In contrast to the main stream of studying the chaotic behavior of a given map, constructing chaotic map has attracted much less attention. In this paper, we propose simple analytical method for constructing one dimen...In contrast to the main stream of studying the chaotic behavior of a given map, constructing chaotic map has attracted much less attention. In this paper, we propose simple analytical method for constructing one dimensional continuous maps with certain specific periodic points.Further, we obtain results for the existence of chaotic phenomenon in the sense of Li and Yorke. Some examples are also analyzed using the proposed methods.展开更多
The dynamics of a unidirectional nonlinear delayed-coupling chaos system is investigated. Based on the local Hopf bifurcation at the zero equilibrium, we prove the global existence of periodic solutions using a global...The dynamics of a unidirectional nonlinear delayed-coupling chaos system is investigated. Based on the local Hopf bifurcation at the zero equilibrium, we prove the global existence of periodic solutions using a global Hopf bifurcation result due to Wu and a Bendixson’s criterion for higher dimensional ordinary differential equations due to Li & Muldowney.展开更多
The K-V beam through a hackle periodic-focusing magnetic field is studied using the particle-core model. The beam halo-chaos is found, and a power function controller is proposed based on mechanism of halo formation a...The K-V beam through a hackle periodic-focusing magnetic field is studied using the particle-core model. The beam halo-chaos is found, and a power function controller is proposed based on mechanism of halo formation and strategy of controlling halo-chaos. Multiparticle simulation was performed to control the halo by using the power function control method. The results show that the halo-chaos and its regeneration can be eliminated effectively. We also find that the radial particle density evolvement is of uniformity at the beam’s centre as long as appropriate parameters are chosen.展开更多
This paper studies the Kapchinsky-Vladimirsky (K-V) beam through a triangle periodic-focusing magnetic field by using the particle-core model. The beam halo-chaos is found,and an idea of Gauss function controller is p...This paper studies the Kapchinsky-Vladimirsky (K-V) beam through a triangle periodic-focusing magnetic field by using the particle-core model. The beam halo-chaos is found,and an idea of Gauss function controller is proposed based on the strategy of controlling the halo-chaos. It performs multiparticle simulation to control the halo by using the Gauss function control method. The numerical results show that the halo-chaos and its regeneration can be eliminated effectively,and that the radial particle density is uniform at the centre of the beam as long as the control method and appropriate parameter are chosen.展开更多
The Kapchinsky Vladimirsky(K-V)beam through a hackle periodic-focusing magnetic field is studiedusing the particle-core model.The beam halo-chaos is found,and an idea of fraction power-law function controller ispropos...The Kapchinsky Vladimirsky(K-V)beam through a hackle periodic-focusing magnetic field is studiedusing the particle-core model.The beam halo-chaos is found,and an idea of fraction power-law function controller isproposed based on the mechanism of halo formation and the strategy of controlling halo-chaos.The method is appliedto the multi-particle simulation to control the halo.The numerical results show that the halo-chaos and its regenerationcan be eliminated effectively by using the fraction power-law function control method.At the same time,the radialparticle density is uniform at the beam's center as long as the control method and appropriate parameter are chosen.展开更多
The equations of the asymmetrical periodic motion in a two-degree-of-freedom vibrating system with two rigid constraints are constructed analytically.Its Poincaré mapping equation is established too.Periodic moti...The equations of the asymmetrical periodic motion in a two-degree-of-freedom vibrating system with two rigid constraints are constructed analytically.Its Poincaré mapping equation is established too.Periodic motions of the system and their routes to chaos are also illustrated by numerical simulation.The ranges of the system excited frequency from periodic motions to chaotic motions are obtained.The chaotic motions of the system are shown by diagrams of Poincaré mapping,phase portraits and diagrams of bifurcation.The chaos controlling methods by the addition of constant load and the addition of phase are dissertated and analyzed numerically by the numerical solution.The chaos of the system is controlled by the two methods.The allowable range controlling variables and the steady orbits of the controlled system are obtained.展开更多
Based on the Silnikov criterion,this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions.Using the Cardano formula,it obtains the exact range of the value of the para...Based on the Silnikov criterion,this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions.Using the Cardano formula,it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory.By calculating the manifold near the equilibrium point,the series expression of the homoclinic orbit is also obtained.The space trajectory and Lyapunov exponent are investigated via numerical simulation,which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system.The results obtained here mean that chaos occurred in the exact range given in this paper.Numerical simulations also verify the analytical results.展开更多
Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new m...Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.展开更多
In a recent paper from Trans. Amer. Math. Soc., 351(1)(1999), 343—351, the maps f and g from [0,n] to itself are constructed, which have periodic points of period 2a+3 . In this note we prove that they have no period...In a recent paper from Trans. Amer. Math. Soc., 351(1)(1999), 343—351, the maps f and g from [0,n] to itself are constructed, which have periodic points of period 2a+3 . In this note we prove that they have no periodic points of period 2a+1.展开更多
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 rigorou...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.展开更多
Several different definitions of transitivity and their relationship are carefullydiscussed for general spaces, and it is proved that a continuous map on a metric space ischaotic in the sense of Devaney if and only if...Several different definitions of transitivity and their relationship are carefullydiscussed for general spaces, and it is proved that a continuous map on a metric space ischaotic in the sense of Devaney if and only if it is periodic orbit transitive or periodic orbitstrongly transitive.展开更多
To enhance the anti-breaking performance of privacy information, this article proposes a new encryption method utilizing the leaping peculiarity of the periodic orbits of chaos systems. This method maps the secret seq...To enhance the anti-breaking performance of privacy information, this article proposes a new encryption method utilizing the leaping peculiarity of the periodic orbits of chaos systems. This method maps the secret sequence to several chaos periodic orbits, and a short sequence obtained by evolving the system parameters of the periodic orbits in another nonlinear system will be the key to reconstruct these periodic orbits. In the decryption end, the shadowing method of chaos trajectory based on the modified Newton-Raphson algorithm is adopted to restore these system parameters. Through deciding which orbit each pair coordinate falls on, the original digital sequence can be decrypted.展开更多
Various nonlinear electric circuits have been designed to exhibit the roads to chaos (for a recent review see Ref. [6]). It is noticed that most of the experiments have only shown clearly the roads of the period-doubl...Various nonlinear electric circuits have been designed to exhibit the roads to chaos (for a recent review see Ref. [6]). It is noticed that most of the experiments have only shown clearly the roads of the period-doubling bifurcations and the intermittency while the road to chaos via quasi-periodicity is not reported yet. This presentation is to report its observation.展开更多
We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means o...We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits,Lyapunov exponents,and Poincarémaps.Based on several periodic orbits with different sizes and shapes,they are encoded systematically with two letters or four letters for two different sets of parameters.The periodic orbits outside the attractor with complex topology are discovered by accident.In addition,the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively.In this process,the rule of orbital period changing with parameters is also investigated.The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.展开更多
With a reasonable parameter configuration,the passive dynamic walking model has a stable,efficient,natural periodic gait,which depends only on gravity and inertia when walking down a slight slope.In fact,there is a de...With a reasonable parameter configuration,the passive dynamic walking model has a stable,efficient,natural periodic gait,which depends only on gravity and inertia when walking down a slight slope.In fact,there is a delicate balance in the energy conversion in the stable periodic gait,making the gait adjustable by changing the model parameters.Poincaré mapping is combined with Newton-Raphson iteration to obtain the numerical solution of the final state of the passive dynamic walking model.In addition,a simulation on the walking gait of the model is performed by increasing the slope step by step,thereby fixing the model's parameters synchronously.Then,the gait features obtained in the different slope stages are analyzed and discussed,the intrinsic laws are revealed in depth.The results indicate that the gait can present features of a single period,doubling period,the entrance of chaos,merging of sub-bands,and so on,because of the high sensitivity of the passive dynamic walking to the slope.From a global viewpoint,the gait becomes chaotic by way of period doubling bifurcation,with a self-similar Feigenbaum fractal structure in the process.At the entrance of chaos,the gait sequence comprises a Cantor set,and during the chaotic stage,sub-bands in the final-state diagram of the robot system present as a mirror of the period doubling bifurcation.展开更多
文摘In contrast to the main stream of studying the chaotic behavior of a given map, constructing chaotic map has attracted much less attention. In this paper, we propose simple analytical method for constructing one dimensional continuous maps with certain specific periodic points.Further, we obtain results for the existence of chaotic phenomenon in the sense of Li and Yorke. Some examples are also analyzed using the proposed methods.
