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.
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.
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.
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 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.
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.
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.
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.
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.
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 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.
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.
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.
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 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 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.
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.
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.
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 WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZA...ijait
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.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
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.
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
are considered, and the design goal is to drive the sum of their respective states to zero asymptotically. This
paper derives new results for the anti-synchronization of hyperchaotic Yang system (2009) and
hyperchaotic Pang system (2011) with uncertain parameters via adaptive control. Hyperchaotic systems
are nonlinear chaotic systems withtwo or more positive Lyapunov exponents and they have applications in
areas like neural networks, encryption, secure data transmission and communication. The main results
derived in this paper are illustrated with MATLAB simulations.
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.
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.
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLijait
This paper investigates the problem of output regulation of Shimizu-Morioka chaotic system, which is one of the classical chaotic systems proposed by T. Shimizu and N. Morioka (1980). Explicitly, for the constant tracking problem, new state feedback control laws regulating the output of the Shimizu-Morioka chaotic system have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Shimizu-Morioka chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the
control schemes proposed in this paper for the output regulation of the Shimizu-Morioka chaotic system.
An Exponential Observer Design for a Class of Chaotic Systems with Exponentia...ijtsrd
In this paper, a class of generalized chaotic systems with exponential nonlinearity is studied and the state observation problem of such systems is explored. Using differential inequality with time domain analysis, a practical state observer for such generalized chaotic systems is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to demonstrate the validity, effectiveness, and correctness of the obtained result. Yeong-Jeu Sun "An Exponential Observer Design for a Class of Chaotic Systems with Exponential Nonlinearity" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38233.pdf Paper URL : https://www.ijtsrd.com/engineering/electrical-engineering/38233/an-exponential-observer-design-for-a-class-of-chaotic-systems-with-exponential-nonlinearity/yeongjeu-sun
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.
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.
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.
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 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 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.
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.
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.
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 WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZA...ijait
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.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
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.
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
are considered, and the design goal is to drive the sum of their respective states to zero asymptotically. This
paper derives new results for the anti-synchronization of hyperchaotic Yang system (2009) and
hyperchaotic Pang system (2011) with uncertain parameters via adaptive control. Hyperchaotic systems
are nonlinear chaotic systems withtwo or more positive Lyapunov exponents and they have applications in
areas like neural networks, encryption, secure data transmission and communication. The main results
derived in this paper are illustrated with MATLAB simulations.
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.
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.
OUTPUT REGULATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEM BY STATE FEEDBACK CONTROLijait
This paper investigates the problem of output regulation of Shimizu-Morioka chaotic system, which is one of the classical chaotic systems proposed by T. Shimizu and N. Morioka (1980). Explicitly, for the constant tracking problem, new state feedback control laws regulating the output of the Shimizu-Morioka chaotic system have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Shimizu-Morioka chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the
control schemes proposed in this paper for the output regulation of the Shimizu-Morioka chaotic system.
An Exponential Observer Design for a Class of Chaotic Systems with Exponentia...ijtsrd
In this paper, a class of generalized chaotic systems with exponential nonlinearity is studied and the state observation problem of such systems is explored. Using differential inequality with time domain analysis, a practical state observer for such generalized chaotic systems is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to demonstrate the validity, effectiveness, and correctness of the obtained result. Yeong-Jeu Sun "An Exponential Observer Design for a Class of Chaotic Systems with Exponential Nonlinearity" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38233.pdf Paper URL : https://www.ijtsrd.com/engineering/electrical-engineering/38233/an-exponential-observer-design-for-a-class-of-chaotic-systems-with-exponential-nonlinearity/yeongjeu-sun
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.
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.
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 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 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.
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.
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.
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.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control 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 anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
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 SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic 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 global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic 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 global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
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.
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 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 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.
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.
Similar to ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTIVE CONTROL (20)
ANALYSIS OF EXISTING TRAILERS’ CONTAINER LOCK SYSTEMS IJCSEIT Journal
Trailers carry large containers to various destinations in the world. These are manually locked on to
trailers as they move through these long distances. Security mainly refers to the safety of a state,
organization, property, and individuals against criminal activity. The study was made to analyze the
existing trailer locks and the insecurity being experienced currently. The study also focused on creating a
background to building an automated lock system for auto-mobiles. Findings showed that there are various
container types like the General Purpose containers, the Hard-Top containers and the Open-Top among
others. Similarly, the twist locks were the ones revised for this study. The study also discussed the
weaknesses of the twist locks, most especially the non-notification on unsecured locks. This causes leads to
accidents and wastage of lives and property. The study finally proposed an automated lock system to
overcome these weaknesses to some good extent.
