namely increased-order synchronization and reduced-order synchronization in hyper-chaotic systems [41,42]. In this manuscript, we have considered the reduced order scheme for synchronization of the novel chaotic system. ...
Dynamical behavior and reduced-order combination synchronization of a novel chaotic system
Article
Full-text available
  • Sep 2018
This manuscript investigates a novel 3D autonomous chaotic system which generates two strange attractors. The Lyapunov exponent, bifurcation diagram, Poincaré section, Kaplan–Yorke dimension, equilibria and phase portraits are given to justify the chaotic nature of the system. The novel system displays fixed orbit, periodic orbit, chaotic orbit as the parameter value varies. The reduced order combination synchronization is also performed by considering three identical 3D novel chaotic systems in two parts (a) choosing two third order master systems and one second order slave system which is the projection in the 2D plane. (b) choosing one third order master system and two second order slave systems which are the projection in the 2D plane. Numerical simulations justify the validity of the theoretical results discussed.
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... As a result of fast-growing interest in chaos control and synchronisation, various synchronisation types and schemes have been proposed and reported, for instance complete synchronisation [20], phase synchronisation [21], generalised synchronisation [22], lag synchronisation [23], antisynchronisation [24], projective synchronisation [25], modified projective synchronisation [26], function projective synchronisation [27], modified-function projective synchronisation [28] and hybrid synchronisation [29]. It has been discovered a few years ago that some other important forms of synchronisation like the increased-order synchronisation [30] and reduced-order synchronisation in hyperchaotic systems [31]. Currently, generalised synchronisation (GS) was put forward where two chaotic (or hyperchaotic) systems are said to be synchronised if there is a functional relation between the states of both systems. ...
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Synchronization in chaotic system
Article
Full-text available
  • Mar 1990
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
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Generalized Projective Synchronization of the Energy Resource System
Article
Full-text available
In this paper, an active control method is proposed to projective-synchronize two chaotic systems. The proposed technique is applied to achieve generalized projective synchro-nization for the energy resource system, where all state variables are in a proportional way. This property allows us to arbitrarily direct the scaling factor onto a desired value. Feasibility of the proposed control scheme is illustrated through the numerical examples.
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ANTICIPATING AND LAG SYNCHRONIZATION IN CHAOTIC LASER SYSTEM
Article
Full-text available
The chaotic synchronization between two unidirectionally coupled single-mode semiconductor lasers is investigated when the lasers are subjected to external feedback. The dynamics of lasers is described by Lang–Kobayashi equation. Through numerical simulations, it is found that the lag and anticipating synchronization coexist in the system, and the transition occurs between two kinds of synchronization. Further discussion shows that the coexistence and transition process is based on the continuous similarity relation in bifilar helix structure.
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BACKSTEPPING CONTROL AND SYNCHRONIZATION OF PARAMETRICALLY AND EXTERNALLY EXCITED Φ VAN DER POL OSCILLATORS WITH APPLICATION TO SECURE COMMUNICATIONS
Article
Full-text available
In this paper recursive and adaptive backstepping nonlinear controllers are designed to, respectively, control and synchronize the triple-well or Φ⁶ Van der Pol oscillators (VDPOs) with both parametric and external excitations. The designed recursive backstepping nonlinear controller is capable of stabilizing the Φ⁶-VDPO at any position as well as controlling it to track any trajectory that is a smooth function of time. The designed adaptive backstepping nonlinear controller globally synchronizes two Φ⁶-VDPOs evolving from different initial conditions and its application to secure communications is computationally demonstrated. The results are all validated by numerical simulations.
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Projective Synchronization In Three-Dimensional Chaotic Systems
Article
Full-text available
  • Apr 1999
  • PHYS REV LETT
In partially linear systems, such as the Lorenz model, chaotic synchronization is possible in only some of the variables. We show that, for the nonsynchronizing variable, synchronization up to a scale factor is possible. We explain the mechanism for this projective form of chaotic synchronization in three-dimensional systems. Projective synchronization is illustrated for the Lorenz and disk dynamo systems. We also introduce a vector field that can be used to predict the scaling factor. [S0031-9007(99)08897-3].
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Intermittent Synchronization in a Pair of Coupled Chaotic Pendula
Article
Full-text available
  • Jul 1998
  • PHYS REV LETT
Numerical simulations have been carried out for a pair of unidirectionally coupled identical pendula under the action of a common external ac torque. Both the master pendulum and the slave pendulum were in chaotic states. The only form of persistent locking appeared to be a computational artifact; otherwise the synchronization of slave to master was found to be intermittent.
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Generalized projective synchronization of chaotic systems via modified active control
Article
  • May 2011
  • ACTA PHYS SIN-CH ED
Based on Lyapunov stability theory, by introducing a special matrix structure, a modified active control is proposed for the generalized projective synchronization of chaotic system. Compared with the traditional active control, the modified active control is independent of the Routh-Hurwitz criterion, which means the complexity of active control is simplified. The method is successfully applied to the energy resource system and nuclear spin generator system. Compared with other method, the proposed method, which could realize the generalized projective synchronization between both identical systems and different systems, is shown to be simple, direct, stable and efficient. Numerical simulation is provided to show the validity of the theoretical analysis and the effectiveness of the method.
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Circadian Oscillators and Phase Synchronization under a Light-Dark Cycle
Article
  • Jan 2006
This paper aims to analyze the circadian behavior of the population of suprachias-matic nucleus(SCN) neuron by light-dark cycle. Self-sustained oscillation is generated in indi-vidual SCN neuron by a molecular regulatory network. Cells oscillate with different periods, but the SCN neurons display significant synchronization under the effect of the light-dark(LD) cycle. We present a model for a population of circadian oscillations in the SCN. We show that synchronization can be entrained by a 24-h LD cycle. The results of this paper establish not only a theoretical foundation, but also a quantitative basis for understanding the essential coop-erative dynamics.
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Synchronization via active control of parametrically and externally excited Φ6 Van der Pol and Duffing oscillators and application to secure communications
Article
  • Apr 2011
  • J VIB CONTROL
In this paper active controllers based on the Lyapunov stability theory and the Routh-Hurwitz criteria, are designed to completely synchronize two parametrically and externally excited φ6 Van der Pol oscillators, φ6 Duffing oscillators, and a φ6 Van der Pol oscillator with a φ6 Duffing oscillator in the triple-well configuration of the φ6 potential. The coefficient matrix of the error dynamics between each pair of synchronized systems is chosen such that the number of active control functions reduces from two to one, thereby significantly reducing controller complexity in the design. The designed controllers enable the state variables of the response system to synchronize with those of the master system in both the identical and nonidentical cases. The results are validated using numerical simulations. Application to secure communications is computationally demonstrated.
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