三类高维系统的分岔、混沌及控制研究
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摘要
本文先以一个映射系统为例分析了不动点失稳后发生各类分岔、混沌等动力学行为,然后针对一个两自由度碰撞振动系统重点分析了其周期运动失稳后发生两种不同的倍化分岔导致混沌现象,最后以一个平面两自由度双摆模型为例说明了其运动微分方程及具有周期系数的扰动运动微分方程的建立过程以及稳定性、分岔、混沌和分岔控制等研究方法。本文的研究主要有以下方面：
1.从映射系统、碰撞振动系统、周期系数系统三种模型有关稳定性、分岔、混沌、分岔控制等方面的理论研究和工程应用背景出发,综述了部分研究成果、最新发展动态。介绍了论文的研究内容与主要结果。
2.研究了一类三维映射系统,考虑了其不动点失稳后可能发生的几类分岔,根据发生不同类型分岔时特征根应满足的特点,确定了系统参数需要满足的条件。以不动点失稳发生Hopf-Flip分岔为例,利用中心流形-范式方法和投影法说明了三维映射研究分岔的过程,计算了相关范式系数,最后通过数值计算验证其结果。结果表明,该映射存在因不动点失稳而发生典型倍周期分岔导致混沌的过程,也存在因Hopf圈多次发生环面倍化而导致混沌的过程。不动点失稳发生Hopf分岔后,可能经历由光滑不变圈—环面倍化—环面破裂—非光滑不变圈的变化过程。满足4阶强共振Hopf分岔条件时不动点失稳后通过4条映射轨道形成吸引不变圈,随着参数的改变不变圈破裂形成4阶Hopf圈,参数进一步改变,可收敛于周期4点。满足非共振Hopf-Flip分岔条件时映射不动点失稳后会形成2个环状或管状混沌吸引子；满足弱共振(λ06=1) Hopf-Flip分岔条件时不动点失稳后会形成周期6点；满足强共振(λ04=1)Hopf-Flip分岔条件时不动点失稳后会形成4个相连或不相连的带状混沌吸引子或周期4点。
3.研究了一类碰撞振动系统,理论分析及数值验证了系统除存在典型倍周期分岔外,还存在非典型的倍周期分岔。结果表明系统参数在满足非共振的Hopf分岔(但靠近强共振或弱共振区域)条件下,系统周期1-1运动失稳首先形成拟周期运动(Poincare截面上不动点失稳形成Hopf圈),如果系统参数临近共振区,由于Arnold舌的存在,在n阶强(弱)共振点附近不动点失稳后沿n条映射轨道收敛于Hopf圈,当参数变化且穿越n阶强(弱)共振参数区(Arnold舌)时,Poincare截面上不动点失稳发生次谐分岔而形成稳定周期n点,参数进一步改变则经次谐倍化分岔而形成稳定周期2n点,参数再次改变则再经次谐倍化分岔而形成稳定周期4n点,8n点,16n点,…,混沌状态,最终形成n条分支共存形式的n条倍化分岔导致混沌序列。
4.研究了一个平面两自由度双摆力学模型,先根据拉格朗日方程建立了系统的运动微分方程,当端点作给定运动情况下,利用运动过程中的几何边角关系、微分关系将运动微分方程中的微分变量分别用端点的坐标及它们的各阶导数表示,得到系统的周期运动,并推导出非显示形式的扰动运动微分方程。采用渐近逼近法,逐步确定扰动运动微分方程一阶、二阶和三阶近似,最终可以得到扰动运动微分方程的任意阶近似。这样将原系统周期解的稳定性与分岔的分析就转化为对扰动运动微分方程n阶近似的平衡点的稳定性与分岔的分析。对于扰动运动微分方程来讲,其平衡点的稳定性以及失稳后的分岔类型一般是由其前三阶项确定的,即4阶及以上高阶非线性项一般不会在本质上影响分析其局部动力学行为,本文最终推导出扰动微分方程的六阶近似,并且通过数值计算,比较了相同参数下扰动运动微分方程6阶近似和扰动运动微分方程3阶近似在分岔过程上的差异,验证结果表明除运算时间外两者是一致的。
5.介绍了线性周期系数微分系统的稳定性与常系数微分系统稳定性的关系,给出稳定性判据。介绍了非线性周期系数系统的稳定性与对应线性周期系数系统的稳定性的关系,给出稳定性判据。将周期系数微分系统经过一系列ti=(i-1)×T→i×T(i1,…,n)积分可以确定n个状态点,其在Poincare截面(σ={(φ10,φ10,φ20,φ20,t)∈R4×s|t=T})上形成的对应关系构成Poincare映射,利用映射分岔条件给出了周期系数微分系统平衡点失稳后可能发生的几种分岔的分岔条件。介绍了周期系数系统发生Hopf分岔的解析分析过程。最后通过调整系统参变量进行数值计算,最终得到Flip分岔、非共振Hopf分岔(λ0n(ε0)≠1,n≠1,2,3,4,…)、强共振(λ3=1、λ2=1)Hopf分岔、Hopf-Flip分岔等行为,从结果来看周期系数系统与常系数微分方程或差分方程有基本类似的分岔行为。给出了两种不同情形的Hopf-F1ip分岔结果：一种情形是平衡点失稳后可形成稳定2阶Hopf圈或高阶周期点,而另一种情形是平衡点一旦失稳后不能稳定于Hopf圈或周期n点。对于强共振(λ2=1)Hopf分岔,是一种特殊情形,在此种情况中,随参数不断变化特征根在临近-1穿越单位圆时,4个负实根突然分解为2个模小于1的负实根和一对复共轭特征根,最终是一对复特征根在很小参数区间穿越单位圆(另2个仍为模小于1的负实根),穿越后为一对-1根(另2根仍为模小于1的负实根),正是由于特征根穿越单位圆周的特殊性,导致分岔结果的特殊性：稳定不动点—周期2焦点—稳定的Hopf圈—周期2结点—周期4结点、8结点、16结点、…、2n点,直至混沌,并得到此种情况倍周期分岔导致混沌相图。
6.先介绍了针对常系数系统的分岔行为进行控制的线性法、平移法以及状态反馈和参数调整控制法等三种方法。然后将上述三种控制分岔混沌的方法应用于周期系数系统,对周期系数系统平衡点失稳后发生的倍化分岔、Hopf分岔两种情形的分岔行为进行控制,从数值上验证其可操作性,得到了系统在控制前后的分岔图。对于倍化分岔控制：(1)采用线性法,当选取适当的控制参数,可将周期2点控制到周期1点,或控制到Hopf圈；(2)采用状态反馈和参数调整控制法,当选取适当的控制参数,可将周期2点控制到周期1点,或控制到Hopf圈、混沌状态；(3)平移法未能达到控制分岔的目的。对于Hopf分岔控制：(1)采用线性法,当选取适当的控制参数,可将Hopf圈控制到周期1点,或控制到另一个光滑Hopf圈或变形的Hopf分岔圈；(2)采用状态反馈和参数调整控制法,当选取适当的控制参数,可将Hopf圈控制到周期1点,或控制到另一个光滑Hopf圈、变形的Hopf分岔圈、周期7点或混沌状态；(3)采用平移法未能达到控制分岔的目的。
This dissertation considered three high dimensional systems. First as an example, a map was considered to study the bifurcations and chaos when the fixed points lost their stability. Then it focused on two kinds of period-doubling bifurcations and chaotic behaviors in a two-degree-of-freedom-vibro-impact system when the periodic solutions lost its stability. Finally a mechanical system with periodic coefficients was investigated. Its differential equations were established. The perturbed differential equations with periodic coefficients of its periodic motions were derived. Moreover stability, bifurcations, chaos and control were addressed. Numerical simulation results were presented. The main respects of the research are followings:
1. It was surveyed in chapter1that some recent achievements and developments and of the theoretic research and engineering applications on the stability, bifurcations, chaos and control of maps, vibro-impact systems and dynamics systems with periodic coefficients. The main contents and results of the dissertation were introduced.
2. A three-dimensional map was considered. Some kinds of bifurcations were investigated while the fixed points losing their stability. The corresponding analytic conditions which the variant parameters satisfy were derived when different kinds of bifurcations occur. A procedure to compute Hopf-Flip bifurcations of fixed points for3-dimensional maps was established by using projection technique. The normal form coefficients are computed. Finally, numerical simulation results demonstrated it. The typical period-doubling bifurcation leading to chaotic was found while the fixed points losing their stability. There was also a period-doubling bifurcation of tori leading to chaotic via curve doubling when the Hopf invariant circle lost its stability. As the control parameters were varied, it can vary from smooth invariant circle to tori doubling-tori breaking-smooth invariant circle. The fixed point would lock into one attractive invariant circle via four orbits while Hopf bifurcation occurs under strong resonant conditions (q=1). The attractive invariant circle would break into four quasi-periodic invariant circles as the control parameters vary. Moreover the four quasi-periodic invariant circles would lock into four fixed points as the parameters vary. Two chaotic attractor occured under non-resonant Hopf-Flip conditions. Six isolated chaotic attractors or six fixed points occured under weak-resonant Hopf-Flip conditions (λ06=1). Four connected or non-connected belt chaotic attractors occured under strong-resonant Hopf-Flip conditions (λ04＝1).
3. The typical period-doubling bifurcation leading to chaotic in a two-degree-of-freedom-vibro-impact system when the periodic solutions lost its stability was investigated theoretically and by means of numerical simulations. It was show that there was also non-typical period-doubling bifurcation leading to chaotic. Quasi-periodic motion occured while the periodic1-1impact motion losing its stability under non-resonant Hopf conditions (but near strong-resonant or weak-resonant zone, Hopf invariant circle occurs on the Poincare section while the fixed points losing their stability). If the system parameters varied near the resonant point, because of the existence of Arnold tongue, the fixed point would lock into one Hopf invariant circle via n orbits while losing its stability. As the control parameters vary across n's order resonant parameters area (Arnold tongue), the fixed point will lock into n fixed point by sub-harmonic bifurcation. As the parameters vary further, the n fixed point will lock into periodic2n point by sub-harmonic bifurcation. And periodic4n point, periodic8n point, periodic16n point,..., chaotic will occur. Finally there was one period-doubling bifurcation leading to chaotic which had n branches.
