This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
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 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.
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.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
ADAPTIVE 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.
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.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
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.
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 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.
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.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
ADAPTIVE 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.
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.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
Adaptive 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.
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 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.
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.
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.
ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU ...ijait
This paper derives new adaptive results for the hybrid synchronization of hyperchaotic Xi systems (2009)
and hyperchaotic Li systems (2005). 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 unknown and we tackle this problem using adaptive control. The main results of
this research work are proved using adaptive control theory and Lyapunov stability theory. MATLAB
simulations using classical fourth-order Runge-Kutta method are shown for the new adaptive hybrid
synchronization results for the hyperchaotic Xu and hyperchaotic Li systems.
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.
Design of State Estimator for a Class of Generalized Chaotic Systemsijtsrd
In this paper, a class of generalized chaotic systems is considered and the state observation problem of such a system is investigated. Based on the time domain approach with differential inequality, a simple state estimator for such generalized chaotic systems is developed to guarantee 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 show the effectiveness of the obtained result. Yeong-Jeu Sun "Design of State Estimator for a Class of Generalized Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29270.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29270/design-of-state-estimator-for-a-class-of-generalized-chaotic-systems/yeong-jeu-sun
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
This paper investigates the exponential observer problem for the Lotka-Volterra (L-V) systems. We have
applied Sundarapandian’s theorem (2002) for nonlinear observer design to solve the problem of observer
design problem for the L-V systems. The results obtained in this paper are applicable to solve the
problem of nonlinear observer design problem for the L-V models of population ecology in the food webs.
Two species L-V predator-prey models are studied and nonlinear observers are constructed by applying
the design technique prescribed in Sundarapandian’s theorem (2002). Numerical simulations are
provided to describe the nonlinear observers for the L-V systems.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
Foundation and Synchronization of the Dynamic Output Dual Systemsijtsrd
In this paper, the synchronization problem of the dynamic output dual systems is firstly introduced and investigated. Based on the time domain approach, the state variables synchronization of such dual systems can be verified. Meanwhile, the guaranteed exponential convergence rate can be accurately estimated. Finally, some numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result. Yeong-Jeu Sun "Foundation and Synchronization of the Dynamic Output Dual Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29256.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29256/foundation-and-synchronization-of-the-dynamic-output-dual-systems/yeong-jeu-sun
Observer design for descriptor linear systemsMahendra Gupta
In this presentation, A method is given to design the observer for descriptor (singular) systems. This talk was delivered at International Conference of Mathematical Sciences, Sathyama University, Chennai, in July 2014
Robust Exponential Stabilization for a Class of Uncertain Systems via a Singl...ijtsrd
In this paper, the robust stabilization for a class of uncertain chaotic or non chaotic systems with single input is investigated. Based on Lyapunov like Theorem with differential and integral inequalities, a simple linear control is developed to realize the global exponential stabilization of such uncertain systems. In addition, the guaranteed exponential convergence rate can be correctly estimated. Finally, some numerical simulations with circuit realization are provided to show the effectiveness of the obtained result. Yeong-Jeu Sun "Robust Exponential Stabilization for a Class of Uncertain Systems via a Single Input Control and its Circuit Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29322.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29322/robust-exponential-stabilization-for-a-class-of-uncertain-systems-via-a-single-input-control-and-its-circuit-implementation/yeong-jeu-sun
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.
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 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.
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 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.
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.
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.
ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU ...ijait
This paper derives new adaptive results for the hybrid synchronization of hyperchaotic Xi systems (2009)
and hyperchaotic Li systems (2005). 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 unknown and we tackle this problem using adaptive control. The main results of
this research work are proved using adaptive control theory and Lyapunov stability theory. MATLAB
simulations using classical fourth-order Runge-Kutta method are shown for the new adaptive hybrid
synchronization results for the hyperchaotic Xu and hyperchaotic Li systems.
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.
Design of State Estimator for a Class of Generalized Chaotic Systemsijtsrd
In this paper, a class of generalized chaotic systems is considered and the state observation problem of such a system is investigated. Based on the time domain approach with differential inequality, a simple state estimator for such generalized chaotic systems is developed to guarantee 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 show the effectiveness of the obtained result. Yeong-Jeu Sun "Design of State Estimator for a Class of Generalized Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29270.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29270/design-of-state-estimator-for-a-class-of-generalized-chaotic-systems/yeong-jeu-sun
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
This paper investigates the exponential observer problem for the Lotka-Volterra (L-V) systems. We have
applied Sundarapandian’s theorem (2002) for nonlinear observer design to solve the problem of observer
design problem for the L-V systems. The results obtained in this paper are applicable to solve the
problem of nonlinear observer design problem for the L-V models of population ecology in the food webs.
