连续和离散动力系统中两类方程的复杂动态
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摘要
本文应用连续和离散动力系统中的分支理论、二阶平均方法、Melnik-ov方法和混沌理论,首次研究连续和离散动力系统中两类方程当参数变化时不动点的分支、三频率共振解的分支和混沌动态.
对于连续动力系统,首先运用Melnikov方法和二阶平均方法研究受悬挂轴振动和外力作用的物理单摆在周期扰动下与拟周期扰动下的复杂动态,给出在周期扰动下系统产生混沌运动的准则,在拟周期扰动下,仅能给出当Ω=nw+(?)v,n=1,2,3,4时平均系统存在混沌的条件,而当Ω=nw+(?)v,n=5-15时,用平均方法不能给出混沌产生的条件,这里ν和ω之比为无理数.同时通过数值模拟,包括二维参数平面和三维参数空间中的分支图,相应的最大Lyapunov指数图,相图以及Poincare映射,验证了理论结果的正确性,发现了系统的一些复杂动力学行为,其中包括从周期1轨到周期2轨的分支与周期2轨到周期2轨的逆分支；混沌的突然发生；不带周期窗口的全混沌区域,带复杂周期窗口或拟周期窗口的混沌区域；混沌的突然消失,混沌转变成周期1轨；不带周期窗口的全不变环区域或全拟周期轨区域；不变环或拟周期轨突然转变与周期1轨；从一个周期1轨区域到另一个周期1轨区域或从一个拟周期轨区域到另一个拟周期轨区域的突然跳跃；周期1轨的对称断裂；内部危机；发现了许多新颖的混沌吸引子和不变环,等等.数值模拟结果表明：当调整分支参数α,δ,f0与Ω的值时,系统动态从全混沌运动或全不变环或全拟周期轨突然转变为周期轨,这有利于控制物理单摆的运动.
其次运用二阶平均方法研究受悬挂轴振动和外力作用的物理单摆的三频率共振动解的分支与混沌,运用二阶平均方法研究了当系统的固有频率ω0,外力激励频率ω与参数频率Ω之比：ω0：ω：Ω≈1：1：n,1：2：n,1：3：n,2：1：n与3：1：n时共振解的存在与分支.运用Melnikov方法,给出了当w0:w:Ω≈1:m:n时共振解存在的条件,并通过数值模拟进行了验证.通过数值模拟,又发现了系统的许多动态,如：不带周期窗口的全不变环行为,不变环区域的串联,不带周期窗口的纯混沌行为,带复杂周期窗口的混沌行为,全周期轨区域；不变环转变为周期轨,周期轨转变为混沌,一种不变环转变为另一种不变环等动态的跳跃行为；内部危机等动态.这些动态与在周期扰动和拟周期扰动下的动态具有很大的差异,特别发现：当初始点由鞍点改变成中心时,有更多的新的不变环吸引子被找到.
首次用Euler方法将细菌培养呼吸过程模型离散化,运用中心流形定理和分支理论,给出映射发生flip分支,Hopf分支的条件,Marotto意义下的混沌存在的条件,证明映射没有fold分支.运用数值模拟方法(包括分支图,相图,最大Lyapunov指数图,分形维数),不仅验证了理论分析结论的正确性,还发现了该映射的许多动态,如：从周期2轨到周期8轨的逆倍周期分支,从周期1轨到周期4轨的逆倍周期分支,带周期窗口的混沌行为,不带周期窗口的全混沌行为,不带周期窗口的全不变环行为,从混沌转变为不变环,从不变环转变为混沌,从混沌转变为周期轨,从周期轨转变为混沌等动态的跳跃,周期轨与混沌的交替行为等.对这两个动力系统的研究,所得到的动态行为将丰富非线性动力系统的内容,对其它学科,例如,化学、物理、生物学的研究有一定的应用价值.
全文共分三章.
第一章是关于动力系统的分支与混沌的预备知识.简要介绍连续和离散动力系统中的中心流形定理,二阶平均方法、Melnikov方法以及混沌的定义、特征和通向混沌的道路.
