scholarly journals Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

2014 ◽  
Vol 24 (4) ◽  
pp. 409-446 ◽  
Author(s):  
Sundarapandian Vaidyanathan ◽  
Christos Volos ◽  
Viet-Thanh Pham
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Lorenz System ◽  
Adaptive Controller ◽  
The Novel ◽  
Maximal Lyapunov Exponent ◽  
System Parameters ◽  
Quadratic Nonlinearities ◽  
Hyperchaotic Lorenz System ◽  
Unknown System

Abstract In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007) with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007) has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.

scholarly journals Hyperchaos, adaptive control and synchronization of a novel 4-D hyperchaotic system with two quadratic nonlinearities

2016 ◽  
Vol 26 (4) ◽  
pp. 471-495 ◽  
Author(s):  
Sundarapandian Vaidyanathan
Keyword(s):  
Adaptive Control ◽  
Lyapunov Exponents ◽  
Research Work ◽  
Adaptive Controller ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
The Novel ◽  
System Parameters ◽  
Quadratic Nonlinearities ◽  
Unknown System

AbstractThis research work announces an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. We describe the qualitative properties of the novel 4-D hyperchaotic system and illustrate their phase portraits. We show that the novel 4-D hyperchaotic system has two unstable equilibrium points. The novel 4-D hyperchaotic system has the Lyapunov exponents L1= 3.1575, L2= 0.3035, L3= 0 and L4= −33.4180. The Kaplan-Yorke dimension of this novel hyperchaotic system is found as DKY= 3.1026. Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, we deduce that the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to stabilize the novel 4-D hyperchaotic system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 4-D hyperchaotic systems with unknown system parameters. The adaptive control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work.

scholarly journals A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control

2016 ◽  
Vol 26 (1) ◽  
pp. 19-47 ◽  
Author(s):  
Sundarapandian Vaidyanathan
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Control Method ◽  
Adaptive Controller ◽  
Backstepping Control ◽  
Phase Portraits ◽  
The Novel ◽  
Unknown Parameters ◽  
Recursive Procedure ◽  
Quadratic Nonlinearities

This paper announces an eight-term novel 3-D jerk chaotic system with three quadratic nonlinearities. The phase portraits of the novel jerk chaotic system are displayed and the qualitative properties of the jerk system are described. The novel jerk chaotic system has two equilibrium points, which are saddle-foci and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1= 0.20572,L2= 0 and L3= −1.20824. Since the sum of the Lyapunov exponents of the jerk chaotic system is negative, we conclude that the chaotic system is dissipative. The Kaplan-Yorke dimension of the novel jerk chaotic system is derived as DKY= 2.17026. Next, an adaptive controller is designed via backstepping control method to globally stabilize the novel jerk chaotic system with unknown parameters. Moreover, an adaptive controller is also designed via backstepping control method to achieve global chaos synchronization of the identical jerk chaotic systems with unknown parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict feedback systems. MATLAB simulations have been depicted to illustrate the phase portraits of the novel jerk chaotic system and also the adaptive backstepping control results.

scholarly journals Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system

2015 ◽  
Vol 25 (3) ◽  
pp. 333-353 ◽  
Author(s):  
Sundarapandian Vaidyanathan ◽  
Christos Volos
Keyword(s):  
Adaptive Control ◽  
Chaotic System ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Adaptive Controller ◽  
Phase Portraits ◽  
The Novel ◽  
Unknown Parameters ◽  
Mathematical Properties ◽  
Quadratic Nonlinearities

AbstractFirst, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1= 0.0395,L2= 0 and L3= −0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY=3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.

SYNCHRONIZATION BETWEEN TWO DIFFERENT HYPERCHAOTIC SYSTEMS WITH UNCERTAIN PARAMETERS

2013 ◽  
Vol 27 (13) ◽  
pp. 1350044
Author(s):  
XING-YUAN WANG ◽  
YU-HONG YANG ◽  
MING-KU FENG
Keyword(s):  
Lyapunov Stability ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Sufficient Condition ◽  
Uncertain Parameters ◽  
Hyperchaotic Lorenz System ◽  
Hyperchaotic Systems

This paper studies the problem of chaos synchronization between two different hyperchaotic systems with uncertain parameters. Based on the Lyapunov stability theory, we obtain the sufficient condition of synchronization between two different hyperchaotic systems with uncertain parameters. A new adaptive controller with parameter update laws is designed to synchronize these chaotic systems. We proved it in theory with an uncertain hyperchaotic Lorenz system and an uncertain hyperchaotic Rössler system. Numerical results verified the validation of the proposed scheme.

Synchronized Dynamics of Hyperchaotic Flows via an Improved Finite-Time Adaptive Controller Design

2020 ◽  
Vol 48 ◽  
pp. 49-62
Author(s):  
Edwin A. Umoh ◽  
Ogechukwu N. Iloanusi
Keyword(s):  
Uniform Convergence ◽  
Finite Time ◽  
Controller Design ◽  
Adaptive Controller ◽  
Coupled Systems ◽  
Convergence Time ◽  
Adaptive Controllers ◽  
System Parameters ◽  
Unknown System ◽  
Asymptotic Convergence Rate

We proposed a performance-improved finite-time adaptive synchronizing controllers and parameter update laws for coupling the dynamics of identical 4D hyperchaotic flows. The four-dimensional hyperchaotic flows consists of 12 terms and 11 system parameters and possessed very rich dynamics and larger parameter space. The performance of the proposed finite-time adaptive synchronizing controller was enhanced by the introduction of scalar quantities known as global controller strength coefficients and parameter update strength coefficients respectively, into the algebraically-derived control and parameter update structures, in order to constrained overshoots of the trajectories of the coupled systems and accelerate their rate of uniform convergence in finite time. Numerical simulation results obtained confirmed that the uniform asymptotic convergence rate of the coupling trajectories was faster, while the parameter update laws give a stable identification of the unknown system parameters in a global synchronizing time. A comparative analysis of the convergence time of the proposed adaptive controllers with recently published works indicated that the proposed controller has faster rates of uniform convergence of system trajectories.

Research on Mode of Deformation Forcasting Based on the Maximal Lyapunov Exponent

2012 ◽  
Vol 204-208 ◽  
pp. 4443-4448
Author(s):  
Xiu Ping Wang ◽  
Ting Chen Jiang
Keyword(s):  
Lyapunov Exponent ◽  
Power System ◽  
Lyapunov Exponents ◽  
Chaos Theory ◽  
Forecasting Model ◽  
Maximal Lyapunov Exponent ◽  
Engineering Research ◽  
Time Deformation ◽  
Model Based ◽  
Forecasting Method

It is necessary for us to monitor and forecast deformation as harm of deformation is tremendous. At the same time, deformation is seen as power system with chaos features because process of deformation is easily influenced by geology、hydrology、earthquake and activity and so on. On the base of chaos theory, forecasting model based on the maximal Lyapunov exponent is put forward in this paper, and is checked by example of engineering. Research indicate that Chaos phenomenon exist in deformation sequence for Lyapunov exponents is bigger than 0. The forecasting results show that forecasting method based on maximal Lyapunov exponent has high predictive precision, so the method is proved to be feasible and referenced.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Sign in / Sign up

Export Citation Format

Share Document