Two species L-V predator-prey models are studied and nonlinear observers are constructed by applying
the design technique prescribed in Sundarapandian’s theorem (2002). Numerical simulations are
provided to describe the nonlinear observers for the L-V systems.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
Foundation and Synchronization of the Dynamic Output Dual Systemsijtsrd
In this paper, the synchronization problem of the dynamic output dual systems is firstly introduced and investigated. Based on the time domain approach, the state variables synchronization of such dual systems can be verified. Meanwhile, the guaranteed exponential convergence rate can be accurately estimated. Finally, some numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result. Yeong-Jeu Sun "Foundation and Synchronization of the Dynamic Output Dual Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29256.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29256/foundation-and-synchronization-of-the-dynamic-output-dual-systems/yeong-jeu-sun
Observer design for descriptor linear systemsMahendra Gupta
In this presentation, A method is given to design the observer for descriptor (singular) systems. This talk was delivered at International Conference of Mathematical Sciences, Sathyama University, Chennai, in July 2014
Robust Exponential Stabilization for a Class of Uncertain Systems via a Singl...ijtsrd
In this paper, the robust stabilization for a class of uncertain chaotic or non chaotic systems with single input is investigated. Based on Lyapunov like Theorem with differential and integral inequalities, a simple linear control is developed to realize the global exponential stabilization of such uncertain systems. In addition, the guaranteed exponential convergence rate can be correctly estimated. Finally, some numerical simulations with circuit realization are provided to show the effectiveness of the obtained result. Yeong-Jeu Sun "Robust Exponential Stabilization for a Class of Uncertain Systems via a Single Input Control and its Circuit Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29322.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29322/robust-exponential-stabilization-for-a-class-of-uncertain-systems-via-a-single-input-control-and-its-circuit-implementation/yeong-jeu-sun
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.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
Global Chaos Synchronization of Hyperchaotic 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.
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.
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.
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.
DYNAMICS, ADAPTIVE CONTROL AND EXTENDED SYNCHRONIZATION OF HYPERCHAOTIC SYSTE...ijccmsjournal
This work studies the dynamics, control and synchronization of hyperchaotic Lorenz-stenflo system and its
application to secure communication. The proposed designed nonlinear feedback controller control and
globally synchronizes two identical Lorenz-stenflo hyperchaotic systems evolving from different initial
conditions with unknown parameters. Adaptive synchronization results were further applied to secure
communication. The numerical simulation results were presented to verify the effectiveness of the designed
nonlinear controller and its success in secure communication application.
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.
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 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
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
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
HYBRID SLIDING SYNCHRONIZER DESIGN OF IDENTICAL HYPERCHAOTIC XU SYSTEMS ijitjournal
In this paper, new results have been obtained via sliding mode control for the hybrid chaos synchronization
of identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009). In hybrid synchronization of master and
slave systems, the odd states are completely synchronized, while the even states are anti-synchronized. The
stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Xu systems
are established using Lyapunov stability theory. MATLAB simulations have been shown for the numerical
results to illustrate the hybrid synchronization schemes derived for the identical hyperchaotic Xu systems.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
Adaptive 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.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical simulations are presented to demonstrate the effectiveness of the synchronization schemes.
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.
Similar to HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL (20)
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.
Information Technology Convergence Services & AI (ITCAI 2024)IJITCA Journal
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International Conference on Information Technology Convergence Services & AI (ITCAI 2024)
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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.
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.
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.
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.
2 nd International Conference on Soft Computing, Data mining and Data Scienc...IJITCA Journal
2
nd International Conference on Soft Computing, Data mining and Data Science (SCDD
2024) will provide an excellent international forum for sharing knowledge and results in
theory, methodology and applications of Soft Computing, Data mining, and Data Science.
The Conference looks for significant contributions to all major fields of the Soft Computing,
Data mining, and Data Science in theoretical and practical aspects. The aim of the
Conference is to provide a platform to the researchers and practitioners from both academia
as well as industry to meet and share cutting-edge development in the field.
Authors are solicited to contribute to the Conference by submitting articles that illustrate
research results, projects, surveying works and industrial experiences that describe significant
advances in the following areas, but are not limited to.
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.
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.
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.
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.
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.
International Journal of Information Technology, Control and Automation (IJITCA)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.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
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The family offers a choice of engines
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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• Remote control system for accessing CCR and allied system over serial or TCP.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
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Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
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HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL
1. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
DOI:10.5121/ijitca.2012.2102 15
HYBRID CHAOS SYNCHRONIZATION OF
UNCERTAIN LORENZ-STENFLO AND QI 4-D
CHAOTIC SYSTEMS BY ADAPTIVE CONTROL
Sundarapandian Vaidyanathan1
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 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 non-
identical 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 anti-
synchronized 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 adaptive
synchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
KEYWORDS
Adaptive Control, Chaos, Hybrid Synchronization, Lorenz-Stenflo System, Qi System.