4. A two-degree-of-freedom mechanical system was investigated. The differential motion equations were established by making use of Lagrange's equations. When the endpoint of the arm performs a prescribed motion, the undisturbed motion can be given by making use of the relations of edges and angles, and the differential variable denoting by the endpoint's coordinates and their derivatives. The disturbed motion equations of implicit type can be derived while the prescribed motions were disturbed. Assuming the disturbance, the first-order approximate disturbed motion equations of explicit type can be derived by making use of asymptotic method. And the second-order approximate disturbed motion equations, the third-order one,..., arbitrary nth order approximate disturbed motion equations of explicit type can also be obtained. So the analysis of stability and bifurcation of motion was converted to the analysis of stability and bifurcation of the equilibrium points of nth order approximate disturbed motion equations. For the disturbed motion equations, the stability and bifurcation types of the equilibrium points are depended on the first three order nonlinear terms. In other words, the fourth-order term and higher order term do not affect essentially the analysis of dynamical behaviors. The first six order nonlinear approximate disturbed motion equations were derived in this dissertation. The difference of the numerical simulation results between the six-order approximate disturbed motion equations and the third-order one was compared. The simulation results show that the conclusion is correct.
5. The relation of stability between linear differential equations with periodic coefficients and linear differential equations with constant coefficients was introduced. The stability criteria were given. The relation of stability between nonlinear systems with periodic coefficients and its corresponding linear systems was introduced. The stability criteria were given. A suitable map can be constructed making use of a series of integrating differential equations with periodic coefficients over ti=(i-1)xT->iT (i=1,…,n), which is the Poincare map on the Poincare section (σ={(φ10,φ10, φ20,φ20, t)∈R4×S t=T}). Several bifurcations conditions were given when the equilibrium points of the differential equations with periodic coefficients lost their stability. The analysis method studying of bifurcations of the differential equations with periodic coefficients is introduced. Numerical simulation results were presented. The Flip bifurcation, Hopf bifurcation under non-resonant conditions (λ0n(ε0)≠1, n≠2,3,4,…), Hopf bifurcation under strong-resonant conditions (λ3=1、λ2=1), Hopf-Flip bifurcation were obtained. Bifurcations behaviors of the differential systems with periodic coefficients are basically similar to the that of the map with constant coefficients. Two different results of Hopf-Flip bifurcation were given:One case is that it maybe form two quasi-periodic invariant circles or periodic n point when the equilibrium points lose their stability. Another case is that it can not any invariant circles or periodic n point. It is a particularcare when Hopf bifurcation occur under strong-resonant conditions (λ2=1). When the eigenvalues are close the unit circle, four negative real roots will be resolved into two negative real roots which magnitude are smaller then1and a pair of roots of complex is conjugate. Finally the roots of complex conjugate cross the unit circle and transform into a pair of-1root (the other two are negative real roots which magnitude are smaller then1). Because of the particularity of root crossing the unit circle, the bifurcation results are special:stable fixed point-periodic2focus points-stable Hopf invariant circle-periodic2node points-periodic4node points, periodic8node points, periodic16node points,..., periodic2n node points, and chaos. The phase diagram of period-doubling bifurcation leading to chaotic was given.
6. Three methods of control chaos for the dynamical system with constant coefficients were introduced:the variable-parameter linear controller, the translation, the state variables feedback and parameter variation. The control of delaying Flip bifurcation and Hopf bifurcation has been studied. The bifurcation phase diagrams are obtained. For Flip bifurcation control:(1) Adopting the variable-parameter linear controller, periodic2points can be controlled to fixed point taking proper control parameters. Otherwise it may be controlled to Hopf circle.(2) Adopting the state variables feedback and parameter variation, periodic2points can be controlled to fixed point taking suitable control parameters. Otherwise it may be controlled to Hopf circle or chaos state.(3) Adopting the translation, it is failed to achieve the control of Flip bifurcation. For Hopf bifurcation control:(1) Adopting the variable-parameter linear controller, Hopf circle can be controlled to fixed point taking proper control parameters. Otherwise it may be controlled to other smooth Hopf circle or deformed Hopf circle.(2) Adopting the state variables feedback and parameter variation, Hopf circle can be controlled to fixed point taking suitable control parameters. Otherwise it may be controlled to other smooth Hopf circle or deformed Hopf circle or periodic7points or chaos state.(3) Adopting the translation, it is failed to achieve the control of Hopf bifurcation.
1.从映射系统、碰撞振动系统、周期系数系统三种模型有关稳定性、分岔、混沌、分岔控制等方面的理论研究和工程应用背景出发,综述了部分研究成果、最新发展动态。介绍了论文的研究内容与主要结果。
2.研究了一类三维映射系统,考虑了其不动点失稳后可能发生的几类分岔,根据发生不同类型分岔时特征根应满足的特点,确定了系统参数需要满足的条件。以不动点失稳发生Hopf-Flip分岔为例,利用中心流形-范式方法和投影法说明了三维映射研究分岔的过程,计算了相关范式系数,最后通过数值计算验证其结果。结果表明,该映射存在因不动点失稳而发生典型倍周期分岔导致混沌的过程,也存在因Hopf圈多次发生环面倍化而导致混沌的过程。不动点失稳发生Hopf分岔后,可能经历由光滑不变圈—环面倍化—环面破裂—非光滑不变圈的变化过程。满足4阶强共振Hopf分岔条件时不动点失稳后通过4条映射轨道形成吸引不变圈,随着参数的改变不变圈破裂形成4阶Hopf圈,参数进一步改变,可收敛于周期4点。满足非共振Hopf-Flip分岔条件时映射不动点失稳后会形成2个环状或管状混沌吸引子；满足弱共振(λ06=1) Hopf-Flip分岔条件时不动点失稳后会形成周期6点；满足强共振(λ04=1)Hopf-Flip分岔条件时不动点失稳后会形成4个相连或不相连的带状混沌吸引子或周期4点。
3.研究了一类碰撞振动系统,理论分析及数值验证了系统除存在典型倍周期分岔外,还存在非典型的倍周期分岔。结果表明系统参数在满足非共振的Hopf分岔(但靠近强共振或弱共振区域)条件下,系统周期1-1运动失稳首先形成拟周期运动(Poincare截面上不动点失稳形成Hopf圈),如果系统参数临近共振区,由于Arnold舌的存在,在n阶强(弱)共振点附近不动点失稳后沿n条映射轨道收敛于Hopf圈,当参数变化且穿越n阶强(弱)共振参数区(Arnold舌)时,Poincare截面上不动点失稳发生次谐分岔而形成稳定周期n点,参数进一步改变则经次谐倍化分岔而形成稳定周期2n点,参数再次改变则再经次谐倍化分岔而形成稳定周期4n点,8n点,16n点,…,混沌状态,最终形成n条分支共存形式的n条倍化分岔导致混沌序列。
4.研究了一个平面两自由度双摆力学模型,先根据拉格朗日方程建立了系统的运动微分方程,当端点作给定运动情况下,利用运动过程中的几何边角关系、微分关系将运动微分方程中的微分变量分别用端点的坐标及它们的各阶导数表示,得到系统的周期运动,并推导出非显示形式的扰动运动微分方程。采用渐近逼近法,逐步确定扰动运动微分方程一阶、二阶和三阶近似,最终可以得到扰动运动微分方程的任意阶近似。这样将原系统周期解的稳定性与分岔的分析就转化为对扰动运动微分方程n阶近似的平衡点的稳定性与分岔的分析。对于扰动运动微分方程来讲,其平衡点的稳定性以及失稳后的分岔类型一般是由其前三阶项确定的,即4阶及以上高阶非线性项一般不会在本质上影响分析其局部动力学行为,本文最终推导出扰动微分方程的六阶近似,并且通过数值计算,比较了相同参数下扰动运动微分方程6阶近似和扰动运动微分方程3阶近似在分岔过程上的差异,验证结果表明除运算时间外两者是一致的。
5.介绍了线性周期系数微分系统的稳定性与常系数微分系统稳定性的关系,给出稳定性判据。介绍了非线性周期系数系统的稳定性与对应线性周期系数系统的稳定性的关系,给出稳定性判据。将周期系数微分系统经过一系列ti=(i-1)×T→i×T(i1,…,n)积分可以确定n个状态点,其在Poincare截面(σ={(φ10,φ10,φ20,φ20,t)∈R4×s|t=T})上形成的对应关系构成Poincare映射,利用映射分岔条件给出了周期系数微分系统平衡点失稳后可能发生的几种分岔的分岔条件。介绍了周期系数系统发生Hopf分岔的解析分析过程。最后通过调整系统参变量进行数值计算,最终得到Flip分岔、非共振Hopf分岔(λ0n(ε0)≠1,n≠1,2,3,4,…)、强共振(λ3=1、λ2=1)Hopf分岔、Hopf-Flip分岔等行为,从结果来看周期系数系统与常系数微分方程或差分方程有基本类似的分岔行为。给出了两种不同情形的Hopf-F1ip分岔结果：一种情形是平衡点失稳后可形成稳定2阶Hopf圈或高阶周期点,而另一种情形是平衡点一旦失稳后不能稳定于Hopf圈或周期n点。对于强共振(λ2=1)Hopf分岔,是一种特殊情形,在此种情况中,随参数不断变化特征根在临近-1穿越单位圆时,4个负实根突然分解为2个模小于1的负实根和一对复共轭特征根,最终是一对复特征根在很小参数区间穿越单位圆(另2个仍为模小于1的负实根),穿越后为一对-1根(另2根仍为模小于1的负实根),正是由于特征根穿越单位圆周的特殊性,导致分岔结果的特殊性：稳定不动点—周期2焦点—稳定的Hopf圈—周期2结点—周期4结点、8结点、16结点、…、2n点,直至混沌,并得到此种情况倍周期分岔导致混沌相图。
6.先介绍了针对常系数系统的分岔行为进行控制的线性法、平移法以及状态反馈和参数调整控制法等三种方法。然后将上述三种控制分岔混沌的方法应用于周期系数系统,对周期系数系统平衡点失稳后发生的倍化分岔、Hopf分岔两种情形的分岔行为进行控制,从数值上验证其可操作性,得到了系统在控制前后的分岔图。对于倍化分岔控制：(1)采用线性法,当选取适当的控制参数,可将周期2点控制到周期1点,或控制到Hopf圈；(2)采用状态反馈和参数调整控制法,当选取适当的控制参数,可将周期2点控制到周期1点,或控制到Hopf圈、混沌状态；(3)平移法未能达到控制分岔的目的。对于Hopf分岔控制：(1)采用线性法,当选取适当的控制参数,可将Hopf圈控制到周期1点,或控制到另一个光滑Hopf圈或变形的Hopf分岔圈；(2)采用状态反馈和参数调整控制法,当选取适当的控制参数,可将Hopf圈控制到周期1点,或控制到另一个光滑Hopf圈、变形的Hopf分岔圈、周期7点或混沌状态；(3)采用平移法未能达到控制分岔的目的。
This dissertation considered three high dimensional systems. First as an example, a map was considered to study the bifurcations and chaos when the fixed points lost their stability. Then it focused on two kinds of period-doubling bifurcations and chaotic behaviors in a two-degree-of-freedom-vibro-impact system when the periodic solutions lost its stability. Finally a mechanical system with periodic coefficients was investigated. Its differential equations were established. The perturbed differential equations with periodic coefficients of its periodic motions were derived. Moreover stability, bifurcations, chaos and control were addressed. Numerical simulation results were presented. The main respects of the research are followings:
1. It was surveyed in chapter1that some recent achievements and developments and of the theoretic research and engineering applications on the stability, bifurcations, chaos and control of maps, vibro-impact systems and dynamics systems with periodic coefficients. The main contents and results of the dissertation were introduced.