A MODEL FOR REMOTE ACCESS AND PROTECTION OF SMARTPHONES USING SHORT MESSAGE S...IJCSEIT Journal
The smartphone usage among people is increasing rapidly. With the phenomenal growth of smartphone
use, smartphone theft is also increasing. This paper proposes a model to secure smartphones from theft as
well as provides options to access a smartphone through other smartphone or a normal mobile via Short
Message Service. This model provides option to track and secure the mobile by locking it. It also provides
facilities to receive the incoming call and sms information to the remotely connected device and enables the
remote user to control the mobile through SMS. The proposed model is validated by the prototype
implementation in Android platform. Various tests are conducted in the implementation and the results are
discussed.
BIOMETRIC APPLICATION OF INTELLIGENT AGENTS IN FAKE DOCUMENT DETECTION OF JOB...IJCSEIT Journal
The Job selection process in today’s globally competitive economy can be a daunting task for prospective
employees no matter their experience level. Although many years of research has been devoted to job
search and application resulting in good integration with information technology including the internet and
intelligent agent-based architectures, there are still many areas that need to be enhanced. Two such areas
include the quality of jobs associated with applicants in the job search by profiling the needs of employers
against the needs of prospective employees and the security and verifications schemes integrated to reduce
the instances of fraud and identity theft. The integration of mobile, intelligent agent, and cryptography
technologies provide benefits such as improved accessibility wirelessly, intelligent dynamic profiling, and
increased security. With this in mind we propose the intelligent mobile agents instead of human agents to
perform the Job search using fuzzy preferences which is been published elsewhere and application
operations incorporating the use of agents with a trust authority to establish employer trust and validate
applicant identity and accuracy. Our proposed system incorporates design methodologies to use JADELEAP
and Android to provide a robust, secure, user friendly solution.
FACE RECOGNITION USING DIFFERENT LOCAL FEATURES WITH DIFFERENT DISTANCE TECHN...IJCSEIT Journal
A face recognition system using different local features with different distance measures is proposed in this
paper. Proposed method is fast and gives accurate detection. Feature vector is based on Eigen values,
Eigen vectors, and diagonal vectors of sub images. Images are partitioned into sub images to detect local
features. Sub partitions are rearranged into vertically and horizontally matrices. Eigen values, Eigenvector
and diagonal vectors are computed for these matrices. Global feature vector is generated for face
recognition. Experiments are performed on benchmark face YALE database. Results indicate that the
proposed method gives better recognition performance in terms of average recognized rate and retrieval
time compared to the existing methods.
BIOMETRICS AUTHENTICATION TECHNIQUE FOR INTRUSION DETECTION SYSTEMS USING FIN...IJCSEIT Journal
Identifying attackers is a major apprehension to both organizations and governments. Recently, the most
used applications for prevention or detection of attacks are intrusion detection systems. Biometrics
technology is simply the measurement and use of the unique characteristics of living humans to distinguish
them from one another and it is more useful as compare to passwords and tokens as they can be lost or
stolen so we have choose the technique biometric authentication. The biometric authentication provides the
ability to require more instances of authentication in such a quick and easy manner that users are not
bothered by the additional requirements. In this paper, we have given a brief introduction about
biometrics. Then we have given the information regarding the intrusion detection system and finally we
have proposed a method which is based on fingerprint recognition which would allow us to detect more
efficiently any abuse of the computer system that is running.
