This document summarizes research on adaptive stabilization and synchronization of hyperchaotic systems. It specifically focuses on applying these techniques to the hyperchaotic Qi system, a 4-dimensional hyperchaotic system. First, adaptive control laws are designed to stabilize the hyperchaotic Qi system to equilibrium based on adaptive control theory and Lyapunov stability theory. Then, adaptive control laws are derived to achieve global synchronization between identical hyperchaotic Qi systems with unknown parameters. Numerical simulations demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM ijait
The hyperchaotic Newton-Leipnik system (Ghosh and Bhattacharya, 2010) is one of the recently discovered four-dimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Newton-Leipnik system with unknown parameters. First, adaptive
control laws are designed to stabilize the hyperchaotic Newton-Leipnik system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Newton-Leipnik
systems with unknown parameters. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
Adaptive Control and Synchronization of Hyperchaotic 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
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 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 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.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM ijait
The hyperchaotic Newton-Leipnik system (Ghosh and Bhattacharya, 2010) is one of the recently discovered four-dimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Newton-Leipnik system with unknown parameters. First, adaptive
control laws are designed to stabilize the hyperchaotic Newton-Leipnik system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Newton-Leipnik
systems with unknown parameters. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
Adaptive Control and Synchronization of Hyperchaotic 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
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 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 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.
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.
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.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
ACTIVE CONTROLLER DESIGN FOR THE 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.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
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 RÖSSLER PROTOTYPE-4 SYSTEMijait
This paper investigates the adaptive control and synchronization of Rössler prototype-4 system with unknown parameters. The Rössler prototype-4 system is a classical three-dimensional chaotic system studied by O.E. Rössler (1979). First, adaptive control laws are designed to stabilize the Rössler prototype-4 system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical Rössler prototype-4 systems with unknown parameters. Numerical simulations are shown to
validate and illustrate the effectiveness of the proposed adaptive control and synchronization schemes for the Rössler prototype-4 system.
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.
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 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.
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.
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.
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...IJITCA Journal
This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
Margin Parameter Variation for an Adaptive Observer to a Class of SystemsCSCJournals
In this paper, adaptive observer robustness is studies. The adaptive observer is proposed to a nonlinear system linearized by output injected for variable structure systems. Despites, the adaptive observer is suggesting to supply the parameter variation but it isn't usually verified. Unfortunately, the parameter uncertainty impact state convergence. For that a margin parameter variation is determined which the convergence of the state is guaranty. Simulation results show that the adaptive observer is robust only in the definite parameter margin variation.
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.
Simultaneous State and Actuator Fault Estimation With Fuzzy Descriptor PMID a...Waqas Tariq
In this paper, Takagi-Sugeno (T-S) fuzzy descriptor proportional multiple-integral derivative (PMID) and Proportional-Derivative (PD) observer methods that can estimate the system states and actuator faults simultaneously are proposed. T-S fuzzy model is obtained by linearsing satellite/spacecraft attitude dynamics at suitable operating points. For fault estimation, actuator fault is introduced as state vector to develop augmented descriptor system and robust fuzzy PMID and PD observers are developed. Stability analysis is performed using Lyapunov direct method. The convergence conditions of state estimation error are formulated in the form of LMI (linear matrix inequality). Derivative gain, obtained using singular value decomposition of descriptor state matrix (E), gives more design degrees of freedom together with proportional and integral gains obtained from LMI. Simulation study is performed for our proposed methods.
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.
In the present work, Lyapunov stability theory, nonlinear adaptive control law and the parameter update
law were utilized to derive the state of two new chaotic reversal systems after being synchronized by the
function projective method. Using this technique allows for a scaling function instead of a constant thereby
giving a better method in applications in secure communication. Numerical simulations are presented to
demonstrate the effective nature of the proposed scheme of synchronization for the new chaotic reversal
system.
Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Fac...ijrap
we have obtained the analytical solution of Schrödinger wave equation with Mie – type potential
using factorization method. We have also obtained energy eigenvalues of our potential and the
corresponding wave function using an ansatz and then compare the result to standard Laguerre’s
differential equation. Under special cases our potential model reduces two well known potentials such as
Coulomb and the Kratzer Feus potentials.
Adaptive Stabilization and Synchronization of Hyperchaotic QI SystemCSEIJJournal
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of four-
dimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization
of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to
stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control
theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations
are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization
schemes.