1. INTRODUCTION
Chaotic systems are nonlinear dynamical systems that are highly sensitive to initial conditions.
The sensitive nature of chaotic systems is commonly called as the butterfly effect [1].
Chaos is an interesting nonlinear phenomenon and has been extensively and intensively studied
in the last two decades [1-30]. Chaos theory has been applied in many scientific disciplines such
as Mathematics, Computer Science, Microbiology, Biology, Ecology, Economics, Population
Dynamics and Robotics.
In 1990, Pecora and Carroll [2] deployed control techniques to synchronize two identical
chaotic systems and showed that it was possible for some chaotic systems to be completely
synchronized. From then on, chaos synchronization has been widely explored in a variety of
fields including physical systems [3], chemical systems [4-5], ecological systems [6], secure
communications [7-9], etc.
In most of the chaos synchronization approaches, the master-slave or drive-response formalism
is used. If a particular chaotic system is called the master or drive system and another chaotic
system is called the slave or response system, then the idea of the synchronization is to use the
2. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
16
output of the master system to control the slave system so that the output of the slave system
tracks the output of the master system asymptotically.
Since the seminal work by Pecora and Carroll [2], a variety of impressive approaches have been
proposed for the synchronization of chaotic systems such as the OGY method [10], active
control method [11-15], adaptive control method [16-22], sampled-data feedback
synchronization method [23], time-delay feedback method [24], backstepping method [25],
sliding mode control method [26-30], etc.
So far, many types of synchronization phenomenon have been presented such as complete
synchronization [2], phase synchronization [31], generalized synchronization [32], anti-
synchronization [33-34], projective synchronization [35], generalized projective
synchronization [36-37], etc.
Complete synchronization (CS) is characterized by the equality of state variables evolving in
time, while anti-synchronization (AS) is characterized by the disappearance of the sum of
relevant variables evolving in time. Projective synchronization (PS) is characterized by the fact
that the master and slave systems could be synchronized up to a scaling factor, whereas in
generalized projective synchronization (GPS), the responses of the synchronized dynamical
states synchronize up to a constant scaling matrix .α It is easy to see that the complete
synchronization (CS) and anti-synchronization (AS) are special cases of the generalized
projective synchronization (GPS) where the scaling matrix Iα = and ,Iα = − respectively.
In hybrid synchronization of chaotic systems [15], one part of the system is synchronized and
the other part is anti-synchronized so that the complete synchronization (CS) and anti-
synchronization (AS) coexist in the system. The coexistence of CS and AS is highly useful in
secure communication and chaotic encryption schemes.
In this paper, we investigate the hybrid chaos synchronization of 4-D chaotic systems, viz.
identical Lorenz-Stenflo systems ([38], 2001), identical Qi systems ([39], 2005) and non-
identical Lorenz-Stenflo and Qi systems. We deploy adaptive control method because the
system parameters of the 4-D chaotic systems are assumed to be unknown. The adaptive
nonlinear controllers are derived using Lyapunov stability theory for the hybrid chaos
synchronization of the two 4-D chaotic systems when the system parameters are unknown.
This paper is organized as follows. In Section 2, we provide a description of the 4-D chaotic
systems addressed in this paper, viz. Lorenz-Stenflo systems (2001) and Qi systems (2005). In
Section 3, we discuss the hybrid chaos synchronization of identical Lorenz-Stenflo systems. In
Section 4, we discuss the hybrid chaos synchronization of identical Qi systems. In Section 5, we
derive results for the hybrid synchronization of non-identical Lorenz-Stenflo and Qi systems.
2. SYSTEMS DESCRIPTION
The Lorenz-Stenflo system ([38], 2001) is described by
1 2 1 4
2 1 3 2
3 1 2 3
4 1 4
( )
( )
x x x x
x x r x x
x x x x
x x x
α γ
β
α
= − +
= − −
= −
= − −
&
&
&
&
(1)
where 1 2 3 4, , ,x x x x are the state variables and , , ,rα β γ are positive constant parameters of the
3. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
17
system.
The LS system (1) is chaotic when the parameter values are taken as
2.0, 0.7, 1.5α β γ= = = and 26.0.r =
The state orbits of the chaotic LS system (1) are shown in Figure 1.
The Qi system ([29], 2005) is described by
1 2 1 2 3 4
2 1 2 1 3 4
3 3 1 2 4
4 4 1 2 3
( )
( )
x a x x x x x
x b x x x x x
x cx x x x
x dx x x x
= − +
= + −
= − +
= − +
&
&
&
&
(2)
where 1 2 3 4, , ,x x x x are the state variables and , , ,a b c d are positive constant parameters of the
system.