PERFORMANCE ANALYSIS OF FINGERPRINTING EXTRACTION ALGORITHM IN VIDEO COPY DET...IJCSEIT Journal
A video fingerprint is a recognizer that is derived from a piece of video content. The video fingerprinting
methods obtain unique features of a video that differentiates one video clip from another. It aims to identify
whether a query video segment is a copy of video from the video database or not based on the signature of
the video. It is difficult to find whether a video is a copied video or a similar video, since the features of the
content are very similar from one video to the other. The main focus of this paper is to detect that the query
video is present in the video database with robustness depending on the content of video and also by fast
search of fingerprints. The Fingerprint Extraction Algorithm and Fast Search Algorithms are adopted in
this paper to achieve robust, fast, efficient and accurate video copy detection. As a first step, the
Fingerprint Extraction algorithm is employed which extracts a fingerprint through the features from the
image content of video. The images are represented as Temporally Informative Representative Images
(TIRI). Then, the second step is to find the presence of copy of a query video in a video database, in which
a close match of its fingerprint in the corresponding fingerprint database is searched using inverted-filebased
method. The proposed system is tested against various attacks like noise, brightness, contrast,
rotation and frame drop. Thus the performance of the proposed system on an average shows high true
positive rate of 98% and low false positive rate of 1.3% for different attacks.
Effect of Interleaved FEC Code on Wavelet Based MC-CDMA System with Alamouti ...IJCSEIT Journal
In this paper, the impact of Forward Error Correction (FEC) code namely Trellis code with interleaver on
the performance of wavelet based MC-CDMA wireless communication system with the implementation of
Alamouti antenna diversity scheme has been investigated in terms of Bit Error Rate (BER) as a function of
Signal-to-Noise Ratio (SNR) per bit. Simulation of the system under proposed study has been done in M-ary
modulation schemes (MPSK, MQAM and DPSK) over AWGN and Rayleigh fading channel incorporating
Walsh Hadamard code as orthogonal spreading code to discriminate the message signal for individual
user. It is observed via computer simulation that the performance of the interleaved coded based proposed
system outperforms than that of the uncoded system in all modulation schemes over Rayleigh fading
channel.
FUZZY WEIGHTED ASSOCIATIVE CLASSIFIER: A PREDICTIVE TECHNIQUE FOR HEALTH CARE...IJCSEIT Journal
In this paper we extend the problem of classification using Fuzzy Association Rule Mining and propose the
concept of Fuzzy Weighted Associative Classifier (FWAC). Classification based on Association rules is
considered to be effective and advantageous in many cases. Associative classifiers are especially fit to
applications where the model may assist the domain experts in their decisions. Weighted Associative
Classifiers that takes advantage of weighted Association Rule Mining is already being proposed. However,
there is a so-called "sharp boundary" problem in association rules mining with quantitative attribute
domains. This paper proposes a new Fuzzy Weighted Associative Classifier (FWAC) that generates
classification rules using Fuzzy Weighted Support and Confidence framework. The naïve approach can be
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ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTIVE CONTROL
1. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
DOI : 10.5121/ijcseit.2012.2306 81
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO
AND HYPERCHAOTIC XU SYSTEMS VIA
ACTIVE CONTROL
Sundarapandian Vaidyanathan
1
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 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.
KEYWORDS
Active Control, Anti-Synchronization, Hyperchaos, Hyperchaotic Bao System, Hyperchaotic Xu System.
1. INTRODUCTION
Chaos is a complex nonlinear phenomenon. The first chaotic attractor was discovered by Lorenz
([1], 1963), when he was studying the atmospheric convection in the weather models. Since then,
chaos has been received extensive investigations and chaos phenomenon has been observed in a
variety of fields including physical systems [2], chemical systems [3], ecological systems [4],
secure communications [5-6], etc.
The seminal work on chaos synchronization problem was published by Pecora and Carroll ([7],
1990). From then on, chaos synchronization has been extensively and intensively studied in the
last three decades [8-30].
In most of the anti-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. Thus,
when anti-synchronization is achieved, the sum of the states of the master and slave systems tends
to zero asymptotically with time.
2. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
82
In the recent years, various schemes have been deployed for chaos synchronization such as PC
method [2], OGY method [8], active control [9-12], adaptive control [13-15], backstepping
design [16], sampled-data feedback [17], sliding mode control [18-20], etc. Recently, active
control method has been applied to achieve anti-synchronization of two identical chaotic systems
[21-22].
Hyperchaotic system is usually defined as a chaotic system with at least two positive Lyapunov
exponents, implying that its dynamics are expanded in several different directions simultaneously.
Thus, the hyperchaotic systems have more complex dynamical behaviour which can be used to
improve the security of a chaotic communication system. Hence, the theoretical design and circuit
realization of various hyperchaotic signals have become important research topics [23-27].
