This paper discusses the design of active controllers for achieving generalized projective synchronization (GPS) of identical hyperchaotic Lü systems (Chen, Lu, Lü and Yu, 2006), identical hyperchaotic Cai systems (Wang and Cai, 2009) and non-identical hyperchaotic Lü and hyperchaotic Cai systems. The synchronization results (GPS) for the hyperchaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for achieving the GPS of the
hyperchaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF TH...ijait
This paper discusses the design of active controllers for generalized projective synchronization (GPS) of identical Wang 3-scroll chaotic systems (Wang, 2009), identical Dadras 3-scroll chaotic systems (Dadras and Momeni, 2009) and non-identical Wang 3-scroll system and Dadras 3-scroll system. The synchronization results (GPS) derived in this paper for the 3 scroll chaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for
achieving the generalized projective synchronization (GPS) of the 3-scroll chaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF SPROTT-I SYSTEM WITH UNKNOWN PARAMETERSijscai
This paper derives new results for the adaptive control and synchronization design of the Sprott-I chaotic system (1994), when the system parameters are unknown. First, we build an adaptive controller to stabilize the Sprott-I chaotic system to its unstable equilibrium at the origin. Then we build an adaptive
synchronizer to achieve global chaos synchronization of the identical Sprott-I chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Sprott-I chaotic system have been established using adaptive control theory and Lyapunov stability
theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Sprott-I chaotic system.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM ijait
The hyperchaotic Newton-Leipnik system (Ghosh and Bhattacharya, 2010) is one of the recently discovered four-dimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Newton-Leipnik system with unknown parameters. First, adaptive
control laws are designed to stabilize the hyperchaotic Newton-Leipnik system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Newton-Leipnik
systems with unknown parameters. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems.
Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF TH...ijait
This paper discusses the design of active controllers for generalized projective synchronization (GPS) of identical Wang 3-scroll chaotic systems (Wang, 2009), identical Dadras 3-scroll chaotic systems (Dadras and Momeni, 2009) and non-identical Wang 3-scroll system and Dadras 3-scroll system. The synchronization results (GPS) derived in this paper for the 3 scroll chaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for
achieving the generalized projective synchronization (GPS) of the 3-scroll chaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF SPROTT-I SYSTEM WITH UNKNOWN PARAMETERSijscai
This paper derives new results for the adaptive control and synchronization design of the Sprott-I chaotic system (1994), when the system parameters are unknown. First, we build an adaptive controller to stabilize the Sprott-I chaotic system to its unstable equilibrium at the origin. Then we build an adaptive
synchronizer to achieve global chaos synchronization of the identical Sprott-I chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Sprott-I chaotic system have been established using adaptive control theory and Lyapunov stability
theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Sprott-I chaotic system.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM ijait
The hyperchaotic Newton-Leipnik system (Ghosh and Bhattacharya, 2010) is one of the recently discovered four-dimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Newton-Leipnik system with unknown parameters. First, adaptive
control laws are designed to stabilize the hyperchaotic Newton-Leipnik system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Newton-Leipnik
systems with unknown parameters. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems.
Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
Adaptive Control and Synchronization of Hyperchaotic Cai Systemijctcm
The hyperchaotic Cai system (Wang, Cai, Miao and Tian, 2010) is one of the important paradigms of fourdimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Cai system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Cai system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Cai systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM cseij
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of fourdimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...IJITCA Journal
This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZHEN...ijscai
This paper deals with a new research problem in the chaos literature, viz. hybrid synchronization of a
pair of chaotic systems called the master and slave systems. In the hybrid synchronization design of
master and slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS),
while the other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS coexist in the process of synchronization. This research work deals with the hybrid synchronization of
hyperchaotic Zheng systems (2010) and hyperchaotic Yu systems (2012). The main results of this hybrid
synchronization research work have been proved using Lyapunov stability theory. Numerical examples of
the hybrid synchronization results are shown along with MATLAB simulations for the hyperchaotic
Zheng and hyperchaotic Yu systems.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AN...Zac Darcy
The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
of master and slave systems asymptotically. In the hybrid synchronization design of master and slave
systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the other
part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process
of synchronization. This research work deals with the hybrid synchronization of hyperchaotic Xi systems
(2009) and hyperchaotic Li systems (2005). The main results of this hybrid research work are established
with Lyapunov stability theory. MATLAB simulations of the hybrid synchronization results are shown for
the hyperchaotic Xu and Li systems.
In the present work, Lyapunov stability theory, nonlinear adaptive control law and the parameter update
law were utilized to derive the state of two new chaotic reversal systems after being synchronized by the
function projective method. Using this technique allows for a scaling function instead of a constant thereby
giving a better method in applications in secure communication. Numerical simulations are presented to
demonstrate the effective nature of the proposed scheme of synchronization for the new chaotic reversal
system.
