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.
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
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 SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
HYBRID SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEMS VIA SLIDING MODE CONTROLijccmsjournal
This paper derives new results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu,
Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems,
the odd states of the two systems are completely synchronized, while their even states are anti-
synchronized. The stability results derived in this paper for the hybrid synchronization of identical
hyperchaotic Liu systems have been proved using Lyapunov stability theory. Since the Lyapunov
exponents are not required for these calculations, the sliding mode control method is very effective and
convenient to achieve anti- synchronization of the identical hyperchaotic Liu systems. Numerical
simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper
for the identical hyperchaotic Liu systems.
SLIDING CONTROLLER DESIGN FOR THE GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL H...ijait
This paper establishes new results for the sliding controller design for the global chaos synchronization of identical hyperchaotic Yujun systems (2010). Hyperchaotic systems are chaotic nonlinear systems having more than one positive Lyapunov exponent. Because of the complex dynamics properties of hyperchaotic system such as high capacity, high security and high efficiency, they are very useful in secure
communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun systems.
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 Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
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 SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
HYBRID SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEMS VIA SLIDING MODE CONTROLijccmsjournal
This paper derives new results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu,
Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems,
the odd states of the two systems are completely synchronized, while their even states are anti-
synchronized. The stability results derived in this paper for the hybrid synchronization of identical
hyperchaotic Liu systems have been proved using Lyapunov stability theory. Since the Lyapunov
exponents are not required for these calculations, the sliding mode control method is very effective and
convenient to achieve anti- synchronization of the identical hyperchaotic Liu systems. Numerical
simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper
for the identical hyperchaotic Liu systems.
SLIDING CONTROLLER DESIGN FOR THE GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL H...ijait
This paper establishes new results for the sliding controller design for the global chaos synchronization of identical hyperchaotic Yujun systems (2010). Hyperchaotic systems are chaotic nonlinear systems having more than one positive Lyapunov exponent. Because of the complex dynamics properties of hyperchaotic system such as high capacity, high security and high efficiency, they are very useful in secure
communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun systems.
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.
On the principle of optimality for linear stochastic dynamic systemijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
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.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Achieve asymptotic stability using Lyapunov's second methodIOSRJM
This paper discusses asymptotic stability for autonomous systems by means of the direct method of Liapunov.Lyapunov stability theory of nonlinear systems is addressed .The paper focuses on the conditions needed in order to guarantee asymptotic stability by Lyapunov'ssecond method in nonlinear dynamic autonomous systems of continuous time and illustrated by examples.
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.
Systems Analysis & Control: Steady State ErrorsJARossiter
In the context of control engineering feedback loops, these slides describe how to find the steady-state error between a target and the system.
Links to more slides at
http://controleducation.group.shef.ac.uk/OER_index.htm
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR 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.
On the principle of optimality for linear stochastic dynamic systemijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
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.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Achieve asymptotic stability using Lyapunov's second methodIOSRJM
This paper discusses asymptotic stability for autonomous systems by means of the direct method of Liapunov.Lyapunov stability theory of nonlinear systems is addressed .The paper focuses on the conditions needed in order to guarantee asymptotic stability by Lyapunov'ssecond method in nonlinear dynamic autonomous systems of continuous time and illustrated by examples.
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.
Systems Analysis & Control: Steady State ErrorsJARossiter
In the context of control engineering feedback loops, these slides describe how to find the steady-state error between a target and the system.
Links to more slides at
http://controleducation.group.shef.ac.uk/OER_index.htm
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR 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.
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.
HYBRID SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEMS VIA SLIDING MODE CONTROLijccmsjournal
This paper derives new results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu, Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while their even states are antisynchronized. The stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Liu systems have been proved using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti- synchronization of the identical hyperchaotic Liu systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Liu systems.
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.
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.
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.
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.
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.
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 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.
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...IJITCA Journal
This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
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.
Modified Projective Synchronization of Chaotic Systems with Noise Disturbance,...IJECEIAES
The synchronization problem of chaotic systems using active modified projective non- linear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
Adaptive Control and Synchronization of 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
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.
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Sliding Mode Control
1. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
DOI : 10.5121/ijctcm.2011.1201 1
HYBRID CHAOS SYNCHRONIZATION OF
HYPERCHAOTIC NEWTON-LEIPNIK SYSTEMS BY
SLIDING MODE 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 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.
KEYWORDS
Hybrid Synchronization, Hyperchaos, Sliding Mode Control, Hyperchaotic Newton-Leipnik System.