Adaptive Controller and Synchronizer Design for Hyperchaotic Zhou System with...Zac Darcy
In this paper, we establish new results for the adaptive controller and synchronizer design for the
hyperchaotic Zhou system (2009), when the parameters of the system are unknown. Using adaptive control theory and Lyapunov stability theory, we first design an adaptive controller to stabilize the hyperchaotic Zhou system to its unstable equilibrium at the origin. Next, using adaptive control theory and Lyapunov stability theory, we design an adaptive controller to achieve global chaos synchronization
of the identical hyperchaotic Zhou systems with unknown parameters. Simulations have been provided for adaptive controller and synchronizer designs to validate and illustrate the effectiveness of the schemes.
GLOBAL CHAOS SYNCHRONIZATION OF 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.
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.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
ACTIVE CONTROLLER DESIGN FOR THE 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.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
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 RÖSSLER PROTOTYPE-4 SYSTEMijait
This paper investigates the adaptive control and synchronization of Rössler prototype-4 system with unknown parameters. The Rössler prototype-4 system is a classical three-dimensional chaotic system studied by O.E. Rössler (1979). First, adaptive control laws are designed to stabilize the Rössler prototype-4 system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical Rössler prototype-4 systems with unknown parameters. Numerical simulations are shown to
validate and illustrate the effectiveness of the proposed adaptive control and synchronization schemes for the Rössler prototype-4 system.
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.
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 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.
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.
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.
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...IJITCA Journal
This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
Margin Parameter Variation for an Adaptive Observer to a Class of SystemsCSCJournals
In this paper, adaptive observer robustness is studies. The adaptive observer is proposed to a nonlinear system linearized by output injected for variable structure systems. Despites, the adaptive observer is suggesting to supply the parameter variation but it isn't usually verified. Unfortunately, the parameter uncertainty impact state convergence. For that a margin parameter variation is determined which the convergence of the state is guaranty. Simulation results show that the adaptive observer is robust only in the definite parameter margin variation.
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.
Simultaneous State and Actuator Fault Estimation With Fuzzy Descriptor PMID a...Waqas Tariq
In this paper, Takagi-Sugeno (T-S) fuzzy descriptor proportional multiple-integral derivative (PMID) and Proportional-Derivative (PD) observer methods that can estimate the system states and actuator faults simultaneously are proposed. T-S fuzzy model is obtained by linearsing satellite/spacecraft attitude dynamics at suitable operating points. For fault estimation, actuator fault is introduced as state vector to develop augmented descriptor system and robust fuzzy PMID and PD observers are developed. Stability analysis is performed using Lyapunov direct method. The convergence conditions of state estimation error are formulated in the form of LMI (linear matrix inequality). Derivative gain, obtained using singular value decomposition of descriptor state matrix (E), gives more design degrees of freedom together with proportional and integral gains obtained from LMI. Simulation study is performed for our proposed methods.
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.
In the present work, Lyapunov stability theory, nonlinear adaptive control law and the parameter update
law were utilized to derive the state of two new chaotic reversal systems after being synchronized by the
function projective method. Using this technique allows for a scaling function instead of a constant thereby
giving a better method in applications in secure communication. Numerical simulations are presented to
demonstrate the effective nature of the proposed scheme of synchronization for the new chaotic reversal
system.
Analytical Solution Of Schrödinger Equation With Mie–Type Potential Using Fac...ijrap
we have obtained the analytical solution of Schrödinger wave equation with Mie – type potential
using factorization method. We have also obtained energy eigenvalues of our potential and the
corresponding wave function using an ansatz and then compare the result to standard Laguerre’s
differential equation. Under special cases our potential model reduces two well known potentials such as
Coulomb and the Kratzer Feus potentials.
Adaptive Stabilization and Synchronization of Hyperchaotic QI SystemCSEIJJournal
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of four-
dimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization
of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to
stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control
theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations
are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization
schemes.
Adaptive Controller and Synchronizer Design for Hyperchaotic Zhou System with...Zac Darcy
In this paper, we establish new results for the adaptive controller and synchronizer design for the
hyperchaotic Zhou system (2009), when the parameters of the system are unknown. Using adaptive control theory and Lyapunov stability theory, we first design an adaptive controller to stabilize the hyperchaotic Zhou system to its unstable equilibrium at the origin. Next, using adaptive control theory and Lyapunov stability theory, we design an adaptive controller to achieve global chaos synchronization
of the identical hyperchaotic Zhou systems with unknown parameters. Simulations have been provided for adaptive controller and synchronizer designs to validate and illustrate the effectiveness of the schemes.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
GLOBAL CHAOS SYNCHRONIZATION OF 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 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.