文摘The dynamics of a unidirectional nonlinear delayed-coupling chaos system is investigated. Based on the local Hopf bifurcation at the zero equilibrium, we prove the global existence of periodic solutions using a global Hopf bifurcation result due to Wu and a Bendixson’s criterion for higher dimensional ordinary differential equations due to Li & Muldowney.
基金Supported by the National Natural Science Foundation of China (Grant No. 10247005)the Natural Science Foundation of the Anhui Higher Education Bureau (Grant No. KJ2007B187)the Scientific Research Foundation of China University of Mining and Technology for the Young (Grant No. OK060119).
文摘The K-V beam through a hackle periodic-focusing magnetic field is studied using the particle-core model. The beam halo-chaos is found, and a power function controller is proposed based on mechanism of halo formation and strategy of controlling halo-chaos. Multiparticle simulation was performed to control the halo by using the power function control method. The results show that the halo-chaos and its regeneration can be eliminated effectively. We also find that the radial particle density evolvement is of uniformity at the beam’s centre as long as appropriate parameters are chosen.
基金supported by the National Natural Science Foundation of China (Grant No 10247005)the Natural Science Foundation of the Anhui Higher Education Institutions of China (Grant No KJ2007B187)the Scientific Research Foundation of China University of Mining and Technology for the Young (Grant No OK060119)
文摘This paper studies the Kapchinsky-Vladimirsky (K-V) beam through a triangle periodic-focusing magnetic field by using the particle-core model. The beam halo-chaos is found,and an idea of Gauss function controller is proposed based on the strategy of controlling the halo-chaos. It performs multiparticle simulation to control the halo by using the Gauss function control method. The numerical results show that the halo-chaos and its regeneration can be eliminated effectively,and that the radial particle density is uniform at the centre of the beam as long as the control method and appropriate parameter are chosen.
基金National Natural Science Foundation of China under Crant No.10247005the Natural Science Foundation of the Anhui Higher Education Institutions of China under Grant No.KJ2007B187the Scientific Research Foundation of China University Of Mining and Technology for the Young under Grant No.OK060119
文摘The Kapchinsky Vladimirsky(K-V)beam through a hackle periodic-focusing magnetic field is studiedusing the particle-core model.The beam halo-chaos is found,and an idea of fraction power-law function controller isproposed based on the mechanism of halo formation and the strategy of controlling halo-chaos.The method is appliedto the multi-particle simulation to control the halo.The numerical results show that the halo-chaos and its regenerationcan be eliminated effectively by using the fraction power-law function control method.At the same time,the radialparticle density is uniform at the beam's center as long as the control method and appropriate parameter are chosen.
基金supported by the National Natural Science Foundation of China under Grant No.50475109 and No.10572055by the Natural Science Foundation of Gansu Province under Grant No.0803RJZA012
文摘The equations of the asymmetrical periodic motion in a two-degree-of-freedom vibrating system with two rigid constraints are constructed analytically.Its Poincaré mapping equation is established too.Periodic motions of the system and their routes to chaos are also illustrated by numerical simulation.The ranges of the system excited frequency from periodic motions to chaotic motions are obtained.The chaotic motions of the system are shown by diagrams of Poincaré mapping,phase portraits and diagrams of bifurcation.The chaos controlling methods by the addition of constant load and the addition of phase are dissertated and analyzed numerically by the numerical solution.The chaos of the system is controlled by the two methods.The allowable range controlling variables and the steady orbits of the controlled system are obtained.