1. INTRODUCTION
Chaotic systems are dynamical systems that are highly sensitive to initial conditions. The
sensitive nature of chaotic systems is commonly called as the butterfly effect [1].
Synchronization of chaotic systems is a phenomenon which may occur when two or more chaotic
oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator. Because of
the butterfly effect which causes the exponential divergence of the trajectories of two identical
chaotic systems started with nearly the same initial conditions, synchronizing two chaotic systems
is seemingly a very challenging problem.
A hyperchaotic system is usually characterized as a chaotic system with more than one positive
Lyapunov exponent implying that the dynamics expand in more than one direction giving rise to
“thicker” and “more complex” chaotic dynamics. There has been significant interest in the
research on hyperchaotic systems in the literature. Hyperchaotic systems have important
applications in engineering in areas like secure communication, data encryption, 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 Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
2
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 pioneering work by Pecora and Carroll ([2], 1990), chaos synchronization problem has
been studied extensively and intensively in the literature [2-17]. Chaos theory has been applied to
a variety of fields such as physical systems [3], chemical systems [4], ecological systems [5],
secure communications [6-8], etc.
In the last two decades, various schemes have been successfully applied for chaos
synchronization such as PC method [2], OGY method [9], active control method [10-12],
adaptive control method [13-14], time-delay feedback method [15], backstepping design method
[16], sampled-data feedback method [17], etc.
So far, many types of chaos synchronization phenomenon have been studied in the literature such
as complete synchronization [2], phase synchronization [18], generalized synchronization [19],
anti-synchronization [20-21], projective synchronization [22], generalized projective
synchronization [23], 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 state 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 systems synchronize up to a constant scaling matrix .α It is easy to see
that complete synchronization and anti-synchronization are special cases of the generalized
projective synchronization where the scaling matrix Iα = and ,Iα = − respectively.
In hybrid synchronization of chaotic systems [24-25], one part of the system is completely
synchronized and the other part is anti-synchronized so that complete synchronization (CS) and
anti-synchronization (AS) co-exist in the systems. The coexistence of CS and AS is very useful in
applications such as secure communication, chaotic encryption schemes, etc.
In this paper, we derive new results based on the sliding mode control [26-28] for the hybrid
chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and
Bhattacharya, [29], 2010). In robust control systems, the sliding mode control method is often
adopted due to its inherent advantages of easy realization, fast response and good transient
performance as well as its insensitivity to parameter uncertainties and external disturbances.
This paper has been organized as follows. In Section 2, we describe the problem statement and
our methodology using sliding mode control (SMC). In Section 3, we discuss the global chaos
synchronization of identical hyperchaotic Newton-Leipnik systems. In Section 4, we summarize
the main results obtained in this paper.
2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC
In this section, we describe the problem statement for the global chaos synchronization for
identical chaotic systems and our methodology using sliding mode control (SMC).
Consider the chaotic system described by
( )x Ax f x= +& (1)
where n
x∈R is the state of the system, A is the n n× matrix of the system parameters and
: n n
f →R R is the nonlinear part of the system.
We consider the system (1) as the master or drive system.
As the slave or response system, we consider the following chaotic system described by the
dynamics
3. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
3
( )y Ay f y u= + +& (2)
where n
y ∈R is the state of the system and m
u ∈R is the controller to be designed.
We define the hybrid synchronization error as
, if is odd
, if is even
i i
i
i i
y x i
e
y x i
−
=
+
(3)
Then the error dynamics can be obtained in the form
( , ) ,e Ae x y uη= + +& (4)
The objective of the global chaos synchronization problem is to find a controller u such that
lim ( ) 0
t
e t
→∞
= for all (0) .n
e ∈R (5)
To solve this problem, we first define the control u as
( , )u x y Bvη= − + (6)
where B is a constant gain vector selected such that ( , )A B is controllable.
Substituting (6) into (4), the error dynamics simplifies to
e Ae Bv= +& (7)
which is a linear time-invariant control system with single input .v
Thus, the original global chaos synchronization problem can be replaced by an equivalent
problem of stabilizing the zero solution 0e = of the system (7) by a suitable choice of the sliding
mode control. In the sliding mode control, we define the variable
1 1 2 2( ) n ns e Ce c e c e c e= = + + +L (8)
where [ ]1 2 nC c c c= L is a constant vector to be determined.
In the sliding mode control, we constrain the motion of the system (7) to the sliding manifold
defined by
{ }| ( ) 0n
S x s e= ∈ =R
which is required to be invariant under the flow of the error dynamics (7).