第二章,深入分析与研究受悬挂轴振动和外力作用的物理单摆的复杂动态.第二节至第四节,研究在周期扰动下与拟周期扰动下系统的的动态,运用二阶平均方法与Melnikov方法,给出系统存在混沌的准则,数值模拟不仅验证了理论分析结果的正确性,发现了系统的一些复杂动力学行为,而且显示当Ω= nw+(?)v, n= 7时系统也存在混沌.本部分的结果发表在Acta Mathematica Applicatae Sinaca, English Series, Vol. (26), No.1(2010),55-78.第五节,研究系统的三频率共振动解的分支与混沌,运用二阶平均方法给出了当系统的固有频率ω0,外力激励频率ω与参数频率Ω之比：ω0：w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n与3：1：n时共振解的存在条件与分支.运用Melnikov方法,给出了当wo:w:Ω≈1:m:n时共振解存在的条件,并通过数值模拟进行了验证.数值模拟又发现了系统的许多动态,显示了与在周期扰动和拟周期扰动下的动态的差异,发现：当初始点由鞍点改变成中心时,有更多的新的不变环吸引子被找到.本部分的结果已被Acta Mathematica Applicatae Sinaca, English Series接收,
第三章,研究离散型细菌培养呼吸过程模型.应用欧拉方法将连续型细菌培养呼吸过程模型离散化,运用中心流形定理和分支理论,给出映射发生flip分支,Hopf分支的条件,存在Marotto意义下的混沌的条件,证明系统不存在fold分支.运用数值模拟,验证了理论分析结果的正确性,发现了该映射的许多动态.
In this thesis, we investigates the bifurcation of fixed points and resonant so-lutions and chaos for two types of equations in continuous and discrete dynamical systems, which are not considered yet, as the bifurcation parameters vary by ap-plying bifurcation theories, second-order averaging method, Melnikov method and chaos theory in continuous and discrete dynamical systems.
For the continuous system, the complex dynamics for the physical pendulum equation with suspension axis vibrations are investigated. Firstly, we prove the conditions of existence of chaos under periodic perturbations by using Melnikov's method. By using second-order averaging method and Melinikov's method, we give the conditions of existence of chaos in averaged system under quasi-periodic perturbations forΩ=nw+(?)v, n= 1 - 4, where v is not rational to w, and can't prove the condition of existence of chaos for n= 5 - 15, and can show the chaotic behaviors for n= 5 by numerical simulations. By numerical simulations including bifurcation diagrams, phase portraits, computation of maximum Lyapunov expo-nents and Poincare map, we check up the effect of theoretical analysis and expose the complex dynamical behaviors, including the bifurcation and reverse bifurca-tion from period-one to period-two orbits; and the onset of chaos, and the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly dis-appearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parametersα,δ,f0 andΩ; and the onset of invariant torus or quasi-periodic behaviors, the entire invari-ant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to pe-riodic orbit; and the jumping behaviors which including from period-one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors and invariant torus. In particular, the system shown the entire chaotic region or invariant torus region or entire quasi-periodic region suddenly converting to periodic orbit by adjusting the bifurcation parameters a,α,δ,f0 andΩ, which is beneficial to the control of motions of the pendulum.
Secondly, we investigate the existence and the bifurcations of resonant solu-tion for w0:w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n and 3:1:n by using second-order averaging method and give a criterion for the existence of resonant solution for w0:w:Ω≈1:m:n is given by using Melnikov's method and verify the theoretical analysis by numerical simulations. By numerical simulation, we expose some other interesting dynamical behaviors, including the entire invariant torus region, the cascade of invariant torus behaviors, the entire chaos region with-out periodic windows, chaotic region with complex periodic windows, the entire period-one orbits region; the jumping behaviors which including invariant torus behaviors converting tq period-one orbits, from chaos to invariant torus behaviors or from invariant torus behaviors to chaos, from period-one to chaos, from invariant torus behaviors to another invariant torus behaviors; and the interior crisis; and the different nice invariant torus attractors and chaotic attractors. The numerical results show the difference of dynamical behaviors in the physical pendulum equa-tion with suspension axis vibrations between under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations. It exhibits many nice invariant torus behaviors under the resonant conditions and we find a lot of chaotic behaviors which are different to those under the periodic/quasi-periodic perturbations.