2. A three-dimensional map was considered. Some kinds of bifurcations were investigated while the fixed points losing their stability. The corresponding analytic conditions which the variant parameters satisfy were derived when different kinds of bifurcations occur. A procedure to compute Hopf-Flip bifurcations of fixed points for3-dimensional maps was established by using projection technique. The normal form coefficients are computed. Finally, numerical simulation results demonstrated it. The typical period-doubling bifurcation leading to chaotic was found while the fixed points losing their stability. There was also a period-doubling bifurcation of tori leading to chaotic via curve doubling when the Hopf invariant circle lost its stability. As the control parameters were varied, it can vary from smooth invariant circle to tori doubling-tori breaking-smooth invariant circle. The fixed point would lock into one attractive invariant circle via four orbits while Hopf bifurcation occurs under strong resonant conditions (q=1). The attractive invariant circle would break into four quasi-periodic invariant circles as the control parameters vary. Moreover the four quasi-periodic invariant circles would lock into four fixed points as the parameters vary. Two chaotic attractor occured under non-resonant Hopf-Flip conditions. Six isolated chaotic attractors or six fixed points occured under weak-resonant Hopf-Flip conditions (λ06=1). Four connected or non-connected belt chaotic attractors occured under strong-resonant Hopf-Flip conditions (λ04＝1).
3. The typical period-doubling bifurcation leading to chaotic in a two-degree-of-freedom-vibro-impact system when the periodic solutions lost its stability was investigated theoretically and by means of numerical simulations. It was show that there was also non-typical period-doubling bifurcation leading to chaotic. Quasi-periodic motion occured while the periodic1-1impact motion losing its stability under non-resonant Hopf conditions (but near strong-resonant or weak-resonant zone, Hopf invariant circle occurs on the Poincare section while the fixed points losing their stability). If the system parameters varied near the resonant point, because of the existence of Arnold tongue, the fixed point would lock into one Hopf invariant circle via n orbits while losing its stability. As the control parameters vary across n's order resonant parameters area (Arnold tongue), the fixed point will lock into n fixed point by sub-harmonic bifurcation. As the parameters vary further, the n fixed point will lock into periodic2n point by sub-harmonic bifurcation. And periodic4n point, periodic8n point, periodic16n point,..., chaotic will occur. Finally there was one period-doubling bifurcation leading to chaotic which had n branches.
4. A two-degree-of-freedom mechanical system was investigated. The differential motion equations were established by making use of Lagrange's equations. When the endpoint of the arm performs a prescribed motion, the undisturbed motion can be given by making use of the relations of edges and angles, and the differential variable denoting by the endpoint's coordinates and their derivatives. The disturbed motion equations of implicit type can be derived while the prescribed motions were disturbed. Assuming the disturbance, the first-order approximate disturbed motion equations of explicit type can be derived by making use of asymptotic method. And the second-order approximate disturbed motion equations, the third-order one,..., arbitrary nth order approximate disturbed motion equations of explicit type can also be obtained. So the analysis of stability and bifurcation of motion was converted to the analysis of stability and bifurcation of the equilibrium points of nth order approximate disturbed motion equations. For the disturbed motion equations, the stability and bifurcation types of the equilibrium points are depended on the first three order nonlinear terms. In other words, the fourth-order term and higher order term do not affect essentially the analysis of dynamical behaviors. The first six order nonlinear approximate disturbed motion equations were derived in this dissertation. The difference of the numerical simulation results between the six-order approximate disturbed motion equations and the third-order one was compared. The simulation results show that the conclusion is correct.
5. The relation of stability between linear differential equations with periodic coefficients and linear differential equations with constant coefficients was introduced. The stability criteria were given. The relation of stability between nonlinear systems with periodic coefficients and its corresponding linear systems was introduced. The stability criteria were given. A suitable map can be constructed making use of a series of integrating differential equations with periodic coefficients over ti=(i-1)xT->iT (i=1,…,n), which is the Poincare map on the Poincare section (σ={(φ10,φ10, φ20,φ20, t)∈R4×S t=T}). Several bifurcations conditions were given when the equilibrium points of the differential equations with periodic coefficients lost their stability. The analysis method studying of bifurcations of the differential equations with periodic coefficients is introduced. Numerical simulation results were presented. The Flip bifurcation, Hopf bifurcation under non-resonant conditions (λ0n(ε0)≠1, n≠2,3,4,…), Hopf bifurcation under strong-resonant conditions (λ3=1、λ2=1), Hopf-Flip bifurcation were obtained. Bifurcations behaviors of the differential systems with periodic coefficients are basically similar to the that of the map with constant coefficients. Two different results of Hopf-Flip bifurcation were given:One case is that it maybe form two quasi-periodic invariant circles or periodic n point when the equilibrium points lose their stability. Another case is that it can not any invariant circles or periodic n point. It is a particularcare when Hopf bifurcation occur under strong-resonant conditions (λ2=1). When the eigenvalues are close the unit circle, four negative real roots will be resolved into two negative real roots which magnitude are smaller then1and a pair of roots of complex is conjugate. Finally the roots of complex conjugate cross the unit circle and transform into a pair of-1root (the other two are negative real roots which magnitude are smaller then1). Because of the particularity of root crossing the unit circle, the bifurcation results are special:stable fixed point-periodic2focus points-stable Hopf invariant circle-periodic2node points-periodic4node points, periodic8node points, periodic16node points,..., periodic2n node points, and chaos. The phase diagram of period-doubling bifurcation leading to chaotic was given.
6. Three methods of control chaos for the dynamical system with constant coefficients were introduced:the variable-parameter linear controller, the translation, the state variables feedback and parameter variation. The control of delaying Flip bifurcation and Hopf bifurcation has been studied. The bifurcation phase diagrams are obtained. For Flip bifurcation control:(1) Adopting the variable-parameter linear controller, periodic2points can be controlled to fixed point taking proper control parameters. Otherwise it may be controlled to Hopf circle.(2) Adopting the state variables feedback and parameter variation, periodic2points can be controlled to fixed point taking suitable control parameters. Otherwise it may be controlled to Hopf circle or chaos state.(3) Adopting the translation, it is failed to achieve the control of Flip bifurcation. For Hopf bifurcation control:(1) Adopting the variable-parameter linear controller, Hopf circle can be controlled to fixed point taking proper control parameters. Otherwise it may be controlled to other smooth Hopf circle or deformed Hopf circle.(2) Adopting the state variables feedback and parameter variation, Hopf circle can be controlled to fixed point taking suitable control parameters. Otherwise it may be controlled to other smooth Hopf circle or deformed Hopf circle or periodic7points or chaos state.(3) Adopting the translation, it is failed to achieve the control of Hopf bifurcation.
引文
[1]陆启韶.分岔与奇异性.上海：上海科技教育出版社,1997.
[2]陈士华,陆君安.混沌动力学初步.武汉：武汉水利电力大学出版社,1998.
[3]卢侃.混沌动力学.上海：上海翻译出版公司,1990.
[4]郝柏林.从抛物线谈起—混沌动力学引论.上海科技教育出版社,1995.
[5]陈予恕,唐云,陆启韶等.非线性动力学中的现代分析方法.科学出版社,2000.
[6]李骊.强非线性振动系统的定性理论与定量方法.科学出版社,1997.
[7]胡海岩.应用非线性动力学.北京,航空工业出版社,2000.
[8]张筑生.微分动力系统原理.科学出版社,1999.
[9]Iooss G. Bifurcation of maps and applications. Math. Studies 36, North-Holland,1979.
[10]Kuznetsov Y.A. Elements of applied bifurcation theory (Second edition). Springer-Verlag, New York,1998.
[11]闻邦椿,刘树英.振动机械的理论与动态设计方法.机械工业出版社,2002.
[12]赵统武.冲击钻进动力学.冶金工业出版社,1996.
[13]Paget A.L. Vibration in steam turbine buckets and damping by impact. Engineering. 1937,143:305-307.
[14]Reed W.H. Hanging chain impact dampers-a simple method of damping tall flexible structures, Proceeding of International Research Seminar, Wind Effects on Buildings and Structures, Ottawa.1967,2:283-321.
[15]Cai Y. and Chen S.S. Nonlinear dynamics of loosely supported tubes in crossflow. Journal of Sound and Vibration.1993,168(3):449-468.
[16]Laggiard E., and Runkel J. One-dimensional bimodel of vibration and impacting of instrument tubes in a boiling water reactor. Nuclear Science and Engineering.1993,115: 62-70.
[17]Feng Q. and Pfeiffer F. Stochastic model on a rattling system. Journal of Sound and Vibration.1998,215(3):439-453.
[18]Pfeiffer F. and Prestl W. Hammering in diesel-engine driveline systems. Nonlinear Dynamics.1994,5:477-492.
[19]Babistky V. Theory of vibro-impact systems and applications. Springer:Berlin,1998.
[20]Plaut R.H. and Farmer A.L. Large motion of a moored floating breakwater modeled as an impact oscillator. Nonlinear Dynamics.2000,23:319-334.
[21]Plaut R.H., Fanner A. L. and Holland M.M. Bouncing-ball model of 'dry' motions of a tethered buoy. Journal of Vibration and Acoustics.2001,123:333-339.
[22]Stepanenko Y. and Sankar T.S. Vibro-impact analysis of control systems with mechanical clearance and it application to robotics actuators, Journal of Dynamic System Measurement and Control.1986,108(1):9-16.
[23]Sinha S.C., Redkar S. and Eric A. Butcher. On macromodeling of nonlinear systems with time periodic coefficients. Communications in Nonlinear Science and Numerical Simulation.2006,11:510-530.
[24]May R. Simple mathematical modds with very complicated dymamics. Nature.1976, 261:459-469.
[25]Cushing J.M. An introduction to structured population dynamics. SIMAM, Philadelphia, PA,1998.
[26]Rogers T.D. Chaos in systems in population biology. Prog Theoret Biol.1981,6:1-144.
[27]傅湘陵,蒋艳,王卫兵.具周期系数离散型单物种竞争模型的稳定周期解.湖南科技大学学报(自然科学版).2010,25(4)：126-128.