In this paper, we use active control to derive new results for the anti-synchronization of identical
hyperchaotic Bao systems ([28], 2008), identical hyperchaotic Xu systems ([29], 2009) and non-
identical hyperchaotic Bao and hyperchaotic Xu systems.
This paper is organized as follows. In Section 2, we describe the problem statement and our
methodology using Lyapunov stability theory. In Section 3, we provide a description of the
hyperchaotic systems addressed in this paper. In Section 4, we discuss the anti-synchronization of
identical hyperchaotic Bao systems (2008) using active nonlinear control. In Section 5, we
discuss the anti-synchronization of identical hyperchaotic Xu systems (2009) using active
nonlinear control. In Section 6, we discuss the anti-synchronization of hyperchaotic Bao and
hyperchaotic Xu systems using active nonlinear control. In Section 7, we summarize the main
results obtained in this paper.
2. PROBLEM STATEMENT AND OUR METHODOLOGY
As the master or drive system, we consider the chaotic system described by
( ),x Ax f x= +& (1)
where n
x∈R is the state vector, A is the n n× matrix of system parameters and : n n
f →R R is
the nonlinear part of the system.
As the slave or response system, we consider the following chaotic system described by
( ) ,y By g y u= + +& (2)
where n
y ∈R is the state of the slave system, B is the n n× matrix of system parameters,
: n n
g →R R is the nonlinear part of the system and u is the active controller to be designed.
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.
For the anti-synchronization of the chaotic systems (1) and (2) using active control, we design a
state feedback controller ,u which anti-synchronizes the states of the master system (1) and the
slave system (2) for all initial conditions (0), (0) .n
x y ∈R
Thus, we define the anti-synchronization error as
3. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
83
,e y x= + (3)
Hence, the error dynamics is obtained as
( ) ( )e By Ax g y f x u= + + + +& (4)
Thus, the anti-synchronization problem is essentially to find a feedback controller (active
controller) u so as to stabilize the error dynamics (4) for all initial conditions, i.e.
lim ( ) 0,
t
e t
→∞
= for all (0) n
e ∈R (5)
We use the Lyapunov stability theory as our methodology.
We take as a candidate Lyapunov function
( ) ,T
V e e Pe= (6)
where P is a positive definite matrix. Note that : n
V R→R is a positive definite function by
construction.
If we find a feedback controller u so that
( ) ,T
V e e Qe= −& (7)
where Q is a positive definite matrix, then : n
V →& R R is a negative definite function.
Thus, by Lyapunov stability theory [25], the error dynamics (4) is globally exponentially stable.
Hence, the states of the master system (1) and the slave system (2) will be globally and
exponentially anti-synchronized for all values of the initial conditions (0), (0) .n
x y ∈R
3. SYSTEMS DESCRIPTION
The hyperchaotic Bao system ([28], 2008) is described by the 4D dynamics
1 2 1 4
2 2 1 3
3 1 2 3
4 1 2 3
( )x a x x x
x cx x x
x x x bx
x x dx xε
= − +
= −
= −
= +
&
&
&
&
(8)
where 1 2 3 4, , ,x x x x are the state variables and , , , ,a b c d ε are positive constants.
The four-dimensional Bao system (8) is hyperchaotic when
36, 3, 20, 0.1a b c d= = = = and 21.ε =
4. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
84
The phase portrait of the hyperchaotic Bao system is shown in Figure 1.
Figure 1. Phase Portrait of the Hyperchaotic Bao System
The hyperchaotic Xu system ([29], 2009) is described by the 4D dynamics
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x x x x
x x rx x
x x lx x
x x x mx
α
β
γ
= − +
= +
= − −
= −
&
&
&
&
(9)
where 1 2 3 4, , ,x x x x are the state variables and , , , ,l mα β γ are positive constants.
The four-dimensional Xu system (9) is hyperchaotic when
10, 40, 2.5, 16, 1r lα β γ= = = = = and 2.m =
The phase portrait of the hyperchaotic Xu system is shown in Figure 2.
5. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
85
Figure 2. Phase Portrait of the Hyperchaotic Xu System
3. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO SYSTEMS
In this section, we discuss the anti-synchronization of identical hyperchaotic Bao systems ([28],
2008).
As the master system, we consider the hyperchaotic Bao system described by
1 2 1 4
2 2 1 3
3 1 2 3
4 1 2 3
( )x a x x x
x cx x x
x x x bx
x x dx xε
= − +
= −
= −
= +
&
&
&
&
(10)
where 1 2 3 4, , ,x x x x are the state variables and , , , ,a b c d ε are positive constants.