ADAPTIVESYNCHRONIZER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZH...ijitcs
This paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zheng
systems (2010) and hyperchaotic Yu systems (2012). In the hybrid synchronization design of master and
slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the
other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the
process of synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are not known and we handle this complicate problem using adaptive control. The
main results of this research work are established via adaptive control theory andLyapunov stability
theory. MATLAB plotsusing classical fourth-order Runge-Kutta method have been depictedfor the new
adaptive hybrid synchronization results for the hyperchaotic Zheng and hyperchaotic Yu systems.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF RÖSSLER PROTOTYPE-4 SYSTEMijait
This paper investigates the adaptive control and synchronization of Rössler prototype-4 system with unknown parameters. The Rössler prototype-4 system is a classical three-dimensional chaotic system studied by O.E. Rössler (1979). First, adaptive control laws are designed to stabilize the Rössler prototype-4 system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical Rössler prototype-4 systems with unknown parameters. Numerical simulations are shown to
validate and illustrate the effectiveness of the proposed adaptive control and synchronization schemes for the Rössler prototype-4 system.
A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and C...IJECEIAES
A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
Modified Projective Synchronization of Chaotic Systems with Noise Disturbance,...IJECEIAES
The synchronization problem of chaotic systems using active modified projective non- linear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.
DYNAMICS, ADAPTIVE CONTROL AND EXTENDED SYNCHRONIZATION OF HYPERCHAOTIC SYSTE...ijccmsjournal
This work studies the dynamics, control and synchronization of hyperchaotic Lorenz-stenflo system and its
application to secure communication. The proposed designed nonlinear feedback controller control and
globally synchronizes two identical Lorenz-stenflo hyperchaotic systems evolving from different initial
conditions with unknown parameters. Adaptive synchronization results were further applied to secure
communication. The numerical simulation results were presented to verify the effectiveness of the designed
nonlinear controller and its success in secure communication application.
SLIDING CONTROLLER DESIGN FOR THE GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL H...ijait
This paper establishes new results for the sliding controller design for the global chaos synchronization of identical hyperchaotic Yujun systems (2010). Hyperchaotic systems are chaotic nonlinear systems having more than one positive Lyapunov exponent. Because of the complex dynamics properties of hyperchaotic system such as high capacity, high security and high efficiency, they are very useful in secure
communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun systems.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
Global Chaos Synchronization of Hyperchaotic Pang and Hyperchaotic Wang Syste...CSEIJJournal
This paper investigates the global chaos synchronization of identical hyperchaotic Wang systems, identical
hyperchaotic Pang systems, and non-identical hyperchaotic Wang and hyperchaotic Pang systems via
adaptive control method. Hyperchaotic Pang system (Pang and Liu, 2011) and hyperchaotic Wang system
(Wang and Liu, 2006) are recently discovered hyperchaotic systems. Adaptive control method is deployed
in this paper for the general case when the system parameters are unknown. Sufficient conditions for global
chaos synchronization of identical hyperchaotic Pang systems, identical hyperchaotic Wang systems and
non-identical hyperchaotic Pang and Wang systems are derived via adaptive control theory and Lyapunov
stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control
method is very convenient for the global chaos synchronization of the hyperchaotic systems discussed in
this paper. Numerical simulations are presented to validate and demonstrate the effectiveness of the
proposed synchronization schemes.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Adaptive Control and Synchronization of Hyperchaotic Cai Systemijctcm
The hyperchaotic Cai system (Wang, Cai, Miao and Tian, 2010) is one of the important paradigms of fourdimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Cai system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Cai system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Cai systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM cseij
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of fourdimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...IJITCA Journal
This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
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parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
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This paper deals with a new research problem in the chaos literature, viz. hybrid synchronization of a
pair of chaotic systems called the master and slave systems. In the hybrid synchronization design of
master and slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS),
while the other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS coexist in the process of synchronization. This research work deals with the hybrid synchronization of
hyperchaotic Zheng systems (2010) and hyperchaotic Yu systems (2012). The main results of this hybrid
synchronization research work have been proved using Lyapunov stability theory. Numerical examples of
the hybrid synchronization results are shown along with MATLAB simulations for the hyperchaotic
Zheng and hyperchaotic Yu systems.
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ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AN...Zac Darcy
The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
of master and slave systems asymptotically. In the hybrid synchronization design of master and slave
systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the other
part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process
of synchronization. This research work deals with the hybrid synchronization of hyperchaotic Xi systems
(2009) and hyperchaotic Li systems (2005). The main results of this hybrid research work are established
with Lyapunov stability theory. MATLAB simulations of the hybrid synchronization results are shown for
the hyperchaotic Xu and Li systems.
In the present work, Lyapunov stability theory, nonlinear adaptive control law and the parameter update
law were utilized to derive the state of two new chaotic reversal systems after being synchronized by the
function projective method. Using this technique allows for a scaling function instead of a constant thereby
giving a better method in applications in secure communication. Numerical simulations are presented to
demonstrate the effective nature of the proposed scheme of synchronization for the new chaotic reversal
system.
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This paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zheng
systems (2010) and hyperchaotic Yu systems (2012). In the hybrid synchronization design of master and
slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the
other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the
process of synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are not known and we handle this complicate problem using adaptive control. The
main results of this research work are established via adaptive control theory andLyapunov stability
theory. MATLAB plotsusing classical fourth-order Runge-Kutta method have been depictedfor the new
adaptive hybrid synchronization results for the hyperchaotic Zheng and hyperchaotic Yu systems.