[28]李海龙.一类具周期系数的单种群模型及其最优收获策略.生物数学学报.1999,14(4)：479-483.
[29]Meng F. and Wang K. Optimal harvesting policy for single population with periodic coefficients. Mathematical Biosciences.1998,152:165-177.
[30]Glipin M.E. and Aynala F.J. Global models of growth in biology and competition. Proc Nat Acad Sci.1973,70(12):3590-3593.
[31]马知恩.种群生态学的数学建模与研究.安徽：安徽教育出版社,1996.
[32]傅廷强.时变环境Lotka-Volterra系统的渐近性态.生物数学学报,1996,11(1)：32-37.
[33]熊佐亮.关于三种群第Ⅱ类功能反应周期系数模型的研究.生物数学学报.1998,13(1)：38—42.
[34]Arnold V.I. Geometrical methods in the theory of ordinary differential equations. Springer-Verlag, New York, Berlin Heidelberg Tokyo,1983.
[35]Wan Y.H. Bifurcation into invariant tori at points of resonance. Archive for Rational Mechanics Analysis.1978,68:343-357.
[36]Guckenheimer J. and Holmes P.J. Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer-Verlag, New York,1983.
[37]Chenciner A. Bifurcation de point fixes elliptiques I, Courbes invariants. Publications Mathematiques, Institut des Hautes Etudes Scientifiques.1985,61:67-127.
[38]Los J.E. Non-normally hyperbolic invariant curves for maps in R3 and doubling Bifurcation. Nonlinearity.1989,2(1):149-174.
[39]Iooss G. and Los J.E. Quasi-genericity of bifurcations to high dimensional invariant tori for maps. Communications in Mathematical Physics.1988,119:453-500.
[40]Chenciner A and Iooss G. Bifurcation of a Torus T2 into a Torus T3. Comptes Rendus de l'Academie des Science Paris.1977,284A.
[41]Chenciner A. and Iooss G. Bifurcation de torus invariant. Archive for Rational Mechanics and Analysis.1979,69:109-198.
[42]Iooss G. and Joseph D.D. Bifurcation and Stability of nT-Period Solutions Branching from T-Periodic Solutions at Points of Resonance. Archive for Rational Mechanics and Analysis.1978,66:135-172.
[43]Iooss G. Some codimension-two bifurcations for maps, leading to chaos. In Chaos and Statistical Methods, Kuramoto, Y.(ed.), Springer-Verlag, New York,1984:136-142.
[44]Iooss G. Persistance d'un cercle invariant par une application voisine de "l'application temps r"d'un champ de vecteurs intergrable. Partie I:En dehors de la bifurcation de Hopf. Comptes Rendus de l'Academie des Sciences Paris,Serie I.1983,296:27-30.
[45]Iooss G. Forms normales d'applications:caracterisation globale et methode de calcul. Uninersite de Nice,1987.
[46]Elphick C., Tirapegu E., Brachet M. and Iooss G. A Simple Global Characterization for Normal Forms of Singular Vector Fields. Physic D.1987,29:95-127.
[47]Iooss G, Arneodo A., Coullet P. and Tresser C. Simple computation of bifurcating invariant circles for mapping. Dynamical systems and turbulence. Lecture notes in Math.1981,898:192-211.
[48]Aronson D. G., Chory M.A., Hall G.R. and McGehee R.P. Bifurcations from an invariant circle for two-parameter families of maps of the plane:a computer-assisted study. Communications in Mathematical Physics.1982,83:303-354.
[49]谢建华.三次映射的全局分叉与混沌.西南交通大学学报,1994,29(1)：30-38.
[50]谢建华,罗冠炜.关于平面映射次谐分岔定理的一个注记.力学学报.2002,34(增)：147-150.
[51]Xie J.H., Ding W.C., Dowell E.H., Virgin L.N. Hopf-flip bifurcation of high dimensional maps and application to vibro-irnpact systems. Acta Mech Sinica.2005, 21:402-410.
[52]谢建华.映射的分叉理论及其在碰撞振动系统中的应用.西南交通大学博士学位论文.1993.
[53]Wen G.L., Xu D.L. and Xie J.H. Control of Degenerate Hopf Bifurcations in Three-Dimensional Maps. Chaos.2003,13(2):486-494.
[54]徐慧东,谢建华.高维映射的Hopf-pitchfork分岔及其在碰撞振动系统中的应用.应用力学学报.2008,25(2)：268-273.
[55]徐慧东,谢建华.一类高维映射不变圈的计算.动力学与控制学报.2007,5(3)：204-208.
[56]徐慧东.高维映射的局部余维二分岔(Ⅱ).兰州铁道学院学报(自然科学版).2003,22(4)：13-17.
[57]李恩颖,文桂林,李光耀Henon映射的Hopf分岔分析及PSpice电路模拟.湖南大学学报(自然科学版),2007,34(9)：26-29.
[58]张建刚.高维映射的局部余维二分岔(Ⅰ).兰州铁道学院学报(自然科学版).2003,22(4)：7-12.
[59]Feigenbaum M.J., Kadanoff L.P. and Shenker S.J. Quasi periodicity in dissipative systems:a renormalization group analysis. Phys.1982, D5:370.
[60]Feigenbaum M.J. Quantitative universality for a class of nonlinear transformations. Journaf of Statisticaf Physics.1978,19(1):25-52.
[61]Alexander J. Dynamical systems. New York:Springer,1988.
[62]Sakaguchi H. and Tomita K. Bifurcation of the coupled Logistic map. Progress of Theoret. Phys.1987,78(2):305-309.
[63]Ferretti A. and Rahman N.K. A study of coupled Logistic map and its applications in chemical physics. Chemical Physics.1988,119:275-288.
[64]Welstead S.T. and Cromer T.L. Coloring periodicities of two-dimensional Mappings. Computers & Graphics.1989,14(1):125-126.
[65]孟湛祥.耦合逻辑斯蒂映射中的一个混沌吸引子.天津大学学报.1999,32(6)：711-715.
[66]王兴元,石其江.二维logistic映射中的一种新型激变、回滞和分形.应用力学学报.2005,22(4),501-506.
[67]王兴元,粱庆永.复合Logistic映射中的逆分岔与分形.力学学报.2005,37(4),522-528.
[68]王兴元,骆超.二维Logistic映射的分岔与分形.力学学报.2005,37(3),346-354.
[69]赵明超,王凯华,傅新楚.二维耦合Logistic映射的动力学性质及在流密码生成中的应用.应用数学与计算数学学报.2009,23(1)：87-92.
[70]舒仲周,谢建华.碰撞振动的稳定性.西南交通大学学报.1985,(3)：14-25.
[71]谢建华,舒仲周.一类无穷维动力系统的分叉问题.西南交通大学学报.1994,29(1)：7-13.
[72]谢建华.振动锤的数学模型与全局分叉.力学学报.1997,29(4)：456-463.
[73]谢建华,葛玉梅.冲击式振动落砂机的动态响应与分叉.振动工程学报.1993,6(1)：90-95.
[74]谢建华,罗冠炜.碰撞振动系统的分叉与混沌研究.见：胡海岩,靳征谟主编：《面向21世纪的中国振动工程研究》,航空工业出版社,1999：52-58.
[75]Xie J.H. Codimension Two Bifurcation and Hopf Bifurcation of an Impacting Vibrating System. Applied Mathematics and Mechanics.1996,17:65-75.
[76]谢建华,郑小武.惯性式冲击振动落砂机周期运动的Hopf分叉.振动工程学报.1999,12(3)：297-303.
[77]郑小武.一类碰撞振动系统周期运动的Hopf分叉.西南交通大学硕士学位论文,1997.
[78]谢建华,文桂林,肖健.两自由度碰撞振动系统分叉参数的确定.振动工程学报.2001,14(3)：249-253.
[79]Luo G.W. and Xie J.H. Hopf Bifurcation of Two- Degree-of-Freedom Vibro-Impact System. Journal of Sound and Vibration.1998,213(3):391-408.
[80]Luo G.W. and Xie J.H. Bifurcations and Chaos in a System with Impacts. Physica D. 2001,(148):183-200.
[81]Luo G.W. and Xie J.H. Hopf Bifurcations and Chaos of a Two-Degree-of-Freedom Vibro-Impact System in Two Strong Resonance Cases, International Journal of Non-Linear Mechanics.2002,37(1):19-34.
[82]Luo G.W. and Xie J.H. and Guo SH.L. Periodic Motions and Global Bifurcations of a Two-Degree-of-Freedom System with Plastic Vibro-Impact. Journal of Sound and Vibration.2001,240(5):837-858.
[83]罗冠炜,谢建华,孙训方.单自由度塑性碰撞振动系统的周期运动及其分叉特点.中国机械工程.2001,12(11)：1297-1300.
[84]Luo G.W. and Xie J.H. The Codimension two Bifurcation of Periodic Vibro-Impact and Chaos of a Dual Component System. Physics Letters A.2003,313(4):267-273.
[85]Wen G.L. Codimension-2 Hopf Bifurcation of a Two- Degree-of-Freedom Vibro-Impact System. Journal of Sound and Vibration.2001,242(3):475-485.
[86]文桂林.碰撞振动系统的复杂Hopf分岔与混沌研究.西南交通大学博士学位论文.2001.
[87]罗冠炜,褚衍东,苟向锋.一类碰撞振动系统的概周期运动及混沌形成过程.机械强度.2005,27(4)：445-451.
[88]罗冠炜.两自由度碰撞振动系统的Hopf分岔与混沌研究.西南交通大学博士学 位论文,1998.
[89]罗冠炜,谢建华.碰撞振动系统的周期运动和分岔.科学出版社,2002.
[90]Wen G.L. and Xie J.H. Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-of-Freedom Vibro-Impact System. ASME Journal of Applied Mechanics.2001,68(4):670-674.
[91]丁旺才,谢建华.碰撞振动系统的一类余维二分岔及T2环面分岔.力学学报.2003,34(4)：503-508.
[92]Xie J.H. and Ding W.C. Hopf-Hopf bifurcation and invariant torus T2 of a vibro-impact system. International Journal of Non-Linear Mechanics.2005,40: 531-543.
[93]Ding W.C. and Xie J.H. Torus T2 and its routes to chaos of a vibro-impact system. Physics Letters A.2006,349:324-330.
[94]丁旺才,谢建华,李国芳.三自由度碰撞振动系统的周期运动稳定性与分岔.工程力学.2004,21(3)：123-128.
[95]丁旺才,谢建华,李万祥.碰撞振动系统强共振下的两参数动力学分析.计算力学学报.2004,21(6)：658-664.