As the slave system, we consider the controlled hyperchaotic Bao dynamics described by
1 2 1 4 1
2 2 1 3 2
3 1 2 3 3
4 1 2 3 4
( )y a y y y u
y cy y y u
y y y by u
y y dy y uε
= − + +
= − +
= − +
= + +
&
&
&
&
(11)
where 1 2 3 4, , ,y y y y are the state variables and 1 2 3 4, , ,u u u u are the active controls.
6. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
86
The anti-synchronization error is defined as
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= +
= +
= +
= +
(12)
A simple calculation gives the error dynamics
1 2 1 4 1
2 2 1 3 1 3 2
3 3 1 2 1 2 3
4 1 2 3 2 3 4
( )
( )
e a e e e u
e ce y y x x u
e be y y x x u
e e d y y x x uε
= − + +
= − − +
= − + + +
= + + +
&
&
&
&
(13)
We consider the active nonlinear controller defined by
1 2 1 4 1 1
2 2 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 1 2 3 2 3 4 4
( )
( )
u a e e e k e
u ce y y x x k e
u be y y x x k e
u e d y y x x k eε
= − − − −
= − + + −
= − − −
= − − + −
(14)
where 1 2 3 4, , ,k k k k are positive constants.
Substitution of (14) into (13) yields the linear error dynamics
1 1 1 2 2 2 3 3 3 4 4 4, , ,e k e e k e e k e e k e= − = − = − = −& & & & (15)
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= = + + + (16)
which is a positive definite function on 4
.R
Next, we establish the main result of this section using Lyapunov stability theory.
Theorem 1. The identical hyperchaotic Bao systems (10) and (11) are globally and
exponentially anti-synchronized with the active nonlinear controller (13).
Proof. Differentiating (16) along the trajectories of the error system (15), we get
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (17)
which is a negative definite function on 4
R
7. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
87
Thus, by Lyapunov stability theory [30], the error dynamics (15) is globally
exponentially stable. This completes the proof.
Numerical Simulations:
For the numerical simulations, the fourth order Runge-Kutta method with initial time-step
8
10h −
= is used to solve the two systems of differential equations (10) and (11) with the
active nonlinear controller (14). We take the gains as 5ik = for 1,2,3,4.i =
The parameters of the identical hyperchaotic Bao systems (8) and (9) are selected as
36, 3, 20, 0.1a b c d= = = = and 21.ε =
The initial values for the master system (8) are taken as
1 2 3 4(0) 5, (0) 2, (0) 14, (0) 20x x x x= = − = =
and the initial values for the slave system (9) are taken as
1 2 3 4(0) 24, (0) 17, (0) 18, (0) 22y y y y= = = − = −
Figure 4 depicts the anti-synchronization of the identical hyperchaotic Bao systems (10) and (11).
Figure 4 depicts the time-history of the anti-synchronization error.
Figure 3. Anti-Synchronization of Identical Hyperchaotic Bao Systems
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Figure 4. Time-History of Anti-Synchronization Error 1 2 3 4, , ,e e e e
4. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC XU SYSTEMS
In this section, we discuss the anti-synchronization of identical hyperchaotic Xu systems ([29],
2009).
As the master system, we consider the hyperchaotic Xu system described by
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x x x x
x x rx x
x x lx x
x x x mx
α
β
γ
= − +
= +
= − −
= −
&
&
&
&
(18)
where 1 2 3 4, , ,x x x x are the state variables and , , , , ,r l mα β γ are positive constants.
As the slave system, we consider the controlled hyperchaotic Xu dynamics described by
1 2 1 4 1
2 1 1 3 2
3 3 1 2 3
4 1 3 2 4
( )y y y y u
y y ry y u
y y ly y u
y y y my u
α
β
γ
= − + +
= + +
= − − +
= − +
(19)
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89
where 1 2 3 4, , ,y y y y are the state variables and 1 2 3 4, , ,u u u u are the active controls.