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.
International Journal of Computer Science, Engineering and Information Techno...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
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.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations are presented to demonstrate the effectiveness of the synchronization schemes.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic systems. Generalized back-stepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized back-stepping method expands this class. 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.
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.
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.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
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ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM
1. Computer Science & Engineering: An International Journal (CSEIJ), Vol.1, No.2, June 2011
DOI : 10.5121/cseij.2011.1202 14
ADAPTIVE STABILIZATION AND
SYNCHRONIZATION OF HYPERCHAOTIC QI
SYSTEM
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
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of four-
dimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization
of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to
stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control
theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations
are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization
schemes.
KEYWORDS
Adaptive Control, Stabilization, Chaos Synchronization, Hyperchaos, Hyperchaotic Qi 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]. Since chaos
phenomenon in weather models was first observed by Lorenz in 1961, a large number of chaos
phenomena and chaos behaviour have been discovered in physical, social, economical,
biological and electrical systems.
The control of chaotic systems is to design state feedback control laws that stabilizes the chaotic
systems around the unstable equilibrium points. Active control technique is used when the
system parameters are known and adaptive control technique is used when the system
parameters are unknown [2-4].
Synchronization of chaotic systems is a phenomenon that 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 in the chaos literature [5-16].
In 1990, Pecora and Carroll [5] introduced a method 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 [6], chemical systems [7], ecological systems [8], secure
communications [9-10], etc.
2. Computer Science & Engineering: An International Journal (CSEIJ), Vol.1, No.2, June 2011
15
In most of the chaos synchronization approaches, the master-slave or drive-response formalism
has been 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 synchronization is to use
the 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 [5], a variety of impressive approaches have been
proposed for the synchronization of chaotic systems such as the sampled-data feedback
synchronization method [11], OGY method [12], time-delay feedback method [13],
backstepping method [14], adaptive design method [15], sliding mode control method [16], etc.
This paper is organized as follows. In Section 2, we derive results for the adaptive stabilization
of hyperchaotic Qi system (Chen, Yang, Qi and Yuan, [17], 2007) with unknown parameters. In
Section 3, we derive results for the adaptive synchronization of hyperchaotic Qi systems with
unknown parameters. In Section 4, we summarize the main results obtained in this paper.
2. ADAPTIVE STABILIZATION OF HYPERCHAOTIC QI SYSTEM
2.1 Theoretical Results
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, [17], 2007) is one of the important
models of four-dimensional hyperchaotic systems. It is a novel hyperchaotic system with only
one equilibrium at the origin. The hyperchaotic Qi dynamics is described by
1 2 1 2 3
2 1 1 3 2 4
3 1 2 3
4 2
( )x a x x x x
x c x d x x x x
x x x b x
x f x
ε= − +
= − + +
= −
= −
&
&
&
&
(1)
where ( 1,2,3,4)ix i = are the state variables and , , , , ,a b c d fε are positive constants.
Figure 1. State Orbits of the Hyperchaotic Qi System
3. Computer Science & Engineering: An International Journal (CSEIJ), Vol.1, No.2, June 2011
16
The system (1) is hyperchaotic when the parameter values are taken as
35, 4.9, 25, 5, 35a b c d ε= = = = = and 22.f = (2)
The hyperchaotic state portrait of the system (1) is described in Figure 1.
When the parameter values are taken as in (2), the system (1) is hyperchaotic and the system
linearization matrix at the equilibrium point 0 (0,0,0,0)E = is given by
35 35 0 0
25 1 0 1
0 0 4.9 0
0 22 0 0
A
−
=
−
−
which has the eigenvalues
1 2 30.8989, 16.6257, 4.9λ λ λ= = = − and 4 51.5245λ = −
Since 1 2,λ λ are eigenvalues with positive real part, it is immediate from Lyapunov stability
theory [18] that the system (1) is unstable at the equilibrium point 0 (0,0,0,0).E =
In this section, we design adaptive control law for globally stabilizing the hyperchaotic system
(1) when the parameter values are unknown.
