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Control of period-doubling bifurcation and chaos in a discrete nonlinear system by the feedback of states and parameter adjustment

Luo Xiao-Shu,Chen Guan-Rong,Wang Bing-Hong,Fang Jin-Qing,Zou Yan-Li,Quan Hong-Jun,

物理学报 , 2003,
Abstract: In this paper, the control of delaying period doubling bifurcations and unstable periodic orbits embedded in a chaotic attractor of a discrete nonlinear dynamical system is effectively realized by using the state variables feedback and parameter variation. Moreover, the 2 n periodic orbits of the system can be controlled into the 2 m ( m
Bifurcations and chaos control in discrete small-world networks

Li Ning,Sun Hai-Yi,Zhang Qing-Ling,

中国物理 B , 2012,
Abstract: An impulsive delayed feedback control strategy to control period-doubling bifurcations and chaos is proposed. The control method is then applied to a discrete small-world network model. Qualitative analyses and simulations show that under a generic condition, the bifurcations and the chaos can be delayed or eliminated completely. In addition, the periodic orbits embedded in the chaotic attractor can be stabilized.
Chaos and Control in Coronary Artery System
Yanxiang Shi
Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/631476
Abstract: Two types of coronary artery system N-type and S-type, are investigated. The threshold conditions for the occurrence of Smale horseshoe chaos are obtained by using Melnikov method. Numerical simulations including phase portraits, potential diagram, homoclinic bifurcation curve diagrams, bifurcation diagrams, and Poincaré maps not only prove the correctness of theoretical analysis but also show the interesting bifurcation diagrams and the more new complex dynamical behaviors. Numerical simulations are used to investigate the nonlinear dynamical characteristics and complexity of the two systems, revealing bifurcation forms and the road leading to chaotic motion. Finally the chaotic states of the two systems are effectively controlled by two control methods: variable feedback control and coupled feedback control.
The Time Invariance Principle, Ecological (Non)Chaos, and A Fundamental Pitfall of Discrete Modeling  [PDF]
Bo Deng
Quantitative Biology , 2007,
Abstract: This paper is to show that most discrete models used for population dynamics in ecology are inherently pathological that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to tackle an on-going controversy regarding ecological chaos. Another implication of the result is that all continuous dynamical systems must be modeled by differential equations. As a result it suggests that researches based on discrete modeling must be closely scrutinized and the teaching of calculus and differential equations must be emphasized for students of biology.
Cryptanalyzing a discrete-time chaos synchronization secure communication system  [PDF]
Gonzalo Alvarez,Fausto Montoya,Miguel Romera,Gerardo Pastor
Physics , 2003, DOI: 10.1016/j.chaos.2003.12.013
Abstract: This paper describes the security weakness of a recently proposed secure communication method based on discrete-time chaos synchronization. We show that the security is compromised even without precise knowledge of the chaotic system used. We also make many suggestions to improve its security in future versions.
Energy minimization control for a discrete chaotic system

Liu Ding,Qian Fu-Cai,Ren Hai-Peng,Kong Zhi-Qiang,
刘 丁

物理学报 , 2004,
Abstract: A general framework algorithm is proposed for energy minimization control for a discrete chaotic system. A quadratic performance function is first given and the chaotic system is decomposed into a linear and a nonlinear parts. Then, the two-level algorithm is presented to solve the nonlinear optimal control problem: The first level predicates the nonlinear part of the chaos system; the second level solves a nonlinear quadratic optimization control problem by dynamic programming. The solution is fed back into the first level. The first level re-estimates the nonlinear part according to the solution from the second level. The information has been exchanged between the two levels by this means such that the optimal control law is obtained eventually. This method not only can make the control of chaos system be realized but also makes the energy consumed minimal during the whole control process.Simulations show the effectiveness of this algorithms.
Controlling Unstable Discrete Chaos and Hyperchaos Systems  [PDF]
Xin Li, Suping Qian
Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.411A2001

A method is introduced to stabilize unstable discrete systems, which does not require any adjustable control parameters of the system. 2-dimension discrete Fold system and 3-dimension discrete hyperchaotic system are stabilized to fixed points respectively. Numerical simulations are then provided to show the effectiveness and feasibility of the proposed chaos and hyperchaos controlling scheme.

Adaptive Feedback Control for Chaos Control and Synchronization for New Chaotic Dynamical System
M. M. El-Dessoky,M. T. Yassen
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/347210
Abstract: This paper investigates the problem of chaos control and synchronization for new chaotic dynamical system and proposes a simple adaptive feedback control method for chaos control and synchronization under a reasonable assumption. In comparison with previous methods, the present control technique is simple both in the form of the controller and its application. Based on Lyapunov's stability theory, adaptive control law is derived such that the trajectory of the new system with unknown parameters is globally stabilized to the origin. In addition, an adaptive control approach is proposed to make the states of two identical systems with unknown parameters asymptotically synchronized. Numerical simulations are shown to verify the analytical results.
Bifurcation control and chaos in a linear impulsive system

Jiang Gui-Rong,Xu Bu-Gong,Yang Qi-Gui,

中国物理 B , 2009,
Abstract: Bifurcation control and the existence of chaos in a class of linear impulsive systems are discussed by means of both theoretical and numerical ways. Chaotic behaviour in the sense of Marotto's definition is rigorously proven. A linear impulsive controller, which does not result in any change in one period-1 solution of the original system, is proposed to control and anti-control chaos. The numerical results for chaotic attractor, route leading to chaos, chaos control, and chaos anti-control, which are illustrated with two examples, are in good agreement with the theoretical analysis.
Control Chaos in System with Fractional Order  [PDF]
Yamin Wang, Xiaozhou Yin, Yong Liu
Journal of Modern Physics (JMP) , 2012, DOI: 10.4236/jmp.2012.36067
Abstract: In this paper, by utilizing the fractional calculus theory and computer simulations, dynamics of the fractional order system is studied. Further, we have extended the nonlinear feedback control in ODE systems to fractional order systems, in order to eliminate the chaotic behavior. The results are proved analytically by stability condition for fractional order system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.
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