The system (2) is hyperchaotic when the parameter values are taken as
30, 10, 1a b c= = = and 10.d =
The state orbits of the chaotic Qi system (2) are shown in Figure 2.
Figure 1. State Orbits of the Lorenz-Stenflo Chaotic System
4. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
18
Figure 2. State Orbits of the Qi Chaotic System
3. HYBRID SYNCHRONIZATION OF IDENTICAL LORENZ-STENFLO SYSTEMS
3.1 Theoretical Results
In this section, we discuss the hybrid synchronization of identical Lorenz-Stenflo systems ([38],
2001), where the parameters of the master and slave systems are unknown.
As the master system, we consider the LS dynamics described by
1 2 1 4
2 1 3 2
3 1 2 3
4 1 4
( )
( )
x x x x
x x r x x
x x x x
x x x
α γ
β
α
= − +
= − −
= −
= − −
&
&
&
&
(3)
where 1 2 3 4, , ,x x x x are the states and , , ,rα β γ are unknown real constant parameters of the
system.
5. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
19
As the slave system, we consider the controlled LS dynamics described by
1 2 1 4 1
2 1 3 2 2
3 1 2 3 3
4 1 4 4
( )
( )
y y y y u
y y r y y u
y y y y u
y y y u
α γ
β
α
= − + +
= − − +
= − +
= − − +
&
&
&
&
(4)
where 1 2 3 4, , ,y y y y are the states and 1 2 3 4, , ,u u u u are the nonlinear controllers to be designed.
The hybrid chaos synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= −
= +
= −
= +
(5)
The error dynamics is easily obtained as
1 2 1 2 4 4 1
2 2 1 1 1 3 1 3 2
3 3 1 2 1 2 3
4 1 1 4 4
( 2 ) ( 2 )
( 2 )
( 2 )
e e e x e x u
e e r e x x x y y u
e e y y x x u
e e x e u
α γ
β
α
= − − + − +
= − + + − − +
= − + − +
= − + − +
&
&
&
&
(6)
Let us now define the adaptive control functions
1 2 1 2 4 4 1 1
2 2 1 1 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 1 1 4 4 4
ˆ ˆ( ) ( 2 ) ( 2 )
ˆ( ) ( 2 )
ˆ( )
ˆ( ) 2
u t e e x e x k e
u t e r e x y y x x k e
u t e y y x x k e
u t e x e k e
α γ
β
α
= − − − − − −
= − + + + −
= − + −
= + + −
(7)
where ˆˆ ˆ, ,α β γ and ˆr are estimates of , ,α β γ and ,r respectively, and ,( 1,2,3,4)ik i = are
positive constants.
Substituting (7) into (6), the error dynamics simplifies to
1 2 1 2 4 4 1 1
2 1 1 2 2
3 3 3 3
4 4 4 4
ˆ ˆ( )( 2 ) ( )( 2 )
ˆ( )( 2 )
ˆ( )
ˆ( )
e e e x e x k e
e r r e x k e
e e k e
e e k e
α α γ γ
β β
α α
= − − − + − − −
= − + −
= − − −
= − − −
&
&
&
&
(8)
Let us now define the parameter estimation errors as
ˆˆ ˆ, ,e e eα β γα α β β γ γ= − = − = − and ˆ.re r r= − (9)
6. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
20
Substituting (9) into (8), we obtain the error dynamics as
1 2 1 2 4 4 1 1
2 1 1 2 2
3 3 3 3
4 4 4 4
( 2 ) ( 2 )
( 2 )r
e e e e x e e x k e
e e e x k e
e e e k e
e e e k e
α γ
β
α
= − − + − −
= + −
= − −
= − −
&
&
&
&
(10)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov
approach is used.
We consider the quadratic Lyapunov function defined by
( )2 2 2 2 2 2 2 2
1 2 3 4 1 2 3 4
1
( , , , , , , , )
2
r rV e e e e e e e e e e e e e e e eα β γ α β γ= + + + + + + + (11)
which is a positive definite function on 8
.R
We also note that
ˆˆ ˆ, ,e e eα β γα β γ= − = − = −
&& && & & and ˆre r= −&& (12)
Differentiating (11) along the trajectories of (10) and using (12), we obtain
[ ]
2 2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2 4 3
1 4 4 2 1 1
ˆˆ( 2 )
ˆ( 2 ) ( 2 )r
V k e k e k e k e e e e e x e e e
e e e x e e e x r
α β
γ
α β
γ
= − − − − + − − − − + − −
+ − − + + −
&&&
&&
(13)
In view of Eq. (13), the estimated parameters are updated by the following law:
2
1 2 1 2 4 5
2
3 6
1 4 4 7
2 1 1 8
ˆ ( 2 )
ˆ
ˆ ( 2 )
ˆ ( 2 ) r
e e e x e k e
e k e
e e x k e
r e e x k e
α
β
γ
α
β
γ
= − − − +
= − +
= − +
= + +
&
&
&
&
(14)
where 5 6 7, ,k k k and 8k are positive constants.