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This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
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validate and illustrate the effectiveness of the proposed adaptive control and synchronization schemes for the Rössler prototype-4 system.
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This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
Modified Projective Synchronization of Chaotic Systems with Noise Disturbance,...IJECEIAES
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DYNAMICS, ADAPTIVE CONTROL AND EXTENDED SYNCHRONIZATION OF HYPERCHAOTIC SYSTE...ijccmsjournal
This work studies the dynamics, control and synchronization of hyperchaotic Lorenz-stenflo system and its
application to secure communication. The proposed designed nonlinear feedback controller control and
globally synchronizes two identical Lorenz-stenflo hyperchaotic systems evolving from different initial
conditions with unknown parameters. Adaptive synchronization results were further applied to secure
communication. The numerical simulation results were presented to verify the effectiveness of the designed
nonlinear controller and its success in secure communication application.
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communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun systems.
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Global Chaos Synchronization of Hyperchaotic Pang and Hyperchaotic Wang Syste...CSEIJJournal
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adaptive control method. Hyperchaotic Pang system (Pang and Liu, 2011) and hyperchaotic Wang system
(Wang and Liu, 2006) are recently discovered hyperchaotic systems. Adaptive control method is deployed
in this paper for the general case when the system parameters are unknown. Sufficient conditions for global
chaos synchronization of identical hyperchaotic Pang systems, identical hyperchaotic Wang systems and
non-identical hyperchaotic Pang and Wang systems are derived via adaptive control theory and Lyapunov
stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control
method is very convenient for the global chaos synchronization of the hyperchaotic systems discussed in
this paper. Numerical simulations are presented to validate and demonstrate the effectiveness of the
proposed synchronization schemes.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Adaptive Controller and Synchronizer Design for Hyperchaotic Zhou System with...Zac Darcy
In this paper, we establish new results for the adaptive controller and synchronizer design for the
hyperchaotic Zhou system (2009), when the parameters of the system are unknown. Using adaptive control theory and Lyapunov stability theory, we first design an adaptive controller to stabilize the hyperchaotic Zhou system to its unstable equilibrium at the origin. Next, using adaptive control theory and Lyapunov stability theory, we design an adaptive controller to achieve global chaos synchronization
of the identical hyperchaotic Zhou systems with unknown parameters. Simulations have been provided for adaptive controller and synchronizer designs to validate and illustrate the effectiveness of the schemes.
The International Journal of Information Technology, Control and Automation (...IJITCA Journal
The International Journal of Information Technology, Control and Automation (IJITCA) is a Quarterly open access peer-reviewed journal that publishes articles which contribute new results in all areas of Information Technology (IT), Control Systems and Automation Engineering. The journal focuses on all technical and practical aspects of IT, Control Systems and Automation with applications in real-world engineering and scientific problems. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on information technology, control engineering, automation, modeling concepts and establishing new collaborations in these areas.
Authors are invited to contribute to this journal by submitting articles that illustrate research results, projects, surveying works and industrial experiences that describe significant advances in Information Technology, Control Systems and Automation.
The International Journal of Information Technology, Control and Automation (...IJITCA Journal
The International Journal of Information Technology, Control and Automation (IJITCA) is a Quarterly open access peer-reviewed journal that publishes articles which contribute new results in all areas of Information Technology (IT), Control Systems and Automation Engineering. The journal focuses on all technical and practical aspects of IT, Control Systems and Automation with applications in real-world engineering and scientific problems. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on information technology, control engineering, automation, modeling concepts and establishing new collaborations in these areas.
Authors are invited to contribute to this journal by submitting articles that illustrate research results, projects, surveying works and industrial experiences that describe significant advances in Information Technology, Control Systems and Automation.
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractorTELKOMNIKA JOURNAL
This paper presented stability application for chaos synchronization using a 6-D hyperchaotic system of different controllers and two tools: Lyapunov stability theory and Linearization methods. Synchronization methods based on nonlinear control strategy is used. The selecting controller's methods have been modified by applying complete synchronization. The Linearization methods can achieve convergence according to the of complete synchronization. Numerical simulations are carried out by using MATLAB to validate the effectiveness of the analytical technique.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...ijcseit
This paper investigates the hybrid chaos synchronization of identical 4-D hyperchaotic Liu systems
(2006), 4-D identical hyperchaotic Chen systems (2005) and hybrid synchronization of 4-D hyperchaotic
Liu and hyperchaotic Chen systems. The hyperchaotic Liu system (Wang and Liu, 2005) and hyperchaotic
Chen system (Li, Tang and Chen, 2006) are important models of new hyperchaotic systems. Hybrid
synchronization of the 4-dimensional hyperchaotic systems addressed in this paper is achieved through
complete synchronization of two pairs of states and anti-synchronization of the other two pairs of states
of the underlying systems. Active nonlinear control is the method used for the hybrid synchronization of
identical and different hyperchaotic Liu and hyperchaotic Chen and the stability results have been
established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these
calculations, the proposed nonlinear control method is effective and convenient to achieve hybrid
synchronization of the hyperchaotic Liu and hyperchaotic Chen systems. Numerical simulations are
shown to demonstrate the effectiveness of the proposed chaos synchronization schemes.