[96]丁旺才.多自由度碰撞振动系统的环面分岔与混沌研究.西南交通大学博士学位论文,2004.
[97]乐源,谢建华.一类双面碰撞振子的对称性、尖点分岔与混沌.应用数学和力学.2007,8：991-998.
[98]Yue Y. and Xie J.H. Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides. Applied Mathematics and Mechanics.2007,8:1109-1117.
[99]乐源.多自由度碰撞振动系统的对称性、分岔与混沌研究.西南交通大学博士学位论文.2008.
[100]乐源,谢建华.一类三自由度碰撞振动系统的Poincare映射的对称性,分岔及混沌,四川大学学报(工程科学版).2008,40(1)：27-31.
[101]乐源,谢建华.两自由度碰撞振动系统的Poincare映射的对称性及分岔,振动工程学报.2008,21(4)：376-380.
[102]乐源,谢建华,丁旺才.一类两自由度碰撞振动系统的Hopf分岔和混沌,动力学与控制学报.2004,2(3)：36-41.
[103]徐慧东,谢建华.一类两自由度分段线性非光滑系统的分岔与混沌.振动工程学报.2008,21(3)：279-284.
[104]徐慧东.非光滑动力系统周期解的分岔研究.西南交通大学博士学位论文.2008.
[105]徐慧东,谢建华.单自由度含间隙分段线性系统周期运动的倍化分岔.西南交通大学学报.2008,43(2)：227-231.
[106]张艳龙,徐慧东.冲击振动落砂机在1：4强共振点附近的动力学特性.工程力学.2008,25(8)：194-199.
[107]June Y.L. and Yan J.J. Control of impact oscillator. Chaos, Solitons and Fractals. 2006,28:136-142.
[108]Hubler A. and Luscher E. Resonant stimulation of complex systerms. Helv. Phys. Acts. 1989,62:544.
[109]胡岗,萧井华,郑志刚.混沌控制.上海：上海科技教育出版社,2000.12
[110]Ott E., Grebogi C. and York J.A. Controlling chaos. Phys Rev Lett.1990,64(11): 1196-1199.
[111]Kazuyuki Y. and Tomotsugu U. A new approach for controlling chaotic dynamical system. Physics Letters A.1998,238:349-357.
[112]Hu H.Y. Controlling chaos of a dynamical system with discontinuous vector field. Physics D.1997,106:1-8.
[113]关新平,范正平,张群亮,王益群.连续时间稳定线性系统的混沌反控制研究.物理学报.2002,51(10)：2216-2220.
[114]任海鹏,刘丁,韩崇昭.基于直接延迟反馈的混沌反控制.物理学报.2006,55(6)：2694-2701.
[115]陶建武,石要武,常文.端口受控哈密顿系统的混沌反控制研究.物理学报.2004,53(6)：1682-1686.
[116]丁洪洁,吕和祥.压缩映射—参数微扰控制混沌.振动工程学报.2003,16(2)：212-218.
[117]丁洪洁,吕和祥.小参数干扰反馈控制动力系统中混沌运动.大连理工大学学报.2003,43(2)：132-135.
[118]蔡国梁,谭振梅,周维怀,涂文桃.一个新的混沌系统的动力学分析及混沌控制.物理学报.2007,56(11)：6230-6237.
[119]Ushio T. Chaotic synchronization and controlling chaos based On contraction mappings. Phys. Lett. A.1995,198:14-22.
[120]Fowler T.B. Application of stochastic control technique to chaotic nonlinear system. IEEE Transactions on Automatic Control.1989,34:205-310.
[121]Kapitaniak T. Controlling chaos without feedback. Chaos, Solitons & Fractals.1992, 2:519-530.
[122]Kapitaniak T. Controlling chaos without feedback and control signals. International Journal of Bifurcation and Chaos.1993,3:459-468.
[123]Hu H.Y. Semmi-active vibration control based on a vibration absorber with adjustable clearance. Transactions of Nanjing University of Aeronautics & Astronautics.1996, 13:135-140.
[124]胡海岩.引导混沌运动到周期运动的自适应控制策略.力学学报.1997,29：631-635.
[125]马永靖,丁旺才,杨小刚.碰撞振动系统的参数自调节混沌控制.振动与冲击.2007,26(1)：26-34.
[126]丁旺才,马永靖,王靖岳.碰撞振动系统的状态预测反馈控制.振动工程学报.2007,20(6)：589-593.
[127]Silvio L. T. de Souza and Ibere L. Caldas. Controlling chaotic orbits in mechanical system with impacts. Chaos, Solitoms and Fractals.2004,19:171-178.
[128]徐慧东,谢建华.一类单自由度分段线性系统的分岔与混沌控制.振动与冲击.2008,27(6)：20-24.
[129]于津江,刘学军,韩万强,许海波.平移法控制离散系统的倍周期分岔和混沌.计算物理.2007,24(1)：116-120.
[130]唐驾时,欧阳克俭Logistic模型的倍周期分岔控制.物理学报.2006,55(9)：437-431.
[131]罗晓曙,陈关荣,汪秉宏,方锦清,邹艳丽,全宏俊.状态反馈和参数调整控制离散非线性系统的倍周期分岔和混沌.物理学报.2003,52(4)：790-794.
[132]徐慧东,张建刚,褚衍东.一类Mathieu方程的混沌控制.西安理工大学学报.2006,22(2)：167-170.
[133]褚衍东,李险峰,张建刚.一类特殊的Mathieu方程的分岔及混沌控制.青岛大学学报(自然科学版).2006,19(3)：30-33.
[134]Kan I., Kocak H. and Yorke J. Antimonotonicity:concurrent creation and annihilation of periodic orbits. Annals of Mathematics.1992,136:219-252.
[135]Kuruvilla T. and Nandakumaran V.M. Suppression of chaos through reverse period doubling in coupled directly modulated semiconductor laser. Physics Letters A.1999, 254:59-64.
[136]舒仲周.运动稳定性.峨眉：西南交通大学出版社,1989.
[137]舒仲周,张继业,曹登庆.运动稳定性.北京：中国铁道出版社,2001.
[138]廖晓昕.动力系统的稳定性理论和应用.北京：国防工业出版社,2000.
[139]廖晓昕.稳定性的数学理论和应用.武汉：华中师范大学出版社,1998.
[140]舒仲周,王照林.运动稳定性的研究进展和趋势.力学进展.1993,23(3)： 424-431.
[141]马知恩,周义仓.常微分方程定性与稳定性方法.北京：科学出版社,2001.
[142]高为炳.运动稳定性基础.北京：高等教育出版社,1987.
[143]秦元勋,王慕秋,王联.运动稳定性理论与应用.北京：科学出版社,1981.
[144]秦元勋.运动稳定性的一般问题讲义.北京：科学出版社,1958.
[145]秦元勋,刘永清,王联.带有时滞的动力系统的稳定性.北京：科学出版社,1958.
[146]许淞庆.常微分方程稳定性理论.上海：上海科学技术出版社,1965.
[147]H.г.契塔耶夫.运动的稳定性.北京：国防工业出版社,1965.
[148]Guilhen P.M., Bertheier P., Ferraris G. and Lalanne M. Dynamic behavior of dissymmetric rotor bearings modeled periodic coefficient large system, in: Proceedings of the 58th Shock and Vibration Symposium, Huntsville, AL,1987.
[149]Sinha S.C. and Pandiyan R. Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov-Floquet transformation. International Journal of Nonlinear Mechanics.1994,29(5):687-702.
[150]Sinha S.C., Pandiyan R. and Bibb J.S. Liapunov-Floquet transformation:computation and applications to periodic systems. Journal of Vibration and Acoustics.1996,118: 209-219.
[151]Deshmukh V., Sinha S.C. and Joseph P. Order reduction and control of parametrically excited dynamical systems,Journal of Vibration and Control.2000,6:1017-1028.
[152]Sinha S.C, Redkar S., Butcher E. and Deshmukh V. Order reduction of nonlinear time periodic systems using invariant manifolds, in:The Proceedings of 19th Biennial Conference on Mechanical Vibration and Noise, ASME 2003 DETC, Chicago, IL, USA,2003,48445.
[153]Sinha S.C and Chou C.C. An approximate analysis of transient response of time-dependent linear systems by use of orthogonal polynomials. Journal of Sound and Vibration.1976,49(3):309-326.
[154]Sinha S.C., Chou C.C. and Denman H.H. Stability analysis of systems with periodic coefficients:an approximate approach. Journal of Sound and Vibration.1979,64(4): 515-527.
[155]Sinha S C and Wu D H. An efficient computational scheme for the analysis of periodic systems. Journal of Sound and Vibration.1991,151(1):91-117.
[156]Sinha S.C., Der-Ho Wu, Juneja V. and Joseph P. Analysis of dynamic systems with periodically varying parameters via Chebyshev polynomials. Transactions of the ASME.1993,115:96-102.
[157]Sinha S.C. and Joseph P. Control of general dynamic systems with periodically varying parameters via Liapunov-Floquet transformation. Journal of Dynamic Systems, Measurement, and Control.1994,116:650-658.
[158]Butcher E.A. and Sinha S.C. A hybrid formulation for the analysis of time-periodic linear systems via Chebyshev polynomials. Journal of Sound and Vibration.1996, 195(3):518-527.
[159]Sinha S.C. and Butcher E.A. Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems. Journal of Sound and Vibration.1997, 206(1):61-85.
[160]Sinha S.C., Marghitu D B and Boghiu D. Stability and control of a parameterically excited rotating beam. Journal of Dynamic Systems, Measurement, and Control. 1998,120:462-470.
[161]Alexandra D. and Sinha S.C. Bifurcation control of nonlinear systems with time-periodic coefficients. Proceedings of the American Control Conference.1999, 99:3057-3061.
[162]Sinha S.C., Henrichs J.T. and Ravindra B. A general approach in the design of active controllers for nonlinear systems exhibiting chaos. International Journal of Bifurcation and Chaos.2000,10(1):165-178.
[163]Alexandra D. and Sinha S.C. Versal deformation and local bifurcation analysis of time-periodic nonlinear systems. Nonlinear Dynamics.2000,21:317-336.
[164]Alexandra D. and Sinha S.C. Bifurcation control of nonlinear systems with time-periodic coefficients. Journal of Dynamic Systems, Measurement, and Control. 2003,125:541-548.
[165]Sinha S.C., Redkar S. and Eric A. Butcher. Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. Journal of Sound and Vibration. 2005,284:985-1002.
[166]Sinha S.C., Gourdon E. and Zhang Y. Control of time-periodic system via symbolic computation with application to chaos control. Communications in Nonlinear Science and Numerical Simulation.2005,10:835-854.