Substituting (14) into (12), we obtain
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 rV k e k e k e k e k e k e k e k eα β γ= − − − − − − − −& (15)
which is a negative definite function on 8
.R
7. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
21
Thus, by Lyapunov stability theory [40], it is immediate that the hybrid synchronization error
,( 1,2,3,4)ie i = and the parameter estimation error , , , re e e eα β γ decay to zero exponentially
with time.
Hence, we have proved the following result.
Theorem 1. The identical hyperchaotic Lorenz-Stenflo systems (3) and (4) with unknown
parameters are globally and exponentially hybrid synchronized via the adaptive control law (7),
where the update law for the parameter estimates is given by (14) and ,( 1,2, ,8)ik i = K are
positive constants.
3.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 6
10h −
= is
used to solve the 4-D chaotic systems (3) and (4) with the adaptive control law (14) and the
parameter update law (14) using MATLAB.
We take 4ik = for 1,2, ,8.i = K
For the Lorenz-Stenflo systems (3) and (4), the parameter values are taken as
2.0, 0.7, 1.5α β γ= = = and 26.0.r =
Suppose that the initial values of the parameter estimates are
ˆ ˆˆ ˆ(0) 6, (0) 5, (0) 2, (0) 7.a b r d= = = =
The initial values of the master system (3) are taken as
1 2 3 4(0) 4, (0) 10, (0) 15, (0) 19.x x x x= = = =
The initial values of the slave system (4) are taken as
1 2 3 4(0) 21, (0) 6, (0) 3, (0) 27.y y y y= = = =
Figure 3 depicts the hybrid-synchronization of the identical Lorenz-Stenflo systems (3) and (4).
It may also be noted that the odd states of the two systems are completely synchronized, while
the even states of the two systems are anti-synchronized.
Figure 4 shows that the estimated values of the parameters, viz. ˆˆ ˆ, ,α β γ and ˆr converge to the
system parameters
2.0, 0.7, 1.5α β γ= = = and 26.0.r =
8. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
22
Figure 3. Hybrid-Synchronization of Lorenz-Stenflo Chaotic Systems
Figure 4. Parameter Estimates ˆˆ ˆ ˆ( ), ( ), ( ), ( )t t t r tα β γ
9. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
23
4. HYBRID SYNCHRONIZATION OF IDENTICAL QI SYSTEMS
4.1 Theoretical Results
In this section, we discuss the hybrid synchronization of identical Qi systems ([39], 2005),
where the parameters of the master and slave systems are unknown.
As the master system, we consider the Qi dynamics described by
1 2 1 2 3 4
2 1 2 1 3 4
3 3 1 2 4
4 4 1 2 3
( )
( )
x a x x x x x
x b x x x x x
x cx x x x
x dx x x x
= − +
= + −
= − +
= − +
&
&
&
&
(16)
where 1 2 3 4, , ,x x x x are the states and , , ,a b c d are unknown real constant parameters of the
system.
As the slave system, we consider the controlled Qi dynamics described by
1 2 1 2 3 4 1
2 1 2 1 3 4 2
3 3 1 2 4 3
4 4 1 2 3 4
( )
( )
y a y y y y y u
y b y y y y y u
y cy y y y u
y dy y y y u
= − + +
= + − +
= − + +
= − + +
&
&
&
&
(17)
where 1 2 3 4, , ,y y y y are the states and 1 2 3 4, , ,u u u u are the nonlinear controllers to be designed.
The hybrid synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= −
= +
= −
= +
(18)
The error dynamics is easily obtained as
1 2 1 2 2 3 4 2 3 4 1
2 1 2 1 1 3 4 1 3 4 2
3 3 1 2 4 1 2 4 3
4 4 1 2 3 1 2 3 4
( 2 )
( 2 )
e a e e x y y y x x x u
e b e e x y y y x x x u
e ce y y y x x x u
e de y y y x x x u
= − − + − +
= + + − − +
= − + − +
= − + + +
&
&
&
&
(19)
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Let us now define the adaptive control functions
1 2 1 2 2 3 4 2 3 4 1 1
2 1 2 1 1 3 4 1 3 4 2 2
3 3 1 2 4 1 2 4 3 3
4 4 1 2 3 1 2 3 4 4
ˆ( ) ( 2 )
ˆ( ) ( 2 )
ˆ( )
ˆ( )
u t a e e x y y y x x x k e
u t b e e x y y y x x x k e
u t ce y y y x x x k e
u t de y y y x x x k e
= − − − − + −
= − + + + + −
= − + −
= − − −
(20)
where ˆˆ ˆ, ,a b c and ˆd are estimates of , ,a b c and ,d respectively, and ,( 1,2,3,4)ik i = are
positive constants.