Adaptive Stabilization and Synchronization of Hyperchaotic QI SystemCSEIJJournal
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of four-
dimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization
of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to
stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control
theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations
are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization
schemes.
HYBRID SLIDING SYNCHRONIZER DESIGN OF IDENTICAL HYPERCHAOTIC XU SYSTEMS ijitjournal
In this paper, new results have been obtained via sliding mode control for the hybrid chaos synchronization
of identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009). In hybrid synchronization of master and
slave systems, the odd states are completely synchronized, while the even states are anti-synchronized. The
stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Xu systems
are established using Lyapunov stability theory. MATLAB simulations have been shown for the numerical
results to illustrate the hybrid synchronization schemes derived for the identical hyperchaotic Xu systems.
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
International Journal of Instrumentation and Control Systems (IJICS)ijcisjournal
International Journal of Instrumentation and Control Systems (IJICS) is a Quarterly open access peer-reviewed journal that publishes articles which contribute new results in all areas of Instrumentation Engineering and Control Systems. The journal focuses on all technical and practical aspects of Instrumentation Engineering and Control Systems. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on advanced instrumentation engineering, control systems and automation concepts and establishing new collaborations in these areas.
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ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG ...ijics
In the anti-synchronization of chaotic systems, a pair of chaotic systems called drive and responsesystems
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hyperchaotic Pang system (2011) with uncertain parameters via adaptive control. Hyperchaotic systems
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areas like neural networks, encryption, secure data transmission and communication. The main results
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THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZATION OF HYPERCHAOTIC LÜ AND HYPERCHAOTIC CAI SYSTEMS
1. International Journal of Advanced Information Technology (IJAIT) Vol. 2, No.2, April 2012
DOI : 10.5121/ijait.2012.2206 75
THE ACTIVE CONTROLLER DESIGN FOR
ACHIEVING GENERALIZED PROJECTIVE
SYNCHRONIZATION OF HYPERCHAOTIC LÜ AND
HYPERCHAOTIC CAI SYSTEMS
Sarasu Pakiriswamy1
and Sundarapandian Vaidyanathan1
1
Department of Computer Science and Engineering
Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sarasujivat@gmail.com
2
Research and Development Centre
Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
ABSTRACT
This paper discusses the design of active controllers for achieving generalized projective synchronization
(GPS) of identical hyperchaotic Lü systems (Chen, Lu, Lü and Yu, 2006), identical hyperchaotic Cai
systems (Wang and Cai, 2009) and non-identical hyperchaotic Lü and hyperchaotic Cai systems. The
synchronization results (GPS) for the hyperchaotic systems have been derived using active control method
and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these
calculations, the active control method is very effective and convenient for achieving the GPS of the
hyperchaotic systems addressed in this paper. Numerical simulations are provided to illustrate the
effectiveness of the GPS synchronization results derived in this paper.
KEYWORDS
Active Control, Hyperchaos, Hyperchaotic Systems, Generalized Projective Synchronization, Hyperchaotic
Lü System, Hyperchaotic Cai System.
1. INTRODUCTION
Chaotic systems are nonlinear dynamical systems which are highly sensitive to initial conditions.
This sensitivity of chaotic systems is usually called as the butterfly effect [1]. Small differences in
initial conditions (such as those due to rounding errors in numerical computation) yield widely
diverging outcomes for chaotic systems.
Hyperchaotic system is usually defined as a chaotic system with more than one positive
Lyapunov exponent. The first hyperchaotic system was discovered by O.E. Rössler ([2], 1979).
2. International Journal of Advanced Information Technology (IJAIT) Vol. 2, No.2, April 2012
76
Since hyperchaotic system has the characteristics of high capacity, high security and high
efficiency, it has the potential of broad applications in nonlinear circuits, secure communications,
lasers, neural networks, biological systems and so on. Thus, the studies on hyperchaotic systems,
viz. control, synchronization and circuit implementation are very challenging problems in the
chaos literature [3].
Chaos synchronization problem received great attention in the literature when Pecora and Carroll
[4] published their results on chaos synchronization in 1990. From then on, chaos synchronization
has been extensively and intensively studied in the last three decades [4-37]. Chaos theory has
been explored in a variety of fields including physical systems [5], chemical systems [6],
ecological systems [7], secure communications [8-10], etc.
Synchronization of chaotic systems is a phenomenon that may occur when a chaotic oscillator
drives another chaotic oscillator. Because of the butterfly effect which causes the exponential
divergence of the trajectories of two identical chaotic systems started with nearly the same initial
conditions, synchronizing two chaotic systems is seemingly a very challenging problem.
In most of the chaos synchronization approaches, the master-slave or drive-response formalism is
used. If a particular chaotic system is called the master or drive system and another chaotic
system is called the slave or response system, then the idea of anti-synchronization is to use the
output of the master system to control the slave system so that the states of the slave system have
the same amplitude but opposite signs as the states of the master system asymptotically. In other
words, the sum of the states of the master and slave systems are designed to converge to zero
asymptotically, when anti-synchronization appears.
In the recent years, various schemes have been deployed for chaos synchronization such as PC
method [4], OGY method [11], active control [12-15], adaptive control [16-20], backstepping
design [21-23], sampled-data feedback [24], sliding mode control [25-28], etc.
