Analysis of a 3D chaotic system

In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.

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“…The mathematical properties of the T-system were studied in [10][11][12]. In recent years, the mathematical epidemic models given by Eqs.…”

Section: Introductionmentioning

confidence: 99%

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Harkó

Lobo

Mak

2014

Applied Mathematics and Computation

430

376

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“…In addition, their chaotic behavior are studied. Recently, a new chaotic system, as T chaotic system is introduced in [31], which can be described by means of three dynamical equations with three state variables as follows:…”

Section: Preliminariesmentioning

confidence: 99%

Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems

Tirandaz¹,

Ahmadnia²,

Tavakoli³

2017

IJECE

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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. Keyword:Adaptive method Lyapunov stability theorem Chaos synchronization Projective lag method Copyright © 2017 Institute of Advanced Engineering and Science.All rights reserved. Corresponding Author:Hamed Tirandaz, Electrical and Computer Engineering Department, Hakim Sabzevari University, Sabzevar, Iran. Email: tirandaz@hsu.ac.ir INTRODUCTIONSensitivity to the initial values of the state variables is the main feature of any chaotic system, which causes exponentially different motion trajectories of the state variables with different initial conditions. The problem of chaos synchronization has received a lot of attention in the last three decades due to its potential applications in many different fields such as: physics, chemistry, electrical engineering, economics and secure communications. Until now, many kind of chaos control and synchronization schemes have developed by the researchers. [20] are some of the investigated method during the last recent years. Among these investigated methods, chaos synchronization related to the projective method has considerably noticed during the last few decades due to its proportional feature, which the response chaotic system can be synchronized up to a typical aligned scaling factors. So far many types of projective methods have been studied by the researchers. [30] are some of the projective related method for control and synchronization between two identical/non-identical chaotic systems. But, projective lag hybrid synchronization is rarely investigated by the researchers. So, this paper is devoted to the synchronization problem between T chaotic system and Lorenz chaotic system by designing an appropriate adaptive hybrid projective lag synchronization method.The rest reminder of this paper is constructed as follows: In Section 2, the structure of the T chaotic system and Lorenz chaotic system are described. Then, in Section 3, the problem of chaos synchronization of T chaotic system and Lorenz chaotic system is given. An adaptive projective lag control method is designed

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“…The first famous chaotic system was discovered by Lorenz, when he was designing a weather model in 1963 [4]. Some well-known chaotic systems are Chen system [5], Lü system [6], Cai system [7], Tigan system [8], Sprott systems [9], etc.…”

Section: Introductionmentioning

confidence: 99%

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

Vaidyanathan

2017

Archives of Control Sciences

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A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control SUNDARAPANDIAN VAIDYANATHAN This 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 L 1 = 0.2974, L 2 = 0 and L 3 = −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 D KY = 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.

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Analysis of a 3D chaotic system

Cited by 197 publications

References 15 publications

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems

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

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