When in sliding manifold ,S the system (7) satisfies the following conditions:
( ) 0s e = (9)
which is the defining equation for the manifold S and
( ) 0s e =& (10)
which is the necessary condition for the state trajectory ( )e t of (7) to stay on the sliding manifold
.S
Using (7) and (8), the equation (10) can be rewritten as
[ ]( ) 0s e C Ae Bv= + =& (11)
4. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
4
Solving (11) for ,v we obtain the equivalent control law
1
eq ( ) ( ) ( )v t CB CA e t−
= − (12)
where C is chosen such that 0.CB ≠
Substituting (12) into the error dynamics (7), we obtain the closed-loop dynamics as
1
( )e I B CB C Ae−
= − & (13)
The row vector C is selected such that the system matrix of the controlled dynamics
1
( )I B CB C A−
− is Hurwitz, i.e. it has all eigenvalues with negative real parts. Then the
controlled system (13) is globally asymptotically stable.
To design the sliding mode controller for (7), we apply the constant plus proportional rate
reaching law
sgn( )s q s k s= − −& (14)
where sgn( )⋅ denotes the sign function and the gains 0,q > 0k > are determined such that the
sliding condition is satisfied and sliding motion will occur.
From equations (11) and (14), we can obtain the control ( )v t as
[ ]1
( ) ( ) ( ) sgn( )v t CB C kI A e q s−
= − + + (15)
which yields
[ ]
[ ]
1
1
( ) ( ) , if ( ) 0
( )
( ) ( ) , if ( ) 0
CB C kI A e q s e
v t
CB C kI A e q s e
−
−
− + + >
=
− + − <
(16)
Theorem 1. The master system (1) and the slave system (2) are globally and asymptotically
hybrid synchronized for all initial conditions (0), (0) n
x y R∈ by the feedback control law
( ) ( , ) ( )u t x y Bv tη= − + (17)
where ( )v t is defined by (15) and B is a column vector such that ( , )A B is controllable. Also, the
sliding mode gains ,k q are positive.
Proof. First, we note that substituting (17) and (15) into the error dynamics (4), we obtain the
closed-loop error dynamics as
[ ]1
( ) ( ) sgn( )e Ae B CB C kI A e q s−
= − + +& (18)
To prove that the error dynamics (18) is globally asymptotically stable, we consider the candidate
Lyapunov function defined by the equation
21
( ) ( )
2
V e s e= (19)
which is a positive definite function on .n
R
Differentiating V along the trajectories of (18) or the equivalent dynamics (14), we get
2
( ) ( ) ( ) sgn( )V e s e s e ks q s s= = − −& & (20)
5. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
5
which is a negative definite function on .n
R
This calculation shows that V is a globally defined, positive definite, Lyapunov function for the
error dynamics (18), which has a globally defined, negative definite time derivative .V&
Thus, by Lyapunov stability theory [22], it is immediate that the error dynamics (18) is globally
asymptotically stable for all initial conditions (0) .n
e ∈R
This means that for all initial conditions (0) ,n
e R∈ we have
lim ( ) 0
t
e t
→∞
=
This shows that the hybrid synchronization error vector ( )e t decays to zero asymptotically with
time.Hence, it follows that the master system (1) and the slave system (2) are globally and
asymptotically hybrid synchronized for all initial conditions (0), (0) .n
x y ∈R
This completes the proof.
3. HYBRID CHAOS SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC
NEWTON-LEIPNIK SYSTEMS USING SLIDING MODE CONTROL
3.1 Theoretical Results
In this section, we apply the sliding mode control results derived in Section 2 for the hybrid chaos
synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya,
[29], 2010). In hybrid synchronization of master and slave systems, the first and third states of the
two systems are completely synchronized, while the second and fourth states of the two systems
are anti-synchronized.
Thus, the master system is described by the hyperchaotic Newton-Leipnik dynamics
1 1 2 2 3 4
2 1 2 1 3
3 3 1 2
4 1 3 4
10
0.4 5
5
x ax x x x x
x x x x x
x bx x x
x cx x dx
= − + + +
= − − +
= −
= − +
&
&
&
&
(21)
where 1 2 3 4, , ,x xx x are state variables and , , ,a b c d are positive, constant parameters of the
system.The slave system is described by the controlled hyperchaotic Newton-Leipnik dynamics
1 1 2 2 3 4 1
2 1 2 1 3 2
3 3 1 2 3
4 1 3 4 4
10
0.4 5
5
y ay y y y y u
y y y y y u
y by y y u
y cy y dy u
= − + + + +
= − − + +
= − +
= − + +
&
&
&
&
(22)
where 1 2 3 4, , ,y y y y are state variables and 1 2 3 4, , ,u u u u are the controllers to be designed.