Thus, we consider the controlled hyperchaotic Qi system as follows.
1 2 1 2 3 1
2 1 1 3 2 4 2
3 1 2 3 3
4 2 4
( )x a x x x x u
x c x d x x x x u
x x x b x u
x f x u
ε= − + +
= − + + +
= − +
= − +
&
&
&
&
(3)
where 1 2 3, ,u u u and 4u are feedback controllers to be designed using the states and estimates of
the unknown parameters of the system.
In order to ensure that the controlled system (3) globally converges to the origin asymptotically,
we consider the following adaptive control functions
1 2 1 2 3 1 1
2 1 1 3 2 4 2 2
3 1 2 3 3 3
4 2 4 4
ˆˆ ( )
ˆˆ
ˆ
ˆ
u a x x x x k x
u c x d x x x x k x
u x x b x k x
u f x k x
ε= − − − −
= − + − − −
= − + −
= − −
(4)
where ˆ ˆ ˆˆ ˆ, , , ,a b c d ε and ˆf are estimates of the parameters , , , ,a b c d ε and ,f respectively, and
,( 1,2,3,4)ik i = are positive constants.
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Substituting the control law (4) into the hyperchaotic Qi dynamics (1), we obtain
1 2 1 2 3 1 1
2 1 1 3 2 2
3 3 3 3
4 2 4 4
ˆˆ( ) ( ) ( )
ˆˆ( ) ( )
ˆ( )
ˆ( )
x a a x x x x k x
x c c x d d x x k x
x b b x k x
x f f x k x
ε ε= − − + − −
= − − − −
= − − −
= − − −
&
&
&
&
(5)
Let us now define the parameter errors as
ˆˆ ˆ, ,
ˆ ˆˆ, ,
a b c
d f
e a a e b b e c c
e d d e e f fε ε ε
= − = − = −
= − = − = −
(6)
Using (6), the closed-loop dynamics (5) can be written compactly as
1 2 1 2 3 1 1
2 1 1 3 2 2
3 3 3 3
4 2 4 4
( )a
c d
b
f
x e x x e x x k x
x e x e x x k x
x e x k x
x e x k x
ε= − + −
= − −
= − −
= − −
&
&
&
&
(7)
For the derivation of the update law for adjusting the parameter estimates ˆ ˆ ˆˆˆ ˆ, , , , , ,a b c d fε the
Lyapunov approach is used.
Consider the quadratic Lyapunov function
( )2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c d fV x x x x e e e e e eε= + + + + + + + + + (8)
which is a positive definite function on 10
.R
Note also that
ˆˆ ˆ, ,
ˆ ˆˆ, ,
a b c
d f
e a e b e c
e d e e fε ε
= − = − = −
= − = − = −
&& && & &
& &&& & &
(9)
Differentiating V along the trajectories of (7) and using (9), we obtain
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 3
1 2 1 2 3 1 2 3 2 4
ˆˆ( )
ˆ ˆˆˆ
a b
c d f
V k x k x k x k x e x x x a e x b
e x x c e x x x d e x x x e x x fε ε
= − − − − + − − + − −
+ − + − − + − + − −
&&&
& &&&
(10)
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In view of Eq. (10), the estimated parameters are updated by the following law:
1 2 1 5
2
3 6
1 2 7
1 2 3 8
1 2 3 9
2 4 10
ˆ ( )
ˆ
ˆ
ˆ
ˆ
ˆ
a
b
c
d
f
a x x x k e
b x k e
c x x k e
d x x x k e
x x x k e
f x x k e
εε
= − +
= − +
= +
= − +
= +
= − +
&
&
&
&
&
&
(11)
where ,( 5, ,10)ik i = K are positive constants.
Substituting (11) into (10), we get
2 2 2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 9 10a b c d fV k x k x k x k x k e k e k e k e k e k eε= − − − − − − − − − −& (12)
which is a negative definite function on 10
.R
Thus, by Lyapunov stability theory [18], we obtain the following result.
Theorem 1. The hyperchaotic Qi system (1) with unknown parameters is globally and
exponentially stabilized for all initial conditions 4
(0)x R∈ by the adaptive control law (4),
where the update law for the parameters is given by (11) and , ( 1, ,10)ik i = K are positive
constants.