For the discrete system, the dynamical behaviors of a discreet mathematical model for respiratory process in bacterial culture are investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using cen-ter 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 of the model is also calcu-lated. The numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting complex dynamical behaviors compared with the continuous model, including reverse bifurcation from period-two to period-eight orbits and from period-one orbits to period-four orbits, the cascades of period-doubling bifurcations from period-one orbits to period-eight orbits and from period-three orbits to period-twelve orbits; and the onset of chaos, and the entire chaotic region without periodic windows, chaotic regions with com-plex periodic windows, the entire invariant torus without periodic windows; chaotic behaviors converting to periodic orbits; and the jumping behaviors including from chaos to invariant torus, from invariant torus to chaos and from periodic orbits to chaos; and the interleaving occurrence of periodic orbits and invariant torus behaviors; and the different nice chaotic attractors and invariant torus. The study for them is of fundamental and even practical interest.
The dynamical behaviors of these systems will enrich the content of nonlinear dynamical systems and will be useful in other subjects such as chemistry, physics and biology.
This thesis consists of three chapters as the following.
Chapter 1 is about preparation knowledge. A brief review of center manifold theorems for continuous and discrete dynamical system is presented. At the same time, some definitions and characteristics of chaos as well as some routes to chaos are mentioned.
In chapter 2, the physical pendulum equation with suspension axis vibra-tions is investigated. In section 2.2,2.3 and 2.4, the conditions of existence of chaos under periodic perturbations and under quasi-periodic perturbations are given by using Melnikov's method and second-order averaging method. By nu-merical simulations we not only check up the effect of theoretical analysis and expose the complex dynamical behaviors, but also show the chaotic behaviors as Ω=nw+(?)v, n= 7. In section 2.5, we investigate the existence and the bifurca-tions of resonant solution for w0:w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n and 3:1:n by using second-order averaging method and give a criterion for the exis-tence of resonant solution for w0:w:Ω≈1:m:n is given by using Melnikov's method and verify the theoretical analysis by numerical simulations. By numerical simulation, we expose some other interesting dynamical behaviors. The numerical results show the difference of dynamical behaviors in the physical pendulum equa-tion with suspension axis vibrations between under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations. It exhibits many nice invariant torus behaviors under the resonant conditions and we find a lot of chaotic behaviors which are different to those under the periodic/quasi-periodic perturbations.
In chapter 3, the dynamical behaviors of a discreet mathematical model for the respiratory process in bacterial culture are investigated. The conditions of ex-istence for flip bifurcation and Hopf bifurcation are derived by using center mani-fold theorem and bifurcation theory, and we prove that there is no fold bifurcation. The chaotic existence in the sense of Marotto's definition of chaos is proved. The numerical simulation results display some new and complex dynamical behaviors.
对于连续动力系统,首先运用Melnikov方法和二阶平均方法研究受悬挂轴振动和外力作用的物理单摆在周期扰动下与拟周期扰动下的复杂动态,给出在周期扰动下系统产生混沌运动的准则,在拟周期扰动下,仅能给出当Ω=nw+(?)v,n=1,2,3,4时平均系统存在混沌的条件,而当Ω=nw+(?)v,n=5-15时,用平均方法不能给出混沌产生的条件,这里ν和ω之比为无理数.同时通过数值模拟,包括二维参数平面和三维参数空间中的分支图,相应的最大Lyapunov指数图,相图以及Poincare映射,验证了理论结果的正确性,发现了系统的一些复杂动力学行为,其中包括从周期1轨到周期2轨的分支与周期2轨到周期2轨的逆分支；混沌的突然发生；不带周期窗口的全混沌区域,带复杂周期窗口或拟周期窗口的混沌区域；混沌的突然消失,混沌转变成周期1轨；不带周期窗口的全不变环区域或全拟周期轨区域；不变环或拟周期轨突然转变与周期1轨；从一个周期1轨区域到另一个周期1轨区域或从一个拟周期轨区域到另一个拟周期轨区域的突然跳跃；周期1轨的对称断裂；内部危机；发现了许多新颖的混沌吸引子和不变环,等等.数值模拟结果表明：当调整分支参数α,δ,f0与Ω的值时,系统动态从全混沌运动或全不变环或全拟周期轨突然转变为周期轨,这有利于控制物理单摆的运动.