In generalized projective synchronization (GPS) of chaotic systems [29-30], the chaotic systems
can synchronize up to a constant scaling matrix. Complete synchronization [12-13], anti-
synchronization [31-34], hybrid synchronization [35], projective synchronization [36] and
generalized synchronization [37] are particular cases of generalized projective synchronization.
GPS has important applications in areas like secure communications and secure data encryption.
In this paper, we deploy active control method so as to derive new results for the generalized
projective synchronization (GPS) for identical and different hyperchaotic Lü and hyperchaotic
Cai systems. Explicitly, using active nonlinear control and Lyapunov stability theory, we achieve
generalized projective synchronization for identical hyperchaotic Lü systems (Chen, Lu, Lü and
Yu, [38], 2006), identical hyperchaotic Cai systems (Wang and Cai, [39], 2009) and non-identical
hyperchaotic Lü and hyperchaotic Cai systems.
This paper has been organized as follows. In Section 2, we give the problem statement and our
methodology. In Section 3, we present a description of the hyperchaotic systems considered in
this paper. In Section 4, we derive results for the GPS of two identical hyperchaotic Lü systems.
In Section 5, we derive results for the GPS of two identical hyperchaotic Cai systems. In Section
6, we discuss the GPS of non-identical hyperchaotic Lü and hyperchaotic Cai systems. In Section
7, we summarize the main results derived in this paper.
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2. PROBLEM STATEMENT AND OUR METHODOLOGY
Consider the chaotic system described by the dynamics
( )x Ax f x= +& (1)
where n
x∈R is the state of the system, A is the n n× matrix of the system parameters and
: n n
f →R R is the nonlinear part of the system. We consider the system (1) as the master or
drive system.
As the slave or response system, we consider the chaotic system described by the dynamics
( )y By g y u= + +& (2)
where n
y ∈R is the state of the system, B is the n n× matrix of the system parameters,
: n n
g →R R is the nonlinear part of the system and n
u ∈R is the controller of the slave system.
If A B= and ,f g= then x and y are the states of two identical chaotic systems. If A B≠ or
,f g≠ then x and y are the states of two different chaotic systems.
In the active control approach, we design a feedback controller ,u which achieves the generalized
projective synchronization (GPS) between the states of the master system (1) and the slave
system (2) for all initial conditions (0), (0) .n
x z ∈R
For the GPS of the systems (1) and (2), the synchronization error is defined as
,e y Mx= − (3)
where
1
2
0 0
0 0
0 0 n
M
α
α
α
=
L
L
M M O M
L
(4)
In other words, we have
, ( 1,2, , )i i i ie y x i nα= − = K (5)
From (1)-(3), the error dynamics is easily obtained as
( ) ( )e By MAx g y Mf x u= − + − +& (6)
The aim of GPS is to find a feedback controller u so that
lim ( ) 0
t
e t
→∞
= for all (0) .n
e ∈R (7)
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Thus, the problem of generalized projective synchronization (GPS) between the master system (1)
and slave system (2) can be translated into a problem of how to realize the asymptotic
stabilization of the system (6). So, the objective is to design an active controller u for stabilizing
the error dynamical system (6) at the origin.
We take as a candidate Lyapunov function
( ) ,T
V e e Pe= (8)
where P is a positive definite matrix.
Note that : n
V →R R is a positive definite function by construction.
We assume that the parameters of the master and slave system are known and that the states of
both systems (1) and (2) are measurable.
If we find a feedback controller u so that
( ) ,T
V e e Qe= −& (9)
where Q is a positive definite matrix, then : n
V →& R R is a negative definite function.
Thus, by Lyapunov stability theory [40], the error dynamics (6) is globally exponentially stable
and hence the condition (7) will be satisfied. Hence, GPS is achieved between the states of the
master system (1) and the slave system (2).
3. SYSTEMS DESCRIPTION
The hyperchaotic Lü system ([38], 2006) is described by the dynamics
1 2 1 4
2 2 1 3
3 3 1 2
4 4 1 3
( )x a x x x
x cx x x
x bx x x
x dx x x
= − +
= +
= − +
= +
&
&
&
&
(10)
where 1 2 3 4, , ,x x x x are the states and , , ,a b c d are constant, positive parameters of the system.
The Lü system (10) exhibits a hyperchaotic attractor when the parameter values are taken as
36, 3, 20a b c= = = and 1.3d =
Figure 1 depicts the phase portrait of the hyperchaotic Lü system (10).
The hyperchaotic Cai system ([39], 2009) is described by the dynamics
1 2 1
2 1 2 4 1 3
2
3 3 2
4 1
( )x p x x
x qx rx x x x
x sx x
x xε
= −
= + + −
= − +
= −
&
&
&
&
(11)
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where 1 2 3 4, , ,x x x x are the states and , , , ,p q r s ε are constant, positive parameters of the system.
The Cai dynamics (11) exhibits a hyperchaotic attractor when the parameter values are taken as
27.5, 3, 19.3, 2.9p q r s= = = = and 3.3ε =
Figure 2 depicts the phase portrait of the hyperchaotic Cai system (11).