2.2 Numerical Results
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the
hyperchaotic system (1) with the adaptive control law (4) and the parameter update law (11).
The parameters of the hyperchaotic Qi system (1) are selected as
35, 4.9, 25, 5, 35a b c d ε= = = = = and 22.f =
For the adaptive and update laws, we take 2, ( 1,2, ,10).ik i= = K
Suppose that the initial values of the estimated parameters are
ˆ ˆ ˆˆ ˆ(0) 4, (0) 5, (0) 2, (0) 7, (0) 5a b c d ε= = = = = and ˆ(0) 8.f =
The initial values of the hyperchaotic Qi system (1) are taken as (0) (10,4,8,12).x =
When the adaptive control law (4) and the parameter update law (11) are used, the controlled
hyperchaotic Qi system converges to the equilibrium 0 (0,0,0,0)E = exponentially as shown in
Figure 2. The parameter estimates are shown in Figure 3.
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Figure 2. Time Responses of the Controlled Hyperchaotic Qi System
Figure 3. Parameter Estimates ˆ ˆ ˆˆˆ ˆ( ), ( ), ( ), ( ), ( ), ( )a t b t c t d t t f tε
7. Computer Science & Engineering: An International Journal (CSEIJ), Vol.1, No.2, June 2011
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3. ADAPTIVE SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC QI
SYSTEMS
3.1 Theoretical Results
In this section, we discuss the adaptive synchronization of identical hyperchaotic Qi systems
(Chen, Yang, Qi and Yuan, [17], 2007) with unknown parameters.
As the master system, we consider the hyperchaotic Qi dynamics described by
1 2 1 2 3
2 1 1 3 2 4
3 1 2 3
4 2
( )x a x x x x
x c x d x x x x
x x x b x
x f x
ε= − +
= − + +
= −
= −
&
&
&
&
(13)
where , ( 1,2,3,4)ix i = are the state variables and , , , , ,a b c d fε are unknown system
parameters.
The system (13) is hyperchaotic when the parameter values are taken as
35, 4.9, 25, 5, 35a b c d ε= = = = = and 22.f =
As the slave system, we consider the controlled hyperchaotic Qi dynamics described by
1 2 1 2 3 1
2 1 1 3 2 4 2
3 1 2 3 3
4 2 4
( )y a y y y y u
y c y d y y y y u
y y y b y u
y f y u
ε= − + +
= − + + +
= − +
= − +
&
&
&
&
(14)
where , ( 1,2,3,4)iy i = are the state variables and , ( 1,2,3,4)iu i = are the nonlinear controllers
to be designed.
The synchronization error is defined by
, ( 1,2,3,4)i i ie y x i= − = (15)
Then the error dynamics is obtained as
1 2 1 2 3 2 3 1
2 1 1 3 1 3 2 4 2
3 1 2 1 2 3 3
4 2 4
( ) ( )
( )
e a e e y y x x u
e c e d y y x x e e u
e y y x x be u
e f e u
ε= − + − +
= − − + + +
= − − +
= − +
&
&
&
&
(16)
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Let us now define the adaptive control functions 1 2 3 4( ), ( ), ( ), ( )u t u t u t u t as
1 2 1 2 3 2 3 1 1
2 1 1 3 1 3 2 4 2 2
3 1 2 1 2 3 3 3
4 2 4 4
ˆˆ ( ) ( )
ˆˆ ( )
ˆ
ˆ
u a e e y y x x k e
u c e d y y x x e e k e
u y y x x b e k e
u f e k e
ε= − − − − −
= − + − − − −
= − + + −
= − −
(17)
where ˆ ˆ ˆˆ ˆ, , , ,a b c d ε and ˆf are estimates of the parameters , , , ,a b c d ε and f respectively, and
,( 1,2,3,4)ik i = are positive constants.
Substituting the control law (17) into (16), we obtain the error dynamics as
1 2 1 2 3 2 3 1 1
2 1 1 3 1 3 2 2
3 3 3 3
4 2 4 4
ˆˆ( ) ( ) ( ) ( )
ˆˆ( ) ( )( )
ˆ( )
ˆ( )
e a a e e y y x x k e
e c c e d d y y x x k e
e b b e k e
e f f e k e
ε ε= − − + − − −
= − − − − −
= − − −
= − − −
&
&
&
&
(18)
Let us now define the parameter errors as
ˆ ˆ ˆˆˆ ˆ, , , , ,a b c d fe a a e b b e c c e d d e e f fε ε ε= − = − = − = − = − = − (19)
Substituting (19) into (18), the error dynamics simplifies to
1 2 1 2 3 2 3 1 1
2 1 1 3 1 3 2 2
3 3 3 3
4 2 4 4
( ) ( )
( )
a
c d
b
f
e e e e e y y x x k e
e e e e y y x x k e
e e e k e
e e e k e
ε= − + − −
= − − −
= − −
= − −
&
&
&
&
(20)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov
approach is used.
