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Local Lyapunov Exponents and characteristics of fixed/periodic points embedded within a chaotic attractor

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2005年第4期6卷 296-304页

作者： ALI M SAHA L.M Department of Mathematics Faculty of Mathematical Science Delhi UniversityDepartment of Mathematics Zakir Husain College Delhi University Delhi 110007 IndiaNew Delhi 110002 India

A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1>0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an alternative method to calculate λ1 has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.

关键词： chaotic attractor Largest Lyapunov Exponent Local Lyapunov Exponents

chaotic characteristics and attractor evolution of friction noise during friction process

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Friction 2018年第1期6卷 47-61页

作者： Cong DING Hua ZHU Guodong SUN Yuankai ZHOU Xue ZUO School of Mechatronic Engineering China University of Mining and Technology School of Mechanical Engineering Jiangsu University of Science and Technology Jiangsu Provincial Key Laboratory of Advanced Manufacturing for Marine Mechanical Equipment Jiangsu University of Science and Technology

Friction experiments are conducted on a ring-on-disk tribometer, and friction noise produced during the friction process is extracted by a microphone. The phase trajectory and chaotic parameters of friction noise are obtained by phase-space reconstruction, and its attractor evolution is analyzed. The results indicate that the friction noise is chaotic because the largest Lyapunov exponent is positive. The phase trajectory of the friction noise follows a "convergence-stability-divergence" pattern during the friction process. The friction noise attractor begins forming in the running-in process, and the correlation dimension D increases gradually. In the stable process, the attractor remains steady, and D is stable. In the last step of the process, the attractor gradually disappears, and D decreases. The friction noise attractor is a chaotic attractor. Knowledge of the dynamic evolution of this attractor can help identify wear state changes from the running-in process to the steady and increasing friction processes.

关键词： friction noise phase trajectory chaotic parameters Lyapunov exponent chaotic attractor

IMPULSIVE CONTROL OF chaotic attractorS IN NONLINEAR chaotic SYSTEMS

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Applied Mathematics and Mechanics(English Edition) 2004年第9期25卷 971-976页

作者：马军海任彪陈予恕 Management School Tianjin University Department of Mechanics Tianjin University

Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is *** applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, and with it, the upper bond of the impulse interval for asymptotically stable control was given. Numerical results are presented, which are considered with important reference value for control of chaotic attractors.

关键词： impulsive control chaotic attractor chaos asymptotically stable

A class of two-dimensional rational maps with self-excited and hidden attractors

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Chinese Physics B 2022年第3期31卷 224-233页

作者： Li-Ping Zhang Yang Liu Zhou-Chao Wei Hai-Bo Jiang Qin-Sheng Bi Faculty of Civil Engineering and Mechanics Jiangsu UniversityZhenjang 212013China School of Mathematics and Statistics Yancheng Teachers UniversityYancheng 224002China College of Engineering Mathematics and Physical SciencesUniversity of ExeterExeter EX44QFUK School of Mathematics and Physics China University of GeosciencesWuhan 430074China

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov(Kaplan–Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

关键词： two-dimensional rational map hidden attractors multi-stability a line of fixed points chaotic attractor

The analysis of complex behaviours of a novel three dimensional autonomous system

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Chinese Physics B 2010年第7期19卷 156-166页

作者：董高高郑松田立新杜瑞瑾孙梅 Research Center for Nonlinear Science Faculty of ScienceJiangsu University

This paper introduces a new three dimensional autonomous system with five equilibrium points. It demonstrates complex chaotic behaviours within a wide range of parameters, which are described by phase portraits, Lyapunov exponents, frequency spectrum, etc. Analysis of the bifurcation and Poincar@ map is used to reveal mechanisms of generating these complicated phenomena. The corresponding electronic circuits are designed, exhibiting experimental chaotic attractors in accord with numerical simulations. Since frequency spectrum analysis shows a broad frequency bandwidth, this system has perspective of potential applications in such engineering fields as secure communication.

关键词： three dimensional autonomous system chaotic attractor electronic circuit frequency spectrum

Generating Lorenz-like and Chen-like attractors from a simple algebraic structure

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Science China(Information Sciences) 2014年第7期57卷 214-220页

作者： WANG Xiong CHEN GuanRong Department of Electronic Engineering City University of Hong Kong

This paper reports the finding of a very simple one-parameter family of three-dimensional quadratic autonomous chaotic *** algebraic structure is simple with only four linear terms and two quadratic *** quadratic terms are different from the Lorenz and the Chen ***,by tuning the only parameter,this new family of systems can also generate both Lorenz-like and Chen-like attractors,thus further reveal the close relation between them.

关键词： chaotic attractor Lorenz system Chen system

A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors

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Chinese Physics B 2020年第6期29卷 109-114页

作者： Li-Ping Zhang Yang Liu Zhou-ChaoWei Hai-Bo Jiang Qin-Sheng Bi Faculty of Civil Engineering and Mechanics Jiangsu UniversityZhenjiang 212013China School of Mathematics and Statistics Yancheng Teachers UniversityYancheng 224002China College of Engineering Mathematics and Physical SciencesUniversity of ExeterExeter EX44QFUK School of Mathematics and Physics China University of GeosciencesWuhan 430074China

We study a novel class of two-dimensional maps with infinitely many coexisting ***,the mathematical model of these maps is formulated by introducing a sinusoidal *** existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all *** particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex ***,this map contains infinitely many coexisting attractors which has been rarely reported in the *** studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)*** analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.

关键词： two-dimensional map infinitely many coexisting attractors extreme multi-stability chaotic attractor

Synchronization of time-delay chaotic systems on small-world networks with delayed coupling

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Chinese Physics B 2009年第4期18卷 1404-1408页

作者：祁伟汪映海 Institute of Theoretical Physics Lanzhou University

By using the well-known Ikeda model as the node dynamics, this paper studies synchronization of time-delay systems on small-world networks where the connections between units involve time delays. It shows that, in contrast with the undelayed case, networks with delays can actually synchronize more easily. Specifically, for randomly distributed delays, time-delayed mutual coupling suppresses the chaotic behaviour by stabilizing a fixed point that is unstable for the uncoupled dynamical system.

关键词： chaotic attractor Ikeda model scale-free network

Symmetry,cusp bifurcation and chaos of an impact oscillator between two rigid sides

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Applied Mathematics and Mechanics(English Edition) 2007年第8期28卷 1109-1117页

作者：乐源谢建华 Department of Applied Mechanics and Engineering Southwest Jiaotong University Chengdu 610031P.R.China

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered. The theory of bifurcations of the fixed point is applied to such model, and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincar6 map. The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation. While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences, and bring about two antisymmetric chaotic attractors subse- quently. If the symmetric system is transformed into asymmetric one, bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.

关键词： periodic motion Poincaémap symmetry pitchfork bifurcation chaotic attractor cusp