Figure 1. The Phase Portrait of the Hyperchaotic Lü System
Figure 2. The Phase Portrait of the Hyperchaotic Cai System
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4. GPS OF IDENTICAL HYPERCHAOTIC LÜ SYSTEMS
4.1 Theoretical Results
In this section, we apply the active nonlinear control method for the generalized projective
synchronization (GPS) of two identical hyperchaotic Lü systems ([38], 2006).
Thus, the master system is described by the hyperchaotic Lü dynamics
1 2 1 4
2 2 1 3
3 3 1 2
4 4 1 3
( )x a x x x
x cx x x
x bx x x
x dx x x
= − +
= −
= − +
= +
&
&
&
&
(12)
where 1 2 3 4, , ,x x x x are the states and , , ,a b c d are positive, constant parameters of the system.
The slave system is described by the controlled hyperchaotic Lü dynamics
1 2 1 4 1
2 2 1 3 2
3 3 1 2 3
4 4 1 3 4
( )y a y y y u
y cy y y u
y by y y u
y dy y y u
= − + +
= − +
= − + +
= + +
&
&
&
&
(13)
where 1 2 3 4, , ,y y y y are the states and 1 2 3 4, , ,u u u u are the active controls to be designed.
For the GPS of the systems (12) and (13), the synchronization error e is defined by
, ( 1,2,3,4)i i i ie y x iα= − = (14)
where the scales 1 2 3 4, , ,α α α α are real numbers.
The error dynamics is obtained as
1 1 2 1 2 4 1 4 1
2 2 1 3 2 1 3 2
3 3 1 2 3 1 2 3
4 4 1 3 4 1 3 4
( )e ae a y x y x u
e ce y y x x u
e be y y x x u
e de y y x x u
α α
α
α
α
= − + − + − +
= − + +
= − + − +
= + − +
&
&
&
&
(15)
We choose the nonlinear controller as
1
2
3
4
1 2 1 2 4 1 4 1 1
2 1 3 2 1 3 2 2
3 1 2 3 1 2 3 3
4 1 3 4 1 3 4 4
( )u a
u ce
u
u de
e a y x y x k e
y y x x k e
be y y x x k e
y y x x k e
α
α
α
α α=
= −
= −
= − −
− − − + −
+ − −
+ −
+ −
(16)
where the gains 1 2 3 4, , ,k k k k are positive constants.
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Substituting (16) into (15), the error dynamics simplifies to
1 1
2 2
3 3
4 3
1
2
3
4
e k e
e k e
e k e
e k e
= −
= −
= −
= −
&
&
&
&
(17)
Next, we prove the following result.
Theorem 1. The active feedback controller (16) achieves global chaos generalized projective
synchronization (GPS) between the identical hyperchaotic Lü systems (12) and (13).
Proof. We consider the quadratic Lyapunov function defined by
( )2 2 2 2
1 2 3 4( )
1 1
,
2 2
T
V e e e e e e e= = + + + (18)
which is a positive definite function on 4
.R
Differentiating (18) along the trajectories of (17), we get
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (19)
which is a negative definite function on 4
.R
Thus, by Lyapunov stability theory [40], the error dynamics (17) is globally exponentially stable.
This completes the proof.
4.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method is used to solve the two
systems of differential equations (12) and (13) with the active controller (16).
The parameters of the identical hyperchaotic Lü systems are chosen as
36, 3, 20, 1.3a b c d= = = =
The initial values for the master system (12) are taken as
1 2 3 4(0) 24, (0) 17, (0) 12, (0) 18x x x x= = − = − =
The initial values for the slave system (13) are taken as
1 2 3 4(0) 11, (0) 20, (0) 5, (0) 34y y y y= − = = − =
The GPS scales are taken as 1 2 33.5, 2.9, 0.8,α α α= = − = and 4 1.4.α = −
We take the state feedback gains as 5ik = for 1,2,3,4.i =
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Figure 3 shows the GPS synchronization of the identical hyperchaotic Lü systems. Figure 4
shows the time-history of the GPS errors 1 2 3 4, , ,e e e e for the identical hyperchaotic Lü systems.
Figure 3. GPS Synchronization of the Identical Hyperchaotic Lü Systems
Figure 4. Time History of the GPS Synchronization Error
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5. GPS OF IDENTICAL HYPERCHAOTIC CAI SYSTEMS
5.1 Theoretical Results
In this section, we apply the active nonlinear control method for the generalized projective
synchronization (GPS) of two identical hyperchaotic Cai systems ([39], 2009).
Thus, the master system is described by the hyperchaotic Cai dynamics
1 2 1
2 1 2 4 1 3
2
3 3 2
4 1
( )x p x x
x qx rx x x x
x sx x
x xε
= −
= + + −
= − +
= −
&
&
&
&
(20)
where 1 2 3 4, , ,x x x x are the states and , , , ,p q r s ε are positive, constant parameters of the system.
The slave system is described by the controlled hyperchaotic Cai dynamics
1 2 1 1
2 1 2 4 1 3 2
2
3 3 2 3
4 1 4
( )y p y y u
y qy ry y y y u
y sy y u
y y uε
= − +
= + + − +
= − + +
= − +
&
&
&
&
(21)
where 1 2 3 4, , ,y y y y are the states and 1 2 3 4, , ,u u u u are the active controls to be designed.