The anti-synchronization error is defined as
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= +
= +
= +
= +
(20)
A simple calculation gives the error dynamics
1 2 1 4 1
2 1 1 3 1 3 2
3 3 1 2 1 2 3
4 2 1 3 1 3 4
( )
( )
( )
e e e e u
e e r y y x x u
e e l y y x x u
e me y y x x u
α
β
γ
= − + +
= + + +
= − − + +
= − + + +
&
&
&
&
(21)
We consider the active nonlinear controller defined by
1 2 1 4 1 1
2 1 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 2 1 3 1 3 4 4
( )
( )
( )
u e e e k e
u e r y y x x k e
u e l y y x x k e
u me y y x x k e
α
β
γ
= − − − −
= − − + −
= + + −
= − − −
(22)
where 1 2 3 4, , ,k k k k are positive constants.
Substitution of (22) into (21) yields the linear error dynamics
1 1 1 2 2 2 3 3 3 4 4 4, , ,e k e e k e e k e e k e= − = − = − = −& & & & (23)
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= = + + + (24)
which is a positive definite function on 4
.R
Next, we establish the main result of this section using Lyapunov stability theory.
Theorem 2. The identical hyperchaotic Xu systems (18) and (19) are globally and
exponentially anti-synchronized with the active nonlinear controller (22).
Proof. Differentiating (16) along the trajectories of the error system (15), we get
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90
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (25)
which is a negative definite function on 4
R
Thus, by Lyapunov stability theory [30], the error dynamics (23) is globally
exponentially stable.
Hence, the states of the identical hyperchaotic Xu systems (18) and (19) are globally and
exponentially anti-synchronized for all initial conditions (0), (0) .n
x y ∈R
This completes the proof.
Numerical Simulations:
For the numerical simulations, the fourth order Runge-Kutta method with initial time-step
6
10h −
= is used to solve the two systems of differential equations (18) and (19) with the
active nonlinear controller (22).
We take the feedback gains as
1 2 3 45, 5, 5, 5k k k k= = = =
The parameters of the identical hyperchaotic Xu systems (18) and (19) are selected as
10, 40, 2.5, 16, 1r lα β γ= = = = = and 2.m =
The initial values for the master system (18) are taken as
1 2 3 4(0) 12, (0) 14, (0) 7, (0) 21x x x x= = − = − =
and the initial values for the slave system (19) are taken as
1 2 3 4(0) 26, (0) 10, (0) 18, (0) 4y y y y= = = = −
Figure 5 depicts the anti-synchronization of the identical hyperchaotic Xu systems (18) and (19).
Figure 6 depicts the time-history of the anti-synchronisation error 1 2 3 4, , , .e e e e
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91
Figure 5. Anti-Synchronization of Identical Hyperchaotic Xu Systems
Figure 6. Time-History of the Anti-Synchronization Error 1 2 3 4, , ,e e e e
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92
5. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND
HYPERCHAOTIC XU SYSTEMS
In this section, we consider the anti-synchronization of non-identical hyperchaotic Bao system
([28], 2008) and hyperchaotic Xu system ([29], 2009).
As the master system, we consider the hyperchaotic Bao system described by
1 2 1 4
2 2 1 3
3 1 2 3
4 1 2 3
( )x a x x x
x cx x x
x x x bx
x x dx xε
= − +
= −
= −
= +
&
&
&
&
(26)
where 1 2 3 4, , ,x x x x are the state variables and , , , ,a b c d ε are positive constants.
As the slave system, we consider the controlled hyperchaotic Xu dynamics described by
1 2 1 4 1
2 1 1 3 2
3 3 1 2 3
4 1 3 2 4
( )y y y y u
y y ry y u
y y ly y u
y y y my u
α
β
γ
= − + +
= + +
= − − +
= − +
(27)
where 1 2 3 4, , ,y y y y are the state variables, , , , , ,r l mα β γ are positive constants and
1 2 3 4, , ,u u u u are the active controls.