The systems (21) and (22) are hyperchaotic when
0.4, 0.175, 0.8a b c= = = and 0.01.d =
Figure 1 illustrates the state portrait of the hyperchaotic Newton-Leipnik system (21).
The chaos synchronization error is defined by
6. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
6
1 1 1
2 2 2
3 3 3
4 4 4
e
e
e
e
y x
y x
y x
y x
= −
= +
= −
= +
(23)
Figure 1. State Orbits of the Hyperchaotic Newton-Leipnik System
The error dynamics is easily obtained as
1 1 2 4 2 4 2 3 2 3 1
2 1 2 1 1 3 1 3 2
3 3 1 2 1 2 3
4 4 1 3 1 3 4
2 2 10( )
0.4 2 5( )
5( )
( )
e ae e e x x y y x x u
e e e x y y x x u
e be y y x x u
e de c y y x x u
= − + + − − + − +
= − − − + − +
= − − +
= − − +
&
&
&
&
(24)
We write the error dynamics (24) in the matrix notation as
( , )e Ae x y uη= + +& (25)
where
1 0 1
1 0.4 0 0
,
0 0 0
0 0 0
a
A
b
d
−
− − =
2 4 2 3 2 3
1 1 3 1 3
1 2 1 2
1 3 1 3
2 2 10( )
2 5( )
( , )
5( )
( )
x x y y x x
x y y x x
x y
y y x x
c y y x x
η
− − + −
− + + =
− −
− +
and
1
2
3
4
.
u
u
u
u
u
=
(26)
7. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
7
The sliding mode controller design is carried out as detailed in Section 2.
First, we set u as
( , )u x y Bvη= − + (27)
where B is chosen such that ( , )A B is controllable.
We take B as
1
1
.
1
1
B
=
(28)
In the hyperchaotic case, the parameter values are
0.4, 0.175, 0.8a b c= = = and 0.01.d =
The sliding mode variable is selected as
[ ] 1 2 3 40.4 4 8 0.1 0.4 4 8 0.1s Ce e e e e e= = = + + + (29)
which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 5k = and 0.1.q =
We note that a large value of k can cause chattering and an appropriate value of q is chosen to
speed up the time taken to reach the sliding manifold as well as to reduce the system chattering.
From Eq. (15), we can obtain ( )v t as
1 2 3 4( ) 0.1728 1.5040 3.3120 0.0721 0.0080 sgn( )v t e e e e s= − − − − (30)
Thus, the required sliding mode controller is obtained as
( , )u x y Bvη= − + (31)
where ( , ),x y Bη and ( )v t are defined as in the equations (26), (28) and (30).
By Theorem 1, we obtain the following result.
Theorem 2. The identical hyperchaotic Newton-Leipnik systems (21) and (22) are globally and
asymptotically hybrid synchronized for all initial conditions with the sliding mode controller
u defined by (31).
3.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 6
10h −
= is
used to solve the hyperchaotic Newton-Leipnik systems (21) and (22) with the sliding mode
controller u given by (31) using MATLAB.
The sliding mode gains are chosen as 5k = and 0.1.q =
The initial values of the master system (21) are taken as (0) (6,5,14,7)x = and the initial values
of the slave system (22) are taken as (0) (3,12,18,16).y =
8. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
8
Figure 2 illustrates the hybrid synchronization of the identical hyperchaotic Newton-Leipnik
systems (21) and (22).
Figure 2. Hybrid Synchronization of Identical Hyperchaotic Newton-Leipnik Systems
4. CONCLUSIONS
In this paper, we have deployed sliding mode control (SMC) to achieve hybrid chaos
synchronization for the identical hyperchaotic Newton-Leipnik systems (2010). Our hybrid
synchronization results for the identical hyperchaotic Newton-Leipnik systems have been proved
using Lyapunov stability theory. Since the Lyapunov exponents are not required for these
calculations, the sliding mode control method is very effective and convenient to achieve hybrid
chaos synchronization for the identical hyperchaotic Newton-Leipnik systems. Numerical
simulations are also shown to illustrate the effectiveness of the hybrid synchronization results
derived in this paper using the sliding mode control.
9. International Journal of Control Theory and Computer Modelling (IJCTCM) Vol.1, No.2, September 2011
9
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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 170 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 90 papers in
National Conferences. He has delivered several Key
Note Lectures on Control Systems, Chaos Theory,
Scientific Computing, MATLAB, SCILAB, etc.