其次运用二阶平均方法研究受悬挂轴振动和外力作用的物理单摆的三频率共振动解的分支与混沌,运用二阶平均方法研究了当系统的固有频率ω0,外力激励频率ω与参数频率Ω之比：ω0：ω：Ω≈1：1：n,1：2：n,1：3：n,2：1：n与3：1：n时共振解的存在与分支.运用Melnikov方法,给出了当w0:w:Ω≈1:m:n时共振解存在的条件,并通过数值模拟进行了验证.通过数值模拟,又发现了系统的许多动态,如：不带周期窗口的全不变环行为,不变环区域的串联,不带周期窗口的纯混沌行为,带复杂周期窗口的混沌行为,全周期轨区域；不变环转变为周期轨,周期轨转变为混沌,一种不变环转变为另一种不变环等动态的跳跃行为；内部危机等动态.这些动态与在周期扰动和拟周期扰动下的动态具有很大的差异,特别发现：当初始点由鞍点改变成中心时,有更多的新的不变环吸引子被找到.
首次用Euler方法将细菌培养呼吸过程模型离散化,运用中心流形定理和分支理论,给出映射发生flip分支,Hopf分支的条件,Marotto意义下的混沌存在的条件,证明映射没有fold分支.运用数值模拟方法(包括分支图,相图,最大Lyapunov指数图,分形维数),不仅验证了理论分析结论的正确性,还发现了该映射的许多动态,如：从周期2轨到周期8轨的逆倍周期分支,从周期1轨到周期4轨的逆倍周期分支,带周期窗口的混沌行为,不带周期窗口的全混沌行为,不带周期窗口的全不变环行为,从混沌转变为不变环,从不变环转变为混沌,从混沌转变为周期轨,从周期轨转变为混沌等动态的跳跃,周期轨与混沌的交替行为等.对这两个动力系统的研究,所得到的动态行为将丰富非线性动力系统的内容,对其它学科,例如,化学、物理、生物学的研究有一定的应用价值.
全文共分三章.
第一章是关于动力系统的分支与混沌的预备知识.简要介绍连续和离散动力系统中的中心流形定理,二阶平均方法、Melnikov方法以及混沌的定义、特征和通向混沌的道路.
第二章,深入分析与研究受悬挂轴振动和外力作用的物理单摆的复杂动态.第二节至第四节,研究在周期扰动下与拟周期扰动下系统的的动态,运用二阶平均方法与Melnikov方法,给出系统存在混沌的准则,数值模拟不仅验证了理论分析结果的正确性,发现了系统的一些复杂动力学行为,而且显示当Ω= nw+(?)v, n= 7时系统也存在混沌.本部分的结果发表在Acta Mathematica Applicatae Sinaca, English Series, Vol. (26), No.1(2010),55-78.第五节,研究系统的三频率共振动解的分支与混沌,运用二阶平均方法给出了当系统的固有频率ω0,外力激励频率ω与参数频率Ω之比：ω0：w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n与3：1：n时共振解的存在条件与分支.运用Melnikov方法,给出了当wo:w:Ω≈1:m:n时共振解存在的条件,并通过数值模拟进行了验证.数值模拟又发现了系统的许多动态,显示了与在周期扰动和拟周期扰动下的动态的差异,发现：当初始点由鞍点改变成中心时,有更多的新的不变环吸引子被找到.本部分的结果已被Acta Mathematica Applicatae Sinaca, English Series接收,
第三章,研究离散型细菌培养呼吸过程模型.应用欧拉方法将连续型细菌培养呼吸过程模型离散化,运用中心流形定理和分支理论,给出映射发生flip分支,Hopf分支的条件,存在Marotto意义下的混沌的条件,证明系统不存在fold分支.运用数值模拟,验证了理论分析结果的正确性,发现了该映射的许多动态.