Consider the quadratic Lyapunov function
( )2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c d fV e e e e e e e e e eε= + + + + + + + + + (21)
which is a positive definite function on 10
.R
Note also that
ˆˆ ˆ, ,
ˆ ˆˆ, ,
a b c
d f
e a e b e c
e d e e fε ε
= − = − = −
= − = − = −
&& && & &
& &&& & &
(22)
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22
Differentiating V along the trajectories of (20) and using (22), we obtain
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 3 1 2
2 1 3 1 3 1 2 3 2 3 2 4
ˆˆ ˆ( )
ˆ ˆˆ( ) ( )
a b c
d f
V k e k e k e k e e e e e a e e b e e e c
e e y y x x d e e y y x x e e e fε ε
= − − − − + − − + − − + −
+ − − − + − − + − −
&& &&
& &&
(23)
In view of Eq. (23), the estimated parameters are updated by the following law:
1 2 1 5
2
3 6
1 2 7
2 1 3 1 3 8
1 2 3 2 3 9
2 4 10
ˆ ( )
ˆ
ˆ
ˆ ( )
ˆ ( )
ˆ
a
b
c
d
f
a e e e k e
b e k e
c e e k e
d e y y x x k e
e y y x x k e
f e e k e
εε
= − +
= − +
= +
= − − +
= − +
= − +
&
&
&
&
&
&
(24)
where ,( 5, ,10)ik i = K are positive constants.
Substituting (24) into (23), we get
2 2 2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 9 10a b c d fV k e k e k e k e k e k e k e k e k e k eε= − − − − − − − − − −& (25)
which is a negative definite function on 10
.R
Thus, by Lyapunov stability theory [18], it is immediate that the synchronization error and the
parameter error decay to zero exponentially with time for all initial conditions.
Hence, we have proved the following result.
Theorem 2. The identical hyperchaotic Qi systems (13) and (14) with unknown parameters are
globally and exponentially synchronized for all initial conditions by the adaptive control law
(17), where the update law for parameters is given by (24) and ,( 1, ,10)ik i = K are positive
constants.
3.2 Numerical Results
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the two
systems of differential equations (13) and (14) with the adaptive control law (17) and the
parameter update law (24).
For the adaptive synchronization of the hyperchaotic Qi systems with parameter values
35, 4.9, 25, 5, 35a b c d ε= = = = = and 22,f =
we apply the adaptive control law (17) and the parameter update law (24).
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23
We take the positive constants , ( 1, ,10)ik i = K as 2ik = for 1,2, ,10.i = K
Suppose that the initial values of the estimated parameters are
ˆ ˆ ˆˆ ˆ(0) 5, (0) 4, (0) 3, (0) 6, (0) 4a b c d ε= = = = = and ˆ(0) 2.f =
We take the initial values of the master system (13) as (0) (5,24,18,20).x =
We take the initial values of the slave system (14) as (0) (17,20,8,12).y =
Figure 4 shows the adaptive chaos synchronization of the identical hyperchaotic Qi systems.
Figure 5 shows that the estimated values of the parameters ˆ ˆ ˆˆˆ ˆ, , , , ,a b c d fε converge to the system
parameters 35, 4.9, 25, 5, 35a b c d ε= = = = = and 22.f =
Figure 4. Adaptive Synchronization of the Identical Hyperchaotic Qi Systems
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24
Figure 5. Parameter Estimates ˆ ˆ ˆˆˆ ˆ( ), ( ), ( ), ( ), ( ), ( )a t b t c t d t t f tε
4. CONCLUSIONS
In this paper, we applied adaptive control theory for the stabilization and synchronization of the
hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) with unknown system parameters.