Substituting (20) into (19), the error dynamics simplifies to
1 2 1 2 1 1
2 1 2 1 2 2
3 3 3 3
4 4 4 4
ˆ( )( 2 )
ˆ( )( 2 )
ˆ( )
ˆ( )
e a a e e x k e
e b b e e x k e
e c c e k e
e d d e k e
= − − − −
= − + + −
= − − −
= − − −
&
&
&
&
(21)
Let us now define the parameter estimation errors as
ˆˆ ˆ, ,a b ce a a e b b e c c= − = − = − and ˆ.de d d= − (22)
Substituting (22) into (21), we obtain the error dynamics as
1 2 1 2 1 1
2 1 2 1 2 2
3 3 3 3
4 4 4 4
( 2 )
( 2 )
a
b
c
d
e e e e x k e
e e e e x k e
e e e k e
e e e k e
= − − −
= + + −
= − −
= − −
&
&
&
&
(23)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov
approach is used.
We consider the quadratic Lyapunov function defined by
( )2 2 2 2 2 2 2 2
1 2 3 4 1 2 3 4
1
( , , , , , , , )
2
a b c d a b c dV e e e e e e e e e e e e e e e e= + + + + + + + (24)
which is a positive definite function on 8
.R
We also note that
ˆˆ ˆ, ,a b ce a e b e c= − = − = −
&& && & & and ˆ
de d= −
&
& (25)
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25
Differentiating (24) along the trajectories of (23) and using (25), we obtain
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2
2 2
2 1 2 1 3 4
ˆ( 2 )
ˆ ˆˆ( 2 )
a
b c d
V k e k e k e k e e e e e x a
e e e e x b e e c e e d
= − − − − + − − −
+ + + − + − − + − −
&&
& &&
(26)
In view of Eq. (26), the estimated parameters are updated by the following law:
1 2 1 2 5
2 1 2 1 6
2
3 7
2
4 8
ˆ ( 2 )
ˆ ( 2 )
ˆ
ˆ
a
b
c
d
a e e e x k e
b e e e x k e
c e k e
d e k e
= − − +
= + + +
= − +
= − +
&
&
&
&
(27)
where 5 6 7, ,k k k and 8k are positive constants.
Substituting (27) into (26), we obtain
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8a b c dV k e k e k e k e k e k e k e k e= − − − − − − − −& (28)
which is a negative definite function on 8
.R
Thus, by Lyapunov stability theory [40], it is immediate that the synchronization error
,( 1,2,3,4)ie i = and the parameter estimation error , , ,a b c de e e e decay to zero exponentially
with time.
Hence, we have proved the following result.
Theorem 2. The identical Qi systems (16) and (17) with unknown parameters are globally and
exponentially hybrid synchronized by the adaptive control law (20), where the update law for
the parameter estimates is given by (27) and ,( 1,2, ,8)ik i = K are positive constants.
4.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 6
10h −
= is
used to solve the chaotic systems (16) and (17) with the adaptive control law (14) and the
parameter update law (27) using MATLAB. We take 4ik = for 1,2, ,8.i = K
For the Qi systems (16) and (17), the parameter values are taken as
30, 10, 1,a b c= = = 10.d =
Suppose that the initial values of the parameter estimates are
ˆ ˆˆ ˆ(0) 6, (0) 2, (0) 9, (0) 5a b c d= = = =
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26
The initial values of the master system (16) are taken as
1 2 3 4(0) 4, (0) 8, (0) 17, (0) 20.x x x x= = = =
The initial values of the slave system (17) are taken as
1 2 3 4(0) 10, (0) 21, (0) 7, (0) 15.y y y y= = = =
Figure 5 depicts the complete synchronization of the identical Qi systems (16) and (17).
Figure 6 shows that the estimated values of the parameters, viz. ˆˆ ˆ, ,a b c and ˆd converge to the
system parameters
30, 10, 1, 10.a b c d= = = =
Figure 5. Anti-Synchronization of the Qi Systems
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27
Figure 6. Parameter Estimates ˆ ˆˆ ˆ( ), ( ), ( ), ( )a t b t c t d t
5. HYBRID SYNCHRONIZATION OF NON-IDENTICAL LORENZ-STENFLO
AND QI SYSTEMS
5.1 Theoretical Results
In this section, we discuss the adaptive synchronization of non-identical Lorenz-Stenflo system
([38], 2001) and Qi system ([39], 2005), where the parameters of the master and slave systems
are unknown.