[167]Hsu C.S. On approximating a general linear periodic system. Journal of Mathematical Analysis and Applicationss.1974,45; 234-251.
[168]Friendmann P., Hammond C.E. and Woo T.H. Efficient numerical treatment of periodic systems with application to stability problems. International Journal for Numerical Methods in Engineering.1977,11:1117-1136.
[169]Arneodo A., Coullet P.H. and Spiegel E.A. Cascade of period doublings of tori. Physics Letters.1983,94A(1):1-5.
[170]Yedavalli P.K. and Kolla S.R. Stability analysis of linear time-varying systems. International Journal of Systems SCI.1988,19(9):1853-1858.
[171]Lindtner E., Steindl A. and Troger H. Generic one-parameter bifurcations in the motion of a simple robot. Journal of Computational and Applied Mathematics.1989, 26:199-218.
[172]Guttalu R.S. and Flashner H. Stability analysis of periodic systems by truncated point mappings. Journal of Sound and Vibration.1996,189(1):33-54.
[173]Cai Zh.Q., Gu Y.X. and Zhong W.X. A new approach of computing Floquet transition matrix. Computers and Structures.2001,79:631-635.
[174]Yaghoobi H. and Abed E.H. Local feedback control of the Neimark-Sacker bifurcation. International Journal of Bifurcation and Chaos.2003,13(4):879-893.
[175]Venkatesh Y.V. On the stability of a nonlinear system with periodic coefficients. Proceedings of the IEEE.1974,30:278-279.
[176]Willems J.C. On the asymptotic stability of the null solution of linear differential equations with periodic coefficients. IEEE Transactions on Automatic Control. 1968,AC-13(1):65-72.
[177]Thompson J.M.T. and Ghaffari. Chaos after period-doubling bifurcations in the resonance of an impact oscillator. Physics Letters.1982,91 A(1):5-8.
[178]Thompson J.M.T. Complex dynamics of compliant off-shore structures. Proc. R. Lond.1983, A387:407-427.
[179]Kaneko K. Oscillation and doubling of torus. Progess of Theoretical Physics.1984, 72(2):202-215.
[180]Wu W.T., Griffin J.H. and Wickert J.A. Perturbation method for the Floquet eigenvalues and stsbility boundary of periodic linear systems. Journal of Sound and Vibration.1995,182(2):245-257.
[181]Liu Zh.L. Periodic solutions for nonlinear nth order ordinary differential equations. Journal of Mathematical Analysis and Applications.1996,204:46-64.
[182]Liu B. and Cheng S.S. Note on a nonlinear difference equation with periodic coefficients and constant delays. Appl. Math. Lett.1998,11(5):39-43.
[183]Shnol E.E. Stability of linear almost-Hamiltonian periodic systems. J. Appl. Maths. Mechs.1996,60(6):933-939.
[184]Bose T., Member S., Chen M.Q., Joo K.S. and Xu G.F. Stability of two-dimensional discrete systems with periodic coefficients. IEEE Transactions on Circuits and Systems-II:Analog and Digital Signal Processing.1998,45(7):839-847.
[185]Zevin A. A. Stability regions of linear canonical systems with periodic coefficients. J. Appl. Maths. Mechs.1998,62(1):37-42.
[186]Veovoda A.A, Khassoounekh V. On new equivalent mataix model for systems with periodic coefficients. Proceedings of the IEEE.2000,287:170-172.
[187]Mahmoud G.M., Bountis T. and Ahmed S.A. Stability analysis for systems of nonlinear Hill's equations. Physica A.2000,286:133-146.
[188]Natsiavas S., Theodossiades S. and Goudas I. Dynamic analysis of piecewise linear oscillators with time periodic coefficients. International Journal of non-linear mechanics.2000,35:53-68.
[189]Andrey V.S. Lie-and chronologico-algebraic tools for studying stability of time-varying systems. Systems & Control Letters.2001,43:59-76.
[190]Alexandra Z. and Pinsky M. Stability criteria for linear systems with partly uncertain periodic coefficients. Society for Industrial and Applied Mathematics.2003,42(4): 1185-1197.
[191]Montagnier P., Christopher C.P. and Spiteri R.J. Real Floquet factors of linear time-periodic systems. Systems & Control Letters.2003,50:251-262.
[192]Tyshchenko V.Y. Equivalence of Riccati equations with periodic coefficients. Translated from Differentsial'nye Uravneniya.2003,39(4):565-567.
[193]Hong J.L. and Liu Y. A novel numerical approach to simulating nonlinear Schrodinger equations with varying coefficients. Applied Mathematics Letters.2003, 16:759-765.
[194]Yuan R. On the second-order differential equation with piecewise constant argument and almost periodic coefficients. Nonlinear Analysis.2003,52:1411-1440.
[195]Yenicerioglu F A and Talcinbas S. On the stability of the second-order delay differential equations with variable coefficients. Applied Mathematics and Computation.2004,152:667-673.
[196]Zevin A.A. A new approach to the theory of the stability of linear canonical systems of differential equations with periodic coefficients. J. Appl. Maths. Mechs.2004, 68(2):183-198.
[197]Ma H.T., Deshmukh V, Butcher E and Averina V. Delayed state feedback and chaos control for time-periodic systems via a symbolic approach. Communications in Nonlinear Science and Numerical Simulation.2005,10:479-497.
[198]Gaishun I.V. Stability of linear Hamiltonian systems in total differentials with periodic coefficients. Translated from Differentsial'nye Uravneniya.2005,41(1): 33-40.
[199]Gergely R. Neimark-Sacker bifurcation for periodic delay differential equations. Nonlinear Analysis.2005,60:1025-1044.
[200]Liu Y.J., Ge W.G. and Gui Zh.J. Three positive periodic solutions of nonlinear differential equations with periodic coefficients. Analysis and Applications.2005, 3(2):145-155.
[201]冯兆生,张培军,张后扬.关于周期系数Abel方程的周期解问题.北方交通大学学报.1998,22(2)：9-11.
[202]徐鉴,陈予恕.非自治时滞反馈控制系统的周期解分岔和混沌.力学学报.2003,35(4)：443-451.
[203]黎雄.周期系数的高维Riccati方程的周期解.数学进展.1999,28(4)：313-322.
[204]徐君祥,尤建功.关于几乎周期系数的线性微分方程的可约化性.数学年刊.1996,17A(5)：607-616.
[205]冯立新,马富明.解一类具有周期系数的Helmholtz方程的Galerkin谱方法.吉林大学学报(理学版).2003,41(3)：253-258.
[206]Li X.M. Oscillation of neutral differential-difference equations with positive and negative periodic coefficients. Chinese Quarterly Journal of Mathematics.2000, 15(2):17-22.
[207]Li X.M. and Huang L.H. Oscillation and linearized oscillations for difference equations with positive and negative periodic coefficients. Mathematica Applicata. 2001,14(4):26-30.
[208]Huang Zh.K. and Chen F.D. Global asymptotic stanility of the type II functional response of three-species model with almost periodic coefficients. Journal of Mathetical Study.2003,36(2):124-132.
[209]王天成.线性周期系数系统的稳定性.烟台师范学院学报(自然科学版)1999,15(4)：246-249.
[210]陈茂海.具有周期系数线性微分方程的性质.工科数学.2001,17(1)：94-96.
[211]杨秀香.周期系数非线性差分系统的稳定性.渭南师范学院学报.2000,15(5)：33-34.
[212]徐俊峰,刘名生.一类周期线性微分方程的复振荡.华南师范大学学报(自然 科学版).2004,3：11-15.
[213]朱萌纾.一类双周期系数Riccati方程的单值群的性质.南京师大学报(自然科学版).2002,25(3)：44-48.
[214]汤光宋,熊炎.具有周期系数的几类线性微分系统的周期解.南都学坛.1990,3：46-53.
[215]李雪臣,王鸿燕.具周期系数时滞差分方程的全局渐近稳定性.扬州大学学报(自然科学版).2005,8(1)：14-17.
[216]张海燕,唐友刚,陈芳启.非线性Mathieu方程的局部分岔和在余维2退化点的Hopf分岔.机械强度.2007,29(5)：717-721.
[217]洪灵,徐建学.非线性强迫Mathieu方程的激变和瞬态混沌.力学学报.2001,33(3)：423-429.
[218]Troger H. and Steindl A. Nonlinear stability and bifurcation theory. Springer-verlag, 1991.
[2]陈士华,陆君安.混沌动力学初步.武汉：武汉水利电力大学出版社,1998.
[3]卢侃.混沌动力学.上海：上海翻译出版公司,1990.
[4]郝柏林.从抛物线谈起—混沌动力学引论.上海科技教育出版社,1995.
[5]陈予恕,唐云,陆启韶等.非线性动力学中的现代分析方法.科学出版社,2000.
[6]李骊.强非线性振动系统的定性理论与定量方法.科学出版社,1997.
[7]胡海岩.应用非线性动力学.北京,航空工业出版社,2000.
[8]张筑生.微分动力系统原理.科学出版社,1999.
[9]Iooss G. Bifurcation of maps and applications. Math. Studies 36, North-Holland,1979.
[10]Kuznetsov Y.A. Elements of applied bifurcation theory (Second edition). Springer-Verlag, New York,1998.
[11]闻邦椿,刘树英.振动机械的理论与动态设计方法.机械工业出版社,2002.
[12]赵统武.冲击钻进动力学.冶金工业出版社,1996.
[13]Paget A.L. Vibration in steam turbine buckets and damping by impact. Engineering. 1937,143:305-307.
[14]Reed W.H. Hanging chain impact dampers-a simple method of damping tall flexible structures, Proceeding of International Research Seminar, Wind Effects on Buildings and Structures, Ottawa.1967,2:283-321.
[15]Cai Y. and Chen S.S. Nonlinear dynamics of loosely supported tubes in crossflow. Journal of Sound and Vibration.1993,168(3):449-468.
[16]Laggiard E., and Runkel J. One-dimensional bimodel of vibration and impacting of instrument tubes in a boiling water reactor. Nuclear Science and Engineering.1993,115: 62-70.
[17]Feng Q. and Pfeiffer F. Stochastic model on a rattling system. Journal of Sound and Vibration.1998,215(3):439-453.
[18]Pfeiffer F. and Prestl W. Hammering in diesel-engine driveline systems. Nonlinear Dynamics.1994,5:477-492.
[19]Babistky V. Theory of vibro-impact systems and applications. Springer:Berlin,1998.
[20]Plaut R.H. and Farmer A.L. Large motion of a moored floating breakwater modeled as an impact oscillator. Nonlinear Dynamics.2000,23:319-334.
[21]Plaut R.H., Fanner A. L. and Holland M.M. Bouncing-ball model of 'dry' motions of a tethered buoy. Journal of Vibration and Acoustics.2001,123:333-339.
