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 A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control

2017 ◽  
Vol 27 (3) ◽  
pp. 409-439 ◽  
Author(s):  
Sundarapandian Vaidyanathan
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Backstepping Control ◽  
Phase Portraits ◽  
The Novel ◽  
Adaptive Backstepping ◽  
Unknown Parameters ◽  
Recursive Procedure ◽  
Backstepping Controller ◽  
Adaptive Backstepping Controller

AbstractThis paper presents a new seven-term 3-D jerk chaotic system with two cubic 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 a unique equilibrium at the origin, which is a saddle-focus and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1= 0:2974, L2= 0 and L3= −3:8974. 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 new jerk chaotic system is found as DKY= 2:0763. Next, an adaptive backstepping controller is designed to globally stabilize the new jerk chaotic system with unknown parameters. Moreover, an adaptive backstepping controller is also designed 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 are shown to illustrate all the main results derived in this work.

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.

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 Analysis, Adaptive Control and Synchronization of a Novel 4-D Hyperchaotic Hyperjerk System via Backstepping Control Method

2016 ◽  
Vol 26 (3) ◽  
pp. 311-338 ◽  
Author(s):  
Sundarapandian Vaidyanathan
Keyword(s):  
Control Method ◽  
Research Work ◽  
Adaptive Controller ◽  
Backstepping Control ◽  
The Novel ◽  
Qualitative Properties ◽  
Unknown Parameters ◽  
A Chain ◽  
Control And Synchronization ◽  
Hyperjerk System

AbstractA hyperjerk system is a dynamical system, which is modelled by annth order ordinary differential equation withn≥ 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system ofnfirst order ordinary differential equations withn≥ 4. In this research work, a 4-D novel hyperchaotic hyperjerk system with two nonlinearities has been proposed, and its qualitative properties have been detailed. The novel hyperjerk system has a unique equilibrium at the origin, which is a saddle-focus and unstable. The Lyapunov exponents of the novel hyperjerk system are obtained asL1= 0.14219,L2= 0.04605,L3= 0 andL4= −1.39267. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained asDKY= 3.1348. Next, an adaptive controller is designed via backstepping control method to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive controller is designed via backstepping control method to achieve global synchronization of the identical novel hyperjerk systems with three unknown parameters. MATLAB simulations are shown to illustrate all the main results derived in this research work on a novel hyperjerk system.

scholarly journals A New Chaotic System with Line of Equilibria: Dynamics, Passive Control and Circuit Design

2019 ◽  
Vol 9 (4) ◽  
pp. 2336 ◽  
Author(s):  
Aceng Sambas ◽  
Mustafa Mamat ◽  
Ayman Ali Arafa ◽  
Gamal M Mahmoud ◽  
Mohamad Afendee Mohamed ◽  
...  
Keyword(s):  
Circuit Design ◽  
Chaotic System ◽  
Passive Control ◽  
Electronic Circuit ◽  
Control Method ◽  
Phase Portraits ◽  
New Chaotic System ◽  
Quadratic Nonlinearities ◽  
Line Of Equilibria ◽  
Line Equilibrium

<p>A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the  new chaotic system, An electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.</p>

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 A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yuhua Xu ◽  
Bing Li ◽  
Yuling Wang ◽  
Wuneng Zhou ◽  
Jian-an Fang
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
The Other ◽  
Phase Portraits ◽  
Fractional Dimension ◽  
The Novel ◽  
New Chaotic System ◽  
New System

A new four-scroll chaotic attractor is found by feedback controlling method in this paper. The novel chaotic system can generate four scrolls two of which are transient chaotic and the other two of which are ultimate chaotic. Of particular interest is that this novel chaotic system can generate one-scroll, two 2-scroll and four-scroll chaotic attractor with variation of a single parameter. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map, respectively. The analysis results show clearly that this is a new chaotic system which deserves further detailed investigation.

scholarly journals Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19 ◽  
Author(s):  
Fei Yu ◽  
Li Liu ◽  
Shuai Qian ◽  
Lixiang Li ◽  
Yuanyuan Huang ◽  
...  
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Random Number Generator ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
The Novel ◽  
Multiple Attractors ◽  
Period 2 ◽  
Different Types ◽  
System Designs

Novel memristive hyperchaotic system designs and their engineering applications have received considerable critical attention. In this paper, a novel multistable 5D memristive hyperchaotic system and its application are introduced. The interesting aspect of this chaotic system is that it has different types of coexisting attractors, chaos, hyperchaos, periods, and limit cycles. First, a novel 5D memristive hyperchaotic system is proposed by introducing a flux-controlled memristor with quadratic nonlinearity into an existing 4D four-wing chaotic system as a feedback term. Then, the phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy are used to analyze the basic dynamics of the 5D memristive hyperchaotic system. For a specific set of parameters, we find an unusual metastability, which shows the transition from chaotic to periodic (period-2 and period-3) dynamics. Moreover, its circuit implementation is also proposed. By using the chaoticity of the novel hyperchaotic system, we have developed a random number generator (RNG) for practical image encryption applications. Furthermore, security analyses are carried out with the RNG and image encryption designs.

scholarly journals Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities

2014 ◽  
Vol 24 (3) ◽  
pp. 375-403 ◽  
Author(s):  
Sundarapandian Vaidyanathan ◽  
Christos Volos ◽  
Viet-Thanh Pham ◽  
Kavitha Madhavan ◽  
Babatunde A. Idowu
Keyword(s):  
Chaotic System ◽  
Circuit Simulation ◽  
Research Work ◽  
The Novel ◽  
Adaptive Backstepping ◽  
Unknown Parameters ◽  
Circuit Realization ◽  
Backstepping Controller ◽  
Jerk System ◽  
Adaptive Backstepping Controller

Abstract In this research work, a six-term 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel jerk system are obtained as L1 = 0.07765,L2 = 0, and L3 = −0.87912. The Kaplan-Yorke dimension of the novel jerk system is obtained as DKY = 2.08833. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system with two unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel jerk chaotic systems with two unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model

Sign in / Sign up

Export Citation Format

Share Document