As the master system, we consider the LS dynamics described by
1 2 1 4
2 1 3 2
3 1 2 3
4 1 4
( )
( )
x x x x
x x r x x
x x x x
x x x
α γ
β
α
= − +
= − −
= −
= − −
&
&
&
&
(29)
where 1 2 3 4, , ,x x x x are the states and , , ,rα β γ are unknown real constant parameters of the
system.
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28
As the slave system, we consider the controlled Qi dynamics described by
1 2 1 2 3 4 1
2 1 2 1 3 4 2
3 3 1 2 4 3
4 4 1 2 3 4
( )
( )
y a y y y y y u
y b y y y y y u
y cy y y y u
y dy y y y u
= − + +
= + − +
= − + +
= − + +
&
&
&
&
(30)
where 1 2 3 4, , ,y y y y are the states, , , ,a b c d are unknown real constant parameters of the system
and 1 2 3 4, , ,u u u u are the nonlinear controllers to be designed.
The hybrid chaos synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= −
= +
= −
= +
(31)
The error dynamics is easily obtained as
1 2 1 2 1 4 2 3 4 1
2 1 2 2 1 1 3 1 3 4 2
3 3 3 1 2 4 1 2 3
4 4 4 1 1 2 3 4
( ) ( )
( )
e a y y x x x y y y u
e b y y x rx x x y y y u
e cy x y y y x x u
e dy x x y y y u
α γ
β
α
= − − − − + +
= + − + − − +
= − + + − +
= − − − + +
&
&
&
&
(32)
Let us now define the adaptive control functions
1 2 1 2 1 4 2 3 4 1 1
2 1 2 2 1 1 3 1 3 4 2 2
3 3 3 1 2 4 1 2 3 3
4 4 4 1 1 2 3 4 4
ˆ ˆˆ( ) ( ) ( )
ˆ ˆ( ) ( )
ˆˆ( )
ˆ ˆ( )
u t a y y x x x y y y k e
u t b y y x rx x x y y y k e
u t cy x y y y x x k e
u t dy x x y y y k e
α γ
β
α
= − − + − + − −
= − + + − + + −
= − − + −
= + + − −
(33)
where ˆ ˆ ˆˆ ˆˆ ˆ, , , , , ,a b c d α β γ and ˆr are estimates of , , , , , ,a b c d α β γ and ,r respectively, and
,( 1,2,3,4)ik i = are positive constants.
Substituting (33) into (32), the error dynamics simplifies to
1 2 1 2 1 4 1 1
2 1 2 1 2 2
3 3 3 3 3
4 4 4 4 4
ˆ ˆˆ( )( ) ( )( ) ( )
ˆ ˆ( )( ) ( )
ˆˆ( ) ( )
ˆ ˆ( ) ( )
e a a y y x x x k e
e b b y y r r x k e
e c c y x k e
e d d y x k e
α α γ γ
β β
α α
= − − − − − − − −
= − + + − −
= − − + − −
= − − − − −
&
&
&
&
(34)
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29
Let us now define the parameter estimation errors as
ˆ ˆˆ ˆ, , ,
ˆˆ ˆ ˆ, , ,
a b c d
r
e a a e b b e c c e d d
e e e e r rα β γα α β β γ γ
= − = − = − = −
= − = − = − = −
(35)
Substituting (35) into (32), we obtain the error dynamics as
1 2 1 2 1 4 1 1
2 1 2 1 2 2
3 3 3 3 3
4 4 4 4 4
( ) ( )
( )
a
b r
c
d
e e y y e x x e x k e
e e y y e x k e
e e y e x k e
e e y e x k e
α γ
β
α
= − − − − −
= + + −
= − + −
= − − −
&
&
&
&
(36)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov
approach is used. We consider the quadratic Lyapunov function defined by
( )2 2 2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c d rV e e e e e e e e e e e eα β γ= + + + + + + + + + + + (37)
which is a positive definite function on 12
.R
We also note that
ˆ ˆ ˆˆ ˆˆ ˆ ˆ, , , , , , ,a b c d re a e b e c e d e e e e rα β γα β γ= − = − = − = − = − = − = − = −
& & && && & && & & & & & & & (38)
Differentiating (37) along the trajectories of (36) and using (38), we obtain
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2 1 2
3 3 4 4 1 2 1 4 4
3 3 1 4 2 1
ˆˆ( ) ( )
ˆ ˆˆ ( )
ˆ ˆ ˆ
a b
c d
r
V k e k e k e k e e e y y a e e y y b
e e y c e e y d e e x x e x
e e x e e x e e x r
α
β γ
α
β γ
= − − − − + − − + + −
+ − − + − − + − − − −
+ − + − − + −
&&&
& &&
& & &
(39)
In view of Eq. (39), the estimated parameters are updated by the following law:
1 2 1 5 1 2 1 4 4 9
2 1 2 6 3 3 10
3 3 7 1 4 11
4 4 8 2 1 12
ˆˆ ( ) , ( )
ˆ ˆ( ) ,
ˆˆ ,
ˆ ˆ,
a
b
c
d r
a e y y k e e x x e x k e
b e y y k e e x k e
c e y k e e x k e
d e y k e r e x k e
α
β
γ
α
β
γ
= − + = − − − +
= + + = +
= − + = − +
= − + = +
&&
& &
&&
& &
(40)
where 5 6 7 8 9 10 11, , , , , ,k k k k k k k and 12k are positive constants.