For the GPS of the systems (20) and (21), the synchronization error e is defined by
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
e
e
e
e
y x
y x
y x
y x
α
α
α
α
= −
= −
= −
= −
(22)
where the scales 1 2 3 4, , ,α α α α are real numbers.
The error dynamics is obtained as
1 1 2 1 2 1
2 2 1 2 1 4 2 4 1 3 2 1 3 2
2 2
3 3 2 3 2 3
4 1 4 1 4
( )
( )
( )
e pe p y x u
e re q y x y x y y x x u
e se y x u
e y x u
α
α α α
α
ε α
= − + − +
= + − + − − + +
= − + − +
= − − +
&
&
&
&
(23)
We choose the nonlinear controller as
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1 1 2 1 2 1 1
2 2 1 2 1 4 2 4 1 3 2 1 3 2 2
2 2
3 3 2 3 2 3 3
4 1 4 1 4 4
( )
( )
( )
u pe p y x k e
u re q y x y x y y x x k e
u se y x k e
u y x k e
α
α α α
α
ε α
= − − −
= − − − − + + − −
= − + −
= − −
(24)
where the gains 1 2 3, ,k k k are positive constants.
Substituting (24) into (23), the error dynamics simplifies to
1 1
2 2
3 3
4 4
1
2
3
4
e k e
e k e
e k e
e k e
= −
= −
= −
= −
&
&
&
&
(25)
Next, we prove the following result.
Theorem 2. The active feedback controller (24) achieves global chaos generalized projective
synchronization (GPS) between the identical hyperchaotic Cai systems (20) and (21).
Proof. We consider the quadratic Lyapunov function defined by
( )2 2 2 2
1 2 3 4( )
1 1
,
2 2
T
V e e e e e e e= = + + + (26)
which is a positive definite function on 4
.R
Differentiating (26) along the trajectories of (25), we get
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (27)
which is a negative definite function on 4
.R
Thus, by Lyapunov stability theory [40], the error dynamics (25) is globally exponentially stable.
This completes the proof.
5.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method is used to solve the two
systems of differential equations (20) and (21) with the active controller (24).
The parameters of the identical hyperchaotic Cai systems are chosen as
27.5, 3, 19.3, 2.9, 3.3p q r s ε= = = = =
The initial values for the master system (20) are taken as
1 2 3 4(0) 15, (0) 26, (0) 10, (0) 8x x x x= = = − =
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The initial values for the slave system (21) are taken as
1 2 3 4(0) 21, (0) 17, (0) 34, (0) 12y y y y= − = = − =
The GPS scales are taken as
1 2 3 41.8, 0.6, 3.8, 2.7α α α α= − = = − =
We take the state feedback gains as
5ik = for 1,2,3,4.i =
Figure 5 shows the GPS synchronization of the identical hyperchaotic Cai systems.
Figure 6 shows the time-history of the GPS synchronization errors for the identical hyperchaotic
Cai systems.
Figure 5. GPS Synchronization of the Identical Hyperchaotic Cai Systems
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Figure 6. Time History of the GPS Synchronization Error
6. GPS OF HYPERCHAOTIC LÜ AND HYPERCHAOTIC CAI SYSTEMS
6.1 Theoretical Results
In this section, we apply the active nonlinear control method for the generalized projective
synchronization (GPS) of hyperchaotic Lü and hyperchaotic Cai systems.
Thus, the master system is described by the hyperchaotic Lü dynamics
1 2 1 4
2 2 1 3
3 3 1 2
4 4 1 3
( )x a x x x
x cx x x
x bx x x
x dx x x
= − +
= −
= − +
= +
&
&
&
&
(28)
where 1 2 3 4, , ,x x x x are the states and , , ,a b c d are constant, positive parameters of the system.
The slave system is described by the controlled hyperchaotic Cai dynamics
1 2 1 1
2 1 2 4 1 3 2
2
3 3 2 3
4 1 4
( )y p y y u
y qy ry y y y u
y sy y u
y y uε
= − +
= + + − +
= − + +
= − +
&
&
&
&
(29)
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where 1 2 3 4, , ,y y y y are the states, , , , ,p q r s ε are positive, constant parameters of the system and
1 2 3 4, , ,u u u u are the active nonlinear controls to be designed.
For the GPS of the systems (28) and (29), the synchronization error e is defined by
, ( 1,2,3,4)i i i ie y x iα= − = (30)
where the scales 1 2 3 4, , ,α α α α are real numbers.
The error dynamics is obtained as
[ ]
[ ]
[ ]
[ ]
1 2 1 1 2 1 4 1
2 1 2 4 1 3 2 2 1 3 2
2
3 3 2 3 3 1 2 3
4 1 4 4 1 3 4
( ) ( )e p y y a x x x u
e qy ry y y y cx x x u
e sy y bx x x u
e y dx x x u
α
α
α
ε α
= − − − + +
= + + − − − +
= − + − − + +
= − − + +
&
&
&
&
(31)
We choose the nonlinear controller as
[ ]
[ ]
[ ]
[ ]
1 2 1 1 2 1 4 1 1
2 1 2 4 1 3 2 2 1 3 2 2
2
3 3 2 3 3 1 2 3 3
4 1 4 4 1 3 4 4
( ) ( )u p y y a x x x k e
u qy ry y y y cx x x k e
u sy y bx x x k e
u y dx x x k e
α
α
α
ε α
= − − + − + −
= − − − + + − −
= − + − + −
= + + −
(32)
where the gains 1 2 3 4, , ,k k k k are positive constants.