In this thesis, we investigates the bifurcation of fixed points and resonant so-lutions and chaos for two types of equations in continuous and discrete dynamical systems, which are not considered yet, as the bifurcation parameters vary by ap-plying bifurcation theories, second-order averaging method, Melnikov method and chaos theory in continuous and discrete dynamical systems.
For the continuous system, the complex dynamics for the physical pendulum equation with suspension axis vibrations are investigated. Firstly, we prove the conditions of existence of chaos under periodic perturbations by using Melnikov's method. By using second-order averaging method and Melinikov's method, we give the conditions of existence of chaos in averaged system under quasi-periodic perturbations forΩ=nw+(?)v, n= 1 - 4, where v is not rational to w, and can't prove the condition of existence of chaos for n= 5 - 15, and can show the chaotic behaviors for n= 5 by numerical simulations. By numerical simulations including bifurcation diagrams, phase portraits, computation of maximum Lyapunov expo-nents and Poincare map, we check up the effect of theoretical analysis and expose the complex dynamical behaviors, including the bifurcation and reverse bifurca-tion from period-one to period-two orbits; and the onset of chaos, and the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly dis-appearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parametersα,δ,f0 andΩ; and the onset of invariant torus or quasi-periodic behaviors, the entire invari-ant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to pe-riodic orbit; and the jumping behaviors which including from period-one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors and invariant torus. In particular, the system shown the entire chaotic region or invariant torus region or entire quasi-periodic region suddenly converting to periodic orbit by adjusting the bifurcation parameters a,α,δ,f0 andΩ, which is beneficial to the control of motions of the pendulum.
Secondly, we investigate the existence and the bifurcations of resonant solu-tion for w0:w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n and 3:1:n by using second-order averaging method and give a criterion for the existence of resonant solution for w0:w:Ω≈1:m:n is given by using Melnikov's method and verify the theoretical analysis by numerical simulations. By numerical simulation, we expose some other interesting dynamical behaviors, including the entire invariant torus region, the cascade of invariant torus behaviors, the entire chaos region with-out periodic windows, chaotic region with complex periodic windows, the entire period-one orbits region; the jumping behaviors which including invariant torus behaviors converting tq period-one orbits, from chaos to invariant torus behaviors or from invariant torus behaviors to chaos, from period-one to chaos, from invariant torus behaviors to another invariant torus behaviors; and the interior crisis; and the different nice invariant torus attractors and chaotic attractors. The numerical results show the difference of dynamical behaviors in the physical pendulum equa-tion with suspension axis vibrations between under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations. It exhibits many nice invariant torus behaviors under the resonant conditions and we find a lot of chaotic behaviors which are different to those under the periodic/quasi-periodic perturbations.
For the discrete system, the dynamical behaviors of a discreet mathematical model for respiratory process in bacterial culture are investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using cen-ter 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 of the model is also calcu-lated. The numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting complex dynamical behaviors compared with the continuous model, including reverse bifurcation from period-two to period-eight orbits and from period-one orbits to period-four orbits, the cascades of period-doubling bifurcations from period-one orbits to period-eight orbits and from period-three orbits to period-twelve orbits; and the onset of chaos, and the entire chaotic region without periodic windows, chaotic regions with com-plex periodic windows, the entire invariant torus without periodic windows; chaotic behaviors converting to periodic orbits; and the jumping behaviors including from chaos to invariant torus, from invariant torus to chaos and from periodic orbits to chaos; and the interleaving occurrence of periodic orbits and invariant torus behaviors; and the different nice chaotic attractors and invariant torus. The study for them is of fundamental and even practical interest.
The dynamical behaviors of these systems will enrich the content of nonlinear dynamical systems and will be useful in other subjects such as chemistry, physics and biology.
This thesis consists of three chapters as the following.
Chapter 1 is about preparation knowledge. A brief review of center manifold theorems for continuous and discrete dynamical system is presented. At the same time, some definitions and characteristics of chaos as well as some routes to chaos are mentioned.