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30
Substituting (40) into (39), we obtain
2 2 2 2
1 1 2 2 3 3 4 4V k e k e k e k e= − − − −& 2 2
5 6a bk e k e− − 2 2
7 8c dk e k e− − 2 2
9 10k e k eα β− − 2 2
11 12 rk e k eγ− − (41)
which is a negative definite function on 12
.R
Thus, by Lyapunov stability theory [30], it is immediate that the synchronization error
,( 1,2,3,4)ie i = and the parameter estimation error decay to zero exponentially with time.
Hence, we have proved the following result.
Theorem 3. The non-identical Lorenz-Stenflo system (29) and Qi system (30) with unknown
parameters are globally and exponentially hybrid synchronized by the adaptive control law
(33), where the update law for the parameter estimates is given by (40) and
,( 1,2, ,12)ik i = K are positive constants.
5.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 6
10h −
= is
used to solve the chaotic systems (29) and (30) with the adaptive control law (27) and the
parameter update law (40) using MATLAB. We take 4ik = for 1,2, ,12.i = K
For the Lorenz-Stenflo system (29) and Qi system (30), the parameter values are taken as
30, 10, 1, 10, 2.0, 0.7, 1.5, 26.a b c d rα β γ= = = = = = = = (42)
Suppose that the initial values of the parameter estimates are
ˆ ˆ ˆˆ ˆˆ ˆ ˆ(0) 2, (0) 6, (0) 1, (0) 4, (0) 5, (0) 8, (0) 7, (0) 11a b c d rα β γ= = = = = = = =
The initial values of the master system (29) are taken as
1 2 3 4(0) 6, (0) 15, (0) 20, (0) 9.x x x x= = = =
The initial values of the slave system (30) are taken as
1 2 3 4(0) 20, (0) 18, (0) 14, (0) 27.y y y y= = = =
Figure 7 depicts the complete synchronization of the non-identical Lorenz-Stenflo and Qi
systems.
Figure 8 shows that the estimated values of the parameters, viz. ˆ ˆ ˆˆ ˆˆ ˆ, , , , , ,a b c d α β γ and
ˆr converge to the original values of the parameters given in (42).
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31
Figure 7. Complete Synchronization of Lorenz-Stenflo and Qi Systems
Figure 8. Parameter Estimates ˆ ˆ ˆˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )a t b t c t d t t t t r tα β γ
18. International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
32
6. CONCLUSIONS
In this paper, we have derived new results for the hybrid synchronization of identical Lorenz-
Stenflo systems (2001), identical Qi systems (2005) and non-identical Lorenz-Stenflo and Qi
systems with unknown parameters via adaptive control method. The hybrid synchronization
results 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 a
very effective and convenient for achieving hybrid chaos synchronization for the uncertain 4-D
chaotic systems discussed in this paper. Numerical simulations are shown to demonstrate the
effectiveness of the adaptive synchronization schemes derived in this paper for the hybrid chaos
synchronization of identical and non-identical uncertain Lorenz-Stenflo and Qi systems.
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Author
Dr. V. Sundarapandian is a Professor (Systems and Control
Engineering), Research and Development Centre at Vel Tech Dr. RR
& Dr. SR Technical University, Chennai, India. His current research
areas are: Linear and Nonlinear Control Systems, Chaos Theory,
Dynamical Systems and Stability Theory, Soft Computing, Operations
Research, Numerical Analysis and Scientific Computing, Population
Biology, etc. He has published over 175 research articles in
international journals and two text-books with Prentice-Hall of India,
New Delhi, India. He has published over 50 papers in International
Conferences and 100 papers in National Conferences. He is the
Editor-in-Chief of the AIRCC control journals – International Journal
of Instrumentation and Control Systems, International Journal of
Control Theory and Computer Modeling, International Journal of
Information Technology, Control and Automation. He is an Associate
Editor of the journals – International Journal of Information Sciences
and Techniques, International Journal of Control Theory and
Applications, International Journal of Computer Information Systems,
International Journal of Advances in Science and Technology. He has
delivered several Key Note Lectures on Control Systems, Chaos
Theory, Scientific Computing, MATLAB, SCILAB, etc.