Substituting (32) into (31), the error dynamics simplifies to
1 1
2 2
3 3
4 4
1
2
3
4
e k e
e k e
e k e
e k e
= −
= −
= −
= −
&
&
&
&
(33)
Next, we prove the following result.
Theorem 3. The active feedback controller (32) achieves global chaos generalized projective
synchronization (GPS) between the hyperchaotic Lü system (28) and hyperchaotic Cai system
(29).
Proof. We consider the quadratic Lyapunov function defined by
( )2 2 2 2
1 2 3 4( )
1 1
,
2 2
T
V e e e e e e e= = + + + (34)
which is a positive definite function on 4
.R
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Differentiating (26) along the trajectories of (33), we get
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (35)
which is a negative definite function on 4
.R
Thus, by Lyapunov stability theory [40], the error dynamics (33) is globally exponentially stable.
This completes the proof.
6.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method is used to solve the two
systems of differential equations (28) and (29) with the active controller (32).
The parameters of the hyperchaotic Lü system are chosen as
36, 3, 20, 1.3a b c d= = = =
The parameters of the hyperchaotic Cai system are chosen as
27.5, 3, 19.3, 2.9, 3.3p q r s ε= = = = =
The initial values for the master system (28) are taken as
1 2 3 4(0) 12, (0) 24, (0) 39, (0) 17x x x x= = = − = −
The initial values for the slave system (29) are taken as
1 2 3 4(0) 11, (0) 28, (0) 7, (0) 20y y y y= − = = =
The GPS scales are taken as
1 2 3 42.1, 1.5, 3.6, 0.6α α α α= = − = − =
We take the state feedback gains as
5ik = for 1,2,3,4i =
Figure 7 shows the GPS synchronization of the non-identical hyperchaotic Lü and hyperchaotic
Cai systems.
Figure 8 shows the time-history of the GPS synchronization errors 1 2 3 4, , ,e e e e for the non-
identical hyperchaotic Lü and hyperchaotic Cai systems.
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Figure 7. GPS Synchronization of the Hyperchaotic Lü and Hyperchaotic Cai Systems
Figure 8. Time History of the GPS Synchronization Error
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7. CONCLUSIONS
In this paper, we had derived active control laws for achieving generalized projective
synchronization (GPS) of the following pairs of hyperchaotic systems:
(A) Identical Hyperchaotic Lü Systems (2006)
(B) Identical Hyperchaotic Cai systems (2009)
(C) Non-identical Hyperchaotic Lü and Hyperchaotic Cai systems
The synchronization results (GPS) derived in this paper for the hyperchaotic Lü and hyperchaotic
Cai systems have been proved using Lyapunov stability theory. Since Lyapunov exponents are
not required for these calculations, the proposed active control method is very effective and
suitable for achieving GPS of the hyperchaotic systems addressed in this paper. Numerical
simulations are shown to demonstrate the effectiveness of the GPS synchronization results
derived in this paper for the hyperchaotic Lü and hyperchaotic Cai systems.
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Authors
Mrs. P. Sarasu obtained her M.E. degree in Embedded System Technologies from Anna
University, Tamil Nadu, India in 2007. She obtained her B.E. degree in Computer Science
Engineering from Bharatidasan University, Trichy, India in 1998. She is working as an
Assistant Professor in the Faculty of Computer Science Engineering, Vel Tech Dr. RR &
Dr. SR Technical University, Chennai. She is currently pursuing her Ph.D degree in
Computer Science Engineering from Vel Tech Dr. RR & Dr. SR Technical University,
Chennai. She has published several papers in International Journals and International
Conferences. Her current research interests are chaos, control, secure communication and
embedded systems.
Dr. V. Sundarapandian obtained his Doctor of Science degree in Electrical and
Systems Engineering from Washington University, Saint Louis, USA under the
guidance of Late Dr. Christopher I. Byrnes (Dean, School of Engineering and Applied
Science) in 1996. He is a Professor in the Research and Development Centre at Vel
Tech Dr. RR & Dr. SR Technical University, Chennai, Tamil Nadu, India. He has
published over 240 refereed international publications. He has published over 100
papers in National Conferences and over 60 papers in International Conferences. He is
the Editor-in-Chief of International Journal of Mathematics and Scientific Computing,
International Journal of Instrumentation and Control Systems, International Journal of
Control Systems and Computer Modelling, etc. His research interests are Linear and Nonlinear Control
Systems, Chaos Theory and Control, Soft Computing, Optimal Control, Process Control, Operations
Research, Mathematical Modelling and Scientific Computing. He has delivered several Key Note Lectures
on Linear and No nlinear Control Systems, Chaos Theory and Control, Scientific Computing using
MATLAB/SCILAB, etc.