This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is estab...This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiate map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.展开更多
In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center man...In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.展开更多
Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey model with two different functional responses...Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey model with two different functional responses(Holling type I and II functional responses)is discussed in this paper,which depicts a complex population relationship.The local dynamical behaviors of the interior fixed point of this system are studied.The detailed analysis reveals this system undergoes flip bifurcation and Neimark-Sacker bifurcation.Especially,we prove the existence of Marotto's chaos by analytical method.In addition,the hybrid control method is applied to control the chaos of this system.Numerical simulations are presented to support our research and demonstrate new dynamical behaviors,such as period-10,19,29,39,48 orbits and chaos in the sense of Li-Yorke.展开更多
A nonlinear discrete time Cournot duopoly game is investigated in this paper. The conditions of existence for saddle-node bifurcation, transcritical bifurcation and flip bifurcation are derived using the center manifo...A nonlinear discrete time Cournot duopoly game is investigated in this paper. The conditions of existence for saddle-node bifurcation, transcritical bifurcation and flip bifurcation are derived using the center manifold theorem and the bifurcation theory. We prove that there exists chaotic behavior in the sense of Marotto＇s definition of chaos. The numerical simulations not only show the consistence with our theoretical analysis, but also exhibit the complex but interesting dynamical behaviors of the model. The computation of maximum Lyapunov exponents confirms the theoretical analysis of the dynamical behaviors of the system.展开更多
The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using cen...The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.展开更多
In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using cen...In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes （invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation） to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting （2,3,4） chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.展开更多
In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for b...In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for boundary fixed points. This system inherits the dynamics of one-dimensional Ricker model such as cascade of period-doubling bifurcation, periodic windows and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and proving the existence of snap-back repeller. We use several dynamical systems tools to demonstrate the qualitative behaviors of the system.展开更多
In this paper, an approach for chaotifying a stable controllable linear system via single input state-feedback is presented. The overflow function of the system states is designed as the feedback controller, which can...In this paper, an approach for chaotifying a stable controllable linear system via single input state-feedback is presented. The overflow function of the system states is designed as the feedback controller, which can make the fixed point of the closed-loop system to be a snap-back repeller, thereby yields chaotic dynamics. Based on the Marotto theorem, it proves theoretically that the closed-loop system is chaotic in the sense of Li and Yorke. Finally, the simulation results are used to illustrate the effectiveness of the proposed method.展开更多
为设计性能较好的伪随机数发生器,提出了一个满足修正的马罗驼(Marotto)定理的新二维离散混沌系统(2DCS)。利用离散广义混沌同步理论和2D-CS构造了一个广义同步混沌系统(2D-GCS);通过一个实数域到整数域的变换设计了一个混沌伪随机数生...为设计性能较好的伪随机数发生器,提出了一个满足修正的马罗驼(Marotto)定理的新二维离散混沌系统(2DCS)。利用离散广义混沌同步理论和2D-CS构造了一个广义同步混沌系统(2D-GCS);通过一个实数域到整数域的变换设计了一个混沌伪随机数生成器(CPRNG);利用美国联邦信息处理标准(federal information processing standards,FIPS)提出的FIPS 140-2检测包分别对CPRNG和RC4算法产生的1000个二进制序列的随机性进行检测,结果均通过了检测。检测结果的平均值和方差对比表明CPRNG和RC4算法产生的伪随机序列随机性能相当,相关性检测结果表明该CPRNG在不同的密钥扰动下产生的密钥各组序列几乎完全独立,设计的CPRNG能产生性能良好的伪随机数。展开更多
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10071043)the Hong Kong Research Grants Council under the CERG grant CityU 1115/03E+1 种基金the NSF Shandong Research Funds for Young Scientists(Grant No.03BS094)the Shandong University Scientific Research Funds for Young Staff.The authors are very happy to have this opportunity to thank the referees for helpful remarks.
文摘This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiate map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.
基金Supported by the National Natural Science Foundation of China (No. 11071066)
文摘In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.
基金supported by the National Natural Science Foundation of China(No.12001503)the Project of Beijing Municipal Commission of Education(KM 202110015001)。
文摘Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey model with two different functional responses(Holling type I and II functional responses)is discussed in this paper,which depicts a complex population relationship.The local dynamical behaviors of the interior fixed point of this system are studied.The detailed analysis reveals this system undergoes flip bifurcation and Neimark-Sacker bifurcation.Especially,we prove the existence of Marotto's chaos by analytical method.In addition,the hybrid control method is applied to control the chaos of this system.Numerical simulations are presented to support our research and demonstrate new dynamical behaviors,such as period-10,19,29,39,48 orbits and chaos in the sense of Li-Yorke.
基金Supported by the National Natural Science Foundation of China(Nos.11101021,11372017)the National Scholarship Fund of China(201303070219)
文摘A nonlinear discrete time Cournot duopoly game is investigated in this paper. The conditions of existence for saddle-node bifurcation, transcritical bifurcation and flip bifurcation are derived using the center manifold theorem and the bifurcation theory. We prove that there exists chaotic behavior in the sense of Marotto＇s definition of chaos. The numerical simulations not only show the consistence with our theoretical analysis, but also exhibit the complex but interesting dynamical behaviors of the model. The computation of maximum Lyapunov exponents confirms the theoretical analysis of the dynamical behaviors of the system.
基金Supported by the National Natural Science Foundation of China(10671063 and 10801135)the Scientific Research Foundation of Hunan Provincial Education Department(09C255)
文摘The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.
基金Supported by the National Natural Science Foundation of China (No. 11071066)
文摘In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes （invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation） to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting （2,3,4） chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.
文摘In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for boundary fixed points. This system inherits the dynamics of one-dimensional Ricker model such as cascade of period-doubling bifurcation, periodic windows and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and proving the existence of snap-back repeller. We use several dynamical systems tools to demonstrate the qualitative behaviors of the system.
文摘In this paper, an approach for chaotifying a stable controllable linear system via single input state-feedback is presented. The overflow function of the system states is designed as the feedback controller, which can make the fixed point of the closed-loop system to be a snap-back repeller, thereby yields chaotic dynamics. Based on the Marotto theorem, it proves theoretically that the closed-loop system is chaotic in the sense of Li and Yorke. Finally, the simulation results are used to illustrate the effectiveness of the proposed method.
文摘为设计性能较好的伪随机数发生器,提出了一个满足修正的马罗驼(Marotto)定理的新二维离散混沌系统(2DCS)。利用离散广义混沌同步理论和2D-CS构造了一个广义同步混沌系统(2D-GCS);通过一个实数域到整数域的变换设计了一个混沌伪随机数生成器(CPRNG);利用美国联邦信息处理标准(federal information processing standards,FIPS)提出的FIPS 140-2检测包分别对CPRNG和RC4算法产生的1000个二进制序列的随机性进行检测,结果均通过了检测。检测结果的平均值和方差对比表明CPRNG和RC4算法产生的伪随机序列随机性能相当,相关性检测结果表明该CPRNG在不同的密钥扰动下产生的密钥各组序列几乎完全独立,设计的CPRNG能产生性能良好的伪随机数。
基金Supported by the National Natural Science Foundation of China(No.30970305)the Sichuan Provincial Education Department Scientific Research Project(No.TER2009-14)+1 种基金the Sichuan Provincial Old Revolutionary Base Areas Foundation(No.SLQ2010C-17)the Sichuan University of Arts and Science Natural Scientific Research Project(No.2009B07Z)