[22]Stepanenko Y. and Sankar T.S. Vibro-impact analysis of control systems with mechanical clearance and it application to robotics actuators, Journal of Dynamic System Measurement and Control.1986,108(1):9-16.
[23]Sinha S.C., Redkar S. and Eric A. Butcher. On macromodeling of nonlinear systems with time periodic coefficients. Communications in Nonlinear Science and Numerical Simulation.2006,11:510-530.
[24]May R. Simple mathematical modds with very complicated dymamics. Nature.1976, 261:459-469.
[25]Cushing J.M. An introduction to structured population dynamics. SIMAM, Philadelphia, PA,1998.
[26]Rogers T.D. Chaos in systems in population biology. Prog Theoret Biol.1981,6:1-144.
[27]傅湘陵,蒋艳,王卫兵.具周期系数离散型单物种竞争模型的稳定周期解.湖南科技大学学报(自然科学版).2010,25(4)：126-128.
[28]李海龙.一类具周期系数的单种群模型及其最优收获策略.生物数学学报.1999,14(4)：479-483.
[29]Meng F. and Wang K. Optimal harvesting policy for single population with periodic coefficients. Mathematical Biosciences.1998,152:165-177.
[30]Glipin M.E. and Aynala F.J. Global models of growth in biology and competition. Proc Nat Acad Sci.1973,70(12):3590-3593.
[31]马知恩.种群生态学的数学建模与研究.安徽：安徽教育出版社,1996.
[32]傅廷强.时变环境Lotka-Volterra系统的渐近性态.生物数学学报,1996,11(1)：32-37.
[33]熊佐亮.关于三种群第Ⅱ类功能反应周期系数模型的研究.生物数学学报.1998,13(1)：38—42.
[34]Arnold V.I. Geometrical methods in the theory of ordinary differential equations. Springer-Verlag, New York, Berlin Heidelberg Tokyo,1983.
[35]Wan Y.H. Bifurcation into invariant tori at points of resonance. Archive for Rational Mechanics Analysis.1978,68:343-357.
[36]Guckenheimer J. and Holmes P.J. Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer-Verlag, New York,1983.
[37]Chenciner A. Bifurcation de point fixes elliptiques I, Courbes invariants. Publications Mathematiques, Institut des Hautes Etudes Scientifiques.1985,61:67-127.
[38]Los J.E. Non-normally hyperbolic invariant curves for maps in R3 and doubling Bifurcation. Nonlinearity.1989,2(1):149-174.
[39]Iooss G. and Los J.E. Quasi-genericity of bifurcations to high dimensional invariant tori for maps. Communications in Mathematical Physics.1988,119:453-500.
[40]Chenciner A and Iooss G. Bifurcation of a Torus T2 into a Torus T3. Comptes Rendus de l'Academie des Science Paris.1977,284A.
[41]Chenciner A. and Iooss G. Bifurcation de torus invariant. Archive for Rational Mechanics and Analysis.1979,69:109-198.
[42]Iooss G. and Joseph D.D. Bifurcation and Stability of nT-Period Solutions Branching from T-Periodic Solutions at Points of Resonance. Archive for Rational Mechanics and Analysis.1978,66:135-172.
[43]Iooss G. Some codimension-two bifurcations for maps, leading to chaos. In Chaos and Statistical Methods, Kuramoto, Y.(ed.), Springer-Verlag, New York,1984:136-142.
[44]Iooss G. Persistance d'un cercle invariant par une application voisine de "l'application temps r"d'un champ de vecteurs intergrable. Partie I:En dehors de la bifurcation de Hopf. Comptes Rendus de l'Academie des Sciences Paris,Serie I.1983,296:27-30.
[45]Iooss G. Forms normales d'applications:caracterisation globale et methode de calcul. Uninersite de Nice,1987.
[46]Elphick C., Tirapegu E., Brachet M. and Iooss G. A Simple Global Characterization for Normal Forms of Singular Vector Fields. Physic D.1987,29:95-127.
[47]Iooss G, Arneodo A., Coullet P. and Tresser C. Simple computation of bifurcating invariant circles for mapping. Dynamical systems and turbulence. Lecture notes in Math.1981,898:192-211.
[48]Aronson D. G., Chory M.A., Hall G.R. and McGehee R.P. Bifurcations from an invariant circle for two-parameter families of maps of the plane:a computer-assisted study. Communications in Mathematical Physics.1982,83:303-354.
[49]谢建华.三次映射的全局分叉与混沌.西南交通大学学报,1994,29(1)：30-38.
[50]谢建华,罗冠炜.关于平面映射次谐分岔定理的一个注记.力学学报.2002,34(增)：147-150.
[51]Xie J.H., Ding W.C., Dowell E.H., Virgin L.N. Hopf-flip bifurcation of high dimensional maps and application to vibro-irnpact systems. Acta Mech Sinica.2005, 21:402-410.
[52]谢建华.映射的分叉理论及其在碰撞振动系统中的应用.西南交通大学博士学位论文.1993.
[53]Wen G.L., Xu D.L. and Xie J.H. Control of Degenerate Hopf Bifurcations in Three-Dimensional Maps. Chaos.2003,13(2):486-494.
[54]徐慧东,谢建华.高维映射的Hopf-pitchfork分岔及其在碰撞振动系统中的应用.应用力学学报.2008,25(2)：268-273.
[55]徐慧东,谢建华.一类高维映射不变圈的计算.动力学与控制学报.2007,5(3)：204-208.
[56]徐慧东.高维映射的局部余维二分岔(Ⅱ).兰州铁道学院学报(自然科学版).2003,22(4)：13-17.
[57]李恩颖,文桂林,李光耀Henon映射的Hopf分岔分析及PSpice电路模拟.湖南大学学报(自然科学版),2007,34(9)：26-29.
[58]张建刚.高维映射的局部余维二分岔(Ⅰ).兰州铁道学院学报(自然科学版).2003,22(4)：7-12.
[59]Feigenbaum M.J., Kadanoff L.P. and Shenker S.J. Quasi periodicity in dissipative systems:a renormalization group analysis. Phys.1982, D5:370.
[60]Feigenbaum M.J. Quantitative universality for a class of nonlinear transformations. Journaf of Statisticaf Physics.1978,19(1):25-52.
[61]Alexander J. Dynamical systems. New York:Springer,1988.
[62]Sakaguchi H. and Tomita K. Bifurcation of the coupled Logistic map. Progress of Theoret. Phys.1987,78(2):305-309.
[63]Ferretti A. and Rahman N.K. A study of coupled Logistic map and its applications in chemical physics. Chemical Physics.1988,119:275-288.
[64]Welstead S.T. and Cromer T.L. Coloring periodicities of two-dimensional Mappings. Computers & Graphics.1989,14(1):125-126.
[65]孟湛祥.耦合逻辑斯蒂映射中的一个混沌吸引子.天津大学学报.1999,32(6)：711-715.
[66]王兴元,石其江.二维logistic映射中的一种新型激变、回滞和分形.应用力学学报.2005,22(4),501-506.
[67]王兴元,粱庆永.复合Logistic映射中的逆分岔与分形.力学学报.2005,37(4),522-528.
[68]王兴元,骆超.二维Logistic映射的分岔与分形.力学学报.2005,37(3),346-354.
[69]赵明超,王凯华,傅新楚.二维耦合Logistic映射的动力学性质及在流密码生成中的应用.应用数学与计算数学学报.2009,23(1)：87-92.
[70]舒仲周,谢建华.碰撞振动的稳定性.西南交通大学学报.1985,(3)：14-25.
[71]谢建华,舒仲周.一类无穷维动力系统的分叉问题.西南交通大学学报.1994,29(1)：7-13.
[72]谢建华.振动锤的数学模型与全局分叉.力学学报.1997,29(4)：456-463.
[73]谢建华,葛玉梅.冲击式振动落砂机的动态响应与分叉.振动工程学报.1993,6(1)：90-95.
[74]谢建华,罗冠炜.碰撞振动系统的分叉与混沌研究.见：胡海岩,靳征谟主编：《面向21世纪的中国振动工程研究》,航空工业出版社,1999：52-58.
[75]Xie J.H. Codimension Two Bifurcation and Hopf Bifurcation of an Impacting Vibrating System. Applied Mathematics and Mechanics.1996,17:65-75.
[76]谢建华,郑小武.惯性式冲击振动落砂机周期运动的Hopf分叉.振动工程学报.1999,12(3)：297-303.
[77]郑小武.一类碰撞振动系统周期运动的Hopf分叉.西南交通大学硕士学位论文,1997.
[78]谢建华,文桂林,肖健.两自由度碰撞振动系统分叉参数的确定.振动工程学报.2001,14(3)：249-253.
[79]Luo G.W. and Xie J.H. Hopf Bifurcation of Two- Degree-of-Freedom Vibro-Impact System. Journal of Sound and Vibration.1998,213(3):391-408.
[80]Luo G.W. and Xie J.H. Bifurcations and Chaos in a System with Impacts. Physica D. 2001,(148):183-200.
[81]Luo G.W. and Xie J.H. Hopf Bifurcations and Chaos of a Two-Degree-of-Freedom Vibro-Impact System in Two Strong Resonance Cases, International Journal of Non-Linear Mechanics.2002,37(1):19-34.
[82]Luo G.W. and Xie J.H. and Guo SH.L. Periodic Motions and Global Bifurcations of a Two-Degree-of-Freedom System with Plastic Vibro-Impact. Journal of Sound and Vibration.2001,240(5):837-858.
[83]罗冠炜,谢建华,孙训方.单自由度塑性碰撞振动系统的周期运动及其分叉特点.中国机械工程.2001,12(11)：1297-1300.
[84]Luo G.W. and Xie J.H. The Codimension two Bifurcation of Periodic Vibro-Impact and Chaos of a Dual Component System. Physics Letters A.2003,313(4):267-273.
[85]Wen G.L. Codimension-2 Hopf Bifurcation of a Two- Degree-of-Freedom Vibro-Impact System. Journal of Sound and Vibration.2001,242(3):475-485.
[86]文桂林.碰撞振动系统的复杂Hopf分岔与混沌研究.西南交通大学博士学位论文.2001.
[87]罗冠炜,褚衍东,苟向锋.一类碰撞振动系统的概周期运动及混沌形成过程.机械强度.2005,27(4)：445-451.
[88]罗冠炜.两自由度碰撞振动系统的Hopf分岔与混沌研究.西南交通大学博士学 位论文,1998.
[89]罗冠炜,谢建华.碰撞振动系统的周期运动和分岔.科学出版社,2002.
[90]Wen G.L. and Xie J.H. Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-of-Freedom Vibro-Impact System. ASME Journal of Applied Mechanics.2001,68(4):670-674.