The anti-synchronization error is defined as
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= +
= +
= +
= +
(28)
A simple calculation gives the error dynamics
1 2 1 4 2 1 1
2 1 1 2 1 3 1 3 2
3 3 3 1 2 1 2 3
4 2 1 2 1 3 2 3 4
( ) ( )( )
( )
e e e e a x x u
e e x cx ry y x x u
e e b x ly y x x u
e me x mx y y dx x u
α α
β β
γ γ
ε
= − + − − − +
= − + + − +
= − − − − + +
= − + + + + +
&
&
&
&
(29)
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We consider the active nonlinear controller defined by
1 2 1 4 2 1 1 1
2 1 1 2 1 3 1 3 2 2
3 3 3 1 2 1 2 3 3
4 2 1 2 1 3 2 3 4 4
( ) ( )( )
( )
u e e e a x x k e
u e x cx ry y x x k e
u e b x ly y x x k e
u me x mx y y dx x k e
α α
β β
γ γ
ε
= − − − + − − −
= − + − − + −
= + − + − −
= − − − − −
(30)
Substitution of (30) into (29) yields the linear error dynamics
1 1 1
2 2 2
3 3 3
4 4 4
e k e
e k e
e k e
e k e
= −
= −
= −
= −
&
&
&
&
(31)
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= = + + + (32)
which is a positive definite function on 4
.R
Next, we establish the main result of this section using Lyapunov stability theory.
Theorem 3. The non-identical hyperchaotic Bao system (26) and hyperchaotic Xu system
(27) are globally and exponentially anti-synchronized with the active nonlinear
controller (29).
Proof. Differentiating (32) along the trajectories of the error system (31), we get
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (33)
which is a negative definite function on 4
R since α and γ are positive constants.
Thus, by Lyapunov stability theory [30], the error dynamics (31) is globally
exponentially stable.
Hence, it follows that the states of the hyperchaotic Bao system (26) and hyperchaotic Xu
system (27) are globally and exponentially anti-synchronized for all initial conditions
(0), (0) .n
x y ∈R
This completes the proof.
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94
Numerical Simulations
For the numerical simulations, the fourth order Runge-Kutta method with initial time-step
6
10h −
= is used to solve the two systems of differential equations (26) and (27) with the
active nonlinear controller (30).
We take the feedback gains as
1 2 35, 5, 5k k k= = = and 4 5.k =
The parameters of the identical hyperchaotic Bao system (24) are selected as
36, 3, 20, 0.1a b c d= = = = and 21.ε =
The parameters of the identical hyperchaotic Xu system (25) are selected as
10, 40, 2.5, 16, 1r lα β γ= = = = = and 2.m =
The initial values for the master system (26) are taken as
1 2 3 4(0) 25, (0) 6, (0) 9, (0) 14x x x x= = − = = −
and the initial values for the slave system (27) are taken as
1 2 3 4(0) 12, (0) 15, (0) 20, (0) 8y y y y= = = − = −
Figure 7 depicts the anti-synchronization of the non-identical hyperchaotic systems, viz.
hyperchaotic Bao system (26) and hyperchaotic Xu system (28).
Figure 8 depicts the time-history of the anti-synchronization error.
6. CONCLUSIONS
In this paper, using the active control method, new results have been derived for the anti-
synchronization for the following pairs of hyperchaotic systems:
(A) Identical hyperchaotic Bao systems (2008)
(B) Identical hyperchaotic Xu systems (2009)
(C) Non-identical hyperchaotic Bao and hyperchaotic Xu systems.
The anti-synchronization results derived in this paper have been proved using Lyapunov stability
theory. Since the Lyapunov exponents are not required for these calculations, the active control
method is very convenient and efficient for the anti- synchronization of identical and non-
identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulation results have been
presented to illustrate the anti-synchronization results derived in this paper.
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95
Figure 7. Anti-Synchronization of Hyperchaotic Bao and Hyperchaotic Xu Systems
Figure 8. Time-History of the Anti-Synchronization Error 1 2 3 4, , ,e e e e
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96
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Author
Dr. V. Sundarapandian earned his Doctor of Science degree in Electrical and Systems
Engineering from Washington University, Saint Louis, Missouri, USA. He is currently
Professor in the Research and Development Centre at Vel Tech Dr. RR & Dr. SR
Technical University, Chennai, Tamil Nadu, India. He has published over 260
publications in refereed International journals and over 170 papers in National and
International Conferences. He is the Editor-in-Chief of the AIRCC journals -
International Journal of Instrumentation and Control Systems, International Journal of
Control Systems and Computer Modelling and International Journal of Information
Technology, Control and Automation. His research interests are Linear and Nonlinear Control Systems,
Chaos Theory and Control, Soft Computing, Optimal Control, Process Control, Operations Research,
Mathematical Modelling, Scientific Computing using SCILAB/MATLAB. He has delivered many Keynote
Lectures on Chaos Theory and Control Engineering.