In chapter 2, the physical pendulum equation with suspension axis vibra-tions is investigated. In section 2.2,2.3 and 2.4, the conditions of existence of chaos under periodic perturbations and under quasi-periodic perturbations are given by using Melnikov's method and second-order averaging method. By nu-merical simulations we not only check up the effect of theoretical analysis and expose the complex dynamical behaviors, but also show the chaotic behaviors as Ω=nw+(?)v, n= 7. In section 2.5, we investigate the existence and the bifurca-tions of resonant solution for w0:w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n and 3:1:n by using second-order averaging method and give a criterion for the exis-tence of resonant solution for w0:w:Ω≈1:m:n is given by using Melnikov's method and verify the theoretical analysis by numerical simulations. By numerical simulation, we expose some other interesting dynamical behaviors. The numerical results show the difference of dynamical behaviors in the physical pendulum equa-tion with suspension axis vibrations between under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations. It exhibits many nice invariant torus behaviors under the resonant conditions and we find a lot of chaotic behaviors which are different to those under the periodic/quasi-periodic perturbations.
In chapter 3, the dynamical behaviors of a discreet mathematical model for the respiratory process in bacterial culture are investigated. The conditions of ex-istence for flip bifurcation and Hopf bifurcation are derived by using center mani-fold theorem and bifurcation theory, and we prove that there is no fold bifurcation. The chaotic existence in the sense of Marotto's definition of chaos is proved. The numerical simulation results display some new and complex dynamical behaviors.
引文
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[36]Hale J.K., Kocak H. Dynamics and Bifurcations. Springer-Verlag; 1991.
[37]Harrison D.E.F., Pirt S.J., The Influence of Dissoived Oxygen Concentrationon on The Respiration and Glucose Metabolism of Klebsiella aerogenes during Growth. J. Gen. Microbiol.1967; 46,193-205.
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[44]Jackson E.A., Kodogergiou A. Entraiment and migration controls of two-dimensional maps. Physica D.1992; 54:253-265.
[45]Jing Z.J., Chan K.Y., Xu D.H., Cao H.J. Bifurcations of periodic solutions and chaos in Josephson system. Discrete and Continuous Dynamical Systems.2001; 7(3):573-592.
[46]Jing Z.J., Yang Z.Y., Jiang T. Complex dynamics in Duffing-van der Pol equation. Chaos, Solitons and Fractals.2005; 27:722-747.
[47]Jing Z. J., Cao H.J. Bifurcation of periodic orbits in Josephson equation with a phase shift, Int. J. Bifurcation and Chaos.2002; 12(7),1515-1530.
[48]Jing Z. J., Yang J. P. Complex dynamics in pendulum equation with parametric and external excitations (Ⅰ), Inter J Bif. and Chaos.2006; 16,2889-2902.
[49]Jing Z. J., Yang J. P. Complex dynamics in pendulum equation with parametric and external excitations (Ⅱ), Inter J Bif. and Chaos.2006; 16,3053-3078.
[50]Jing Z.J., Chang Y., Guo B.L. bifurcation and chaos in discrete FitzHugh-Nagumo system. Chaos, Solitons and Fractals.2004; 21:701-720.
[51]Jing Z.J., Jia Z.Y., Wang R.Q. Chaos behavior in the discrete BVP oscillator. Int.J. Bifurcat & Chaos.2002; 12(3):619-627.
[52]Jing Z.J., Yang J.P. Bifurcation and chaos in discrete-time predator-prey system. Chaos, Solitons and Fractals.2006; 27:259-277.
[53]Jing Z.J., Yang J.P., Feng W. Bifurcation and chaos in neural excitable system. Chaos, Solitons and Fractals.2006; 27:197-215.
[54]Jing Z.J., Huang J.C. Bifurcation and chaos in a discrete genetic toggle switch system. Chaos, Solitons and Fractals.2005; 23:887-908.
[55]Jordan D.W., Smith P. Nolinear ordinary differential equations. Second Edition, Clarendon Press,1987.
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