First, we designed adaptive control laws to stabilize the hyperchaotic Qi system to its
equilibrium point at the origin based on the adaptive control theory and Lyapunov stability
theory. Then we derived adaptive synchronization scheme and update law for the estimation of
system parameters for identical hyperchaotic Qi systems with unknown parameters. Our
synchronization schemes were established using Lyapunov stability theory. Since the Lyapunov
exponents are not required for these calculations, the proposed adaptive control method is very
effective and convenient to achieve chaos control and synchronization of the hyperchaotic Qi
system. Numerical simulations are shown to demonstrate the effectiveness of the proposed
adaptive stabilization and synchronization schemes.
REFERENCES
[1] Alligood, K.T., Sauer, T. & Yorke, J.A. (1997) Chaos: An Introduction to Dynamical Systems,
Springer, New York.
[2] Ge, S.S., Wang, C. & Lee, T.H. (2000) “Adaptive backstepping control of a class of chaotic
systems,” Internat. J. Bifur. Chaos, Vol. 10, pp 1149-1156.
[3] Wang, X., Tian, L. & Yu, L. (2006) “Adaptive control and slow manifold analysis of a new
chaotic system,” Internat. J. Nonlinear Science, Vol. 21, pp 43-49.
12. Computer Science & Engineering: An International Journal (CSEIJ), Vol.1, No.2, June 2011
25
[4] Sun, M., Tian, L., Jiang, S. & Xun, J. (2007) “Feedback control and adaptive control of the
energy resource chaotic system,” Chaos, Solitons & Fractals, Vol. 32, pp 168-180.
[5] Pecora, L.M. & Carroll, T.L. (1990) “Synchronization in chaotic systems”, Phys. Rev. Lett., Vol.
64, pp 821-824.
[6] Lakshmanan, M. & Murali, K. (1996) Nonlinear Oscillators: Controlling and Synchronization,
World Scientific, Singapore.
[7] Han, S.K., Kerrer, C. & Kuramoto, Y. (1995) “Dephasing and bursting in coupled neural
oscillators”, Phys. Rev. Lett., Vol. 75, pp 3190-3193.
[8] Blasius, B., Huppert, A. & Stone, L. (1999) “Complex dynamics and phase synchronization in
spatially extended ecological system”, Nature, Vol. 399, pp 354-359.
[9] Feki, M. (2003) “An adaptive chaos synchronization scheme applied to secure communication”,
Chaos, Solitons and Fractals, Vol. 18, pp 141-148.
[10] Murali, K. & Lakshmanan, M. (1998) “Secure communication using a compound signal from
generalized synchronizable chaotic systems”, Phys. Rev. Lett. A, Vol. 241, pp 303-310.
[11] Yang, T. & Chua, L.O. (1999) “Control of chaos using sampled-data feedback control”, Internat.
J. Bifurcat. Chaos, Vol. 9, pp 215-219.
[12] Ott, E., Grebogi, C. & Yorke, J.A. (1990) “Controlling chaos”, Phys. Rev. Lett., Vol. 64, pp
1196-1199.
[13] Park, J.H. & Kwon, O.M. (2003) “A novel criterion for delayed feedback control of time-delay
chaotic systems”, Chaos, Solitons and Fractals, Vol. 17, pp 709-716.
[14] Yu, Y.G. & Zhang, S.C. (2006) “Adaptive backstepping synchronization of uncertain chaotic
systems”, Chaos, Solitons and Fractals, Vol. 27, pp 1369-1375.
[15] Liao, T.L. & Tsai, S.H. (2000) “Adaptive synchronization of chaotic systems and its applications
to secure communications”, Chaos, Solitons and Fractals, Vol. 11, pp 1387-1396.
[16] Konishi, K.., Hirai, M. & Kokame, H. (1998) “Sliding mode control for a class of chaotic
systems”, Phys. Lett. A, Vol. 245, pp 511-517.
[17] Chen, Z., Yang, Y., Qi, G. & Yuan, Z. (2007) “A novel hyperchaotic system only with one
equilibrium,” Physics Letters A, Vol. 360, pp 696-701.
[18] Hahn, W. (1967) The Stability of Motion, Springer, New York.
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, etc.
He has published over 110 research articles in
international journals and two text-books with
Prentice-Hall of India, New Delhi, India. He has
published over 45 papers in International
Conferences and 90 papers in National
Conferences. He has delivered several Key Note
Lectures on Control Systems, Chaos Theory,
Scientific Computing, SCILAB, etc.