基金Project supported by the National Natural Science Foundation of China (Grant No.10872141)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20060056005)
文摘Based on the Silnikov criterion,this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions.Using the Cardano formula,it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory.By calculating the manifold near the equilibrium point,the series expression of the homoclinic orbit is also obtained.The space trajectory and Lyapunov exponent are investigated via numerical simulation,which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system.The results obtained here mean that chaos occurred in the exact range given in this paper.Numerical simulations also verify the analytical results.
文摘Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.
文摘In a recent paper from Trans. Amer. Math. Soc., 351(1)(1999), 343—351, the maps f and g from [0,n] to itself are constructed, which have periodic points of period 2a+3 . In this note we prove that they have no periodic points of period 2a+1.
基金The project supported by the Key Projects of National Natural Science Foundation of China under Grant No. 70431002 and National Natural Science Foundation of China under Grants Nos. 70371068 and 10247005
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10871074 and 10572011)the Natural Science Foundation of Guangxi Province,China (Grant No 0832244)
文摘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.
文摘Several different definitions of transitivity and their relationship are carefullydiscussed for general spaces, and it is proved that a continuous map on a metric space ischaotic in the sense of Devaney if and only if it is periodic orbit transitive or periodic orbitstrongly transitive.
基金This project was supported by the National Natural Science Foundation of Shaan'Xi Province, China (2003F40).
文摘To enhance the anti-breaking performance of privacy information, this article proposes a new encryption method utilizing the leaping peculiarity of the periodic orbits of chaos systems. This method maps the secret sequence to several chaos periodic orbits, and a short sequence obtained by evolving the system parameters of the periodic orbits in another nonlinear system will be the key to reconstruct these periodic orbits. In the decryption end, the shadowing method of chaos trajectory based on the modified Newton-Raphson algorithm is adopted to restore these system parameters. Through deciding which orbit each pair coordinate falls on, the original digital sequence can be decrypted.
文摘Various nonlinear electric circuits have been designed to exhibit the roads to chaos (for a recent review see Ref. [6]). It is noticed that most of the experiments have only shown clearly the roads of the period-doubling bifurcations and the intermittency while the road to chaos via quasi-periodicity is not reported yet. This presentation is to report its observation.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12205257,11647085,and11647086)the Shanxi Province Science Foundation for Youths(Grant No.201901D211252)+1 种基金Fundamental Research Program of Shanxi Province(Grant No.202203021221095)the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi of China(Grant Nos.2019L0505,2019L0554,and 2019L0572)。
文摘We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits,Lyapunov exponents,and Poincarémaps.Based on several periodic orbits with different sizes and shapes,they are encoded systematically with two letters or four letters for two different sets of parameters.The periodic orbits outside the attractor with complex topology are discovered by accident.In addition,the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively.In this process,the rule of orbital period changing with parameters is also investigated.The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.
基金supported by the National Natural Science Foundation of China (60905049)the self-managed Project of State Key Laboratory of Robotic Technology and System in Harbin Institute of Technology(200804C)
文摘With a reasonable parameter configuration,the passive dynamic walking model has a stable,efficient,natural periodic gait,which depends only on gravity and inertia when walking down a slight slope.In fact,there is a delicate balance in the energy conversion in the stable periodic gait,making the gait adjustable by changing the model parameters.Poincaré mapping is combined with Newton-Raphson iteration to obtain the numerical solution of the final state of the passive dynamic walking model.In addition,a simulation on the walking gait of the model is performed by increasing the slope step by step,thereby fixing the model's parameters synchronously.Then,the gait features obtained in the different slope stages are analyzed and discussed,the intrinsic laws are revealed in depth.The results indicate that the gait can present features of a single period,doubling period,the entrance of chaos,merging of sub-bands,and so on,because of the high sensitivity of the passive dynamic walking to the slope.From a global viewpoint,the gait becomes chaotic by way of period doubling bifurcation,with a self-similar Feigenbaum fractal structure in the process.At the entrance of chaos,the gait sequence comprises a Cantor set,and during the chaotic stage,sub-bands in the final-state diagram of the robot system present as a mirror of the period doubling bifurcation.