[91]丁旺才,谢建华.碰撞振动系统的一类余维二分岔及T2环面分岔.力学学报.2003,34(4)：503-508.
[92]Xie J.H. and Ding W.C. Hopf-Hopf bifurcation and invariant torus T2 of a vibro-impact system. International Journal of Non-Linear Mechanics.2005,40: 531-543.
[93]Ding W.C. and Xie J.H. Torus T2 and its routes to chaos of a vibro-impact system. Physics Letters A.2006,349:324-330.
[94]丁旺才,谢建华,李国芳.三自由度碰撞振动系统的周期运动稳定性与分岔.工程力学.2004,21(3)：123-128.
[95]丁旺才,谢建华,李万祥.碰撞振动系统强共振下的两参数动力学分析.计算力学学报.2004,21(6)：658-664.
[96]丁旺才.多自由度碰撞振动系统的环面分岔与混沌研究.西南交通大学博士学位论文,2004.
[97]乐源,谢建华.一类双面碰撞振子的对称性、尖点分岔与混沌.应用数学和力学.2007,8：991-998.
[98]Yue Y. and Xie J.H. Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides. Applied Mathematics and Mechanics.2007,8:1109-1117.
[99]乐源.多自由度碰撞振动系统的对称性、分岔与混沌研究.西南交通大学博士学位论文.2008.
[100]乐源,谢建华.一类三自由度碰撞振动系统的Poincare映射的对称性,分岔及混沌,四川大学学报(工程科学版).2008,40(1)：27-31.
[101]乐源,谢建华.两自由度碰撞振动系统的Poincare映射的对称性及分岔,振动工程学报.2008,21(4)：376-380.
[102]乐源,谢建华,丁旺才.一类两自由度碰撞振动系统的Hopf分岔和混沌,动力学与控制学报.2004,2(3)：36-41.
[103]徐慧东,谢建华.一类两自由度分段线性非光滑系统的分岔与混沌.振动工程学报.2008,21(3)：279-284.
[104]徐慧东.非光滑动力系统周期解的分岔研究.西南交通大学博士学位论文.2008.
[105]徐慧东,谢建华.单自由度含间隙分段线性系统周期运动的倍化分岔.西南交通大学学报.2008,43(2)：227-231.
[106]张艳龙,徐慧东.冲击振动落砂机在1：4强共振点附近的动力学特性.工程力学.2008,25(8)：194-199.
[107]June Y.L. and Yan J.J. Control of impact oscillator. Chaos, Solitons and Fractals. 2006,28:136-142.
[108]Hubler A. and Luscher E. Resonant stimulation of complex systerms. Helv. Phys. Acts. 1989,62:544.
[109]胡岗,萧井华,郑志刚.混沌控制.上海：上海科技教育出版社,2000.12
[110]Ott E., Grebogi C. and York J.A. Controlling chaos. Phys Rev Lett.1990,64(11): 1196-1199.
[111]Kazuyuki Y. and Tomotsugu U. A new approach for controlling chaotic dynamical system. Physics Letters A.1998,238:349-357.
[112]Hu H.Y. Controlling chaos of a dynamical system with discontinuous vector field. Physics D.1997,106:1-8.
[113]关新平,范正平,张群亮,王益群.连续时间稳定线性系统的混沌反控制研究.物理学报.2002,51(10)：2216-2220.
[114]任海鹏,刘丁,韩崇昭.基于直接延迟反馈的混沌反控制.物理学报.2006,55(6)：2694-2701.
[115]陶建武,石要武,常文.端口受控哈密顿系统的混沌反控制研究.物理学报.2004,53(6)：1682-1686.
[116]丁洪洁,吕和祥.压缩映射—参数微扰控制混沌.振动工程学报.2003,16(2)：212-218.
[117]丁洪洁,吕和祥.小参数干扰反馈控制动力系统中混沌运动.大连理工大学学报.2003,43(2)：132-135.
[118]蔡国梁,谭振梅,周维怀,涂文桃.一个新的混沌系统的动力学分析及混沌控制.物理学报.2007,56(11)：6230-6237.
[119]Ushio T. Chaotic synchronization and controlling chaos based On contraction mappings. Phys. Lett. A.1995,198:14-22.
[120]Fowler T.B. Application of stochastic control technique to chaotic nonlinear system. IEEE Transactions on Automatic Control.1989,34:205-310.
[121]Kapitaniak T. Controlling chaos without feedback. Chaos, Solitons & Fractals.1992, 2:519-530.
[122]Kapitaniak T. Controlling chaos without feedback and control signals. International Journal of Bifurcation and Chaos.1993,3:459-468.
[123]Hu H.Y. Semmi-active vibration control based on a vibration absorber with adjustable clearance. Transactions of Nanjing University of Aeronautics & Astronautics.1996, 13:135-140.
[124]胡海岩.引导混沌运动到周期运动的自适应控制策略.力学学报.1997,29：631-635.
[125]马永靖,丁旺才,杨小刚.碰撞振动系统的参数自调节混沌控制.振动与冲击.2007,26(1)：26-34.
[126]丁旺才,马永靖,王靖岳.碰撞振动系统的状态预测反馈控制.振动工程学报.2007,20(6)：589-593.
[127]Silvio L. T. de Souza and Ibere L. Caldas. Controlling chaotic orbits in mechanical system with impacts. Chaos, Solitoms and Fractals.2004,19:171-178.
[128]徐慧东,谢建华.一类单自由度分段线性系统的分岔与混沌控制.振动与冲击.2008,27(6)：20-24.
[129]于津江,刘学军,韩万强,许海波.平移法控制离散系统的倍周期分岔和混沌.计算物理.2007,24(1)：116-120.
[130]唐驾时,欧阳克俭Logistic模型的倍周期分岔控制.物理学报.2006,55(9)：437-431.
[131]罗晓曙,陈关荣,汪秉宏,方锦清,邹艳丽,全宏俊.状态反馈和参数调整控制离散非线性系统的倍周期分岔和混沌.物理学报.2003,52(4)：790-794.
[132]徐慧东,张建刚,褚衍东.一类Mathieu方程的混沌控制.西安理工大学学报.2006,22(2)：167-170.
[133]褚衍东,李险峰,张建刚.一类特殊的Mathieu方程的分岔及混沌控制.青岛大学学报(自然科学版).2006,19(3)：30-33.
[134]Kan I., Kocak H. and Yorke J. Antimonotonicity:concurrent creation and annihilation of periodic orbits. Annals of Mathematics.1992,136:219-252.
[135]Kuruvilla T. and Nandakumaran V.M. Suppression of chaos through reverse period doubling in coupled directly modulated semiconductor laser. Physics Letters A.1999, 254:59-64.
[136]舒仲周.运动稳定性.峨眉：西南交通大学出版社,1989.
[137]舒仲周,张继业,曹登庆.运动稳定性.北京：中国铁道出版社,2001.
[138]廖晓昕.动力系统的稳定性理论和应用.北京：国防工业出版社,2000.
[139]廖晓昕.稳定性的数学理论和应用.武汉：华中师范大学出版社,1998.
[140]舒仲周,王照林.运动稳定性的研究进展和趋势.力学进展.1993,23(3)： 424-431.
[141]马知恩,周义仓.常微分方程定性与稳定性方法.北京：科学出版社,2001.
[142]高为炳.运动稳定性基础.北京：高等教育出版社,1987.
[143]秦元勋,王慕秋,王联.运动稳定性理论与应用.北京：科学出版社,1981.
[144]秦元勋.运动稳定性的一般问题讲义.北京：科学出版社,1958.
[145]秦元勋,刘永清,王联.带有时滞的动力系统的稳定性.北京：科学出版社,1958.
[146]许淞庆.常微分方程稳定性理论.上海：上海科学技术出版社,1965.
[147]H.г.契塔耶夫.运动的稳定性.北京：国防工业出版社,1965.
[148]Guilhen P.M., Bertheier P., Ferraris G. and Lalanne M. Dynamic behavior of dissymmetric rotor bearings modeled periodic coefficient large system, in: Proceedings of the 58th Shock and Vibration Symposium, Huntsville, AL,1987.
[149]Sinha S.C. and Pandiyan R. Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov-Floquet transformation. International Journal of Nonlinear Mechanics.1994,29(5):687-702.
[150]Sinha S.C., Pandiyan R. and Bibb J.S. Liapunov-Floquet transformation:computation and applications to periodic systems. Journal of Vibration and Acoustics.1996,118: 209-219.
[151]Deshmukh V., Sinha S.C. and Joseph P. Order reduction and control of parametrically excited dynamical systems,Journal of Vibration and Control.2000,6:1017-1028.
[152]Sinha S.C, Redkar S., Butcher E. and Deshmukh V. Order reduction of nonlinear time periodic systems using invariant manifolds, in:The Proceedings of 19th Biennial Conference on Mechanical Vibration and Noise, ASME 2003 DETC, Chicago, IL, USA,2003,48445.
[153]Sinha S.C and Chou C.C. An approximate analysis of transient response of time-dependent linear systems by use of orthogonal polynomials. Journal of Sound and Vibration.1976,49(3):309-326.
[154]Sinha S.C., Chou C.C. and Denman H.H. Stability analysis of systems with periodic coefficients:an approximate approach. Journal of Sound and Vibration.1979,64(4): 515-527.
[155]Sinha S C and Wu D H. An efficient computational scheme for the analysis of periodic systems. Journal of Sound and Vibration.1991,151(1):91-117.
[156]Sinha S.C., Der-Ho Wu, Juneja V. and Joseph P. Analysis of dynamic systems with periodically varying parameters via Chebyshev polynomials. Transactions of the ASME.1993,115:96-102.
[157]Sinha S.C. and Joseph P. Control of general dynamic systems with periodically varying parameters via Liapunov-Floquet transformation. Journal of Dynamic Systems, Measurement, and Control.1994,116:650-658.
[158]Butcher E.A. and Sinha S.C. A hybrid formulation for the analysis of time-periodic linear systems via Chebyshev polynomials. Journal of Sound and Vibration.1996, 195(3):518-527.
[159]Sinha S.C. and Butcher E.A. Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems. Journal of Sound and Vibration.1997, 206(1):61-85.
[160]Sinha S.C., Marghitu D B and Boghiu D. Stability and control of a parameterically excited rotating beam. Journal of Dynamic Systems, Measurement, and Control. 1998,120:462-470.
[161]Alexandra D. and Sinha S.C. Bifurcation control of nonlinear systems with time-periodic coefficients. Proceedings of the American Control Conference.1999, 99:3057-3061.
[162]Sinha S.C., Henrichs J.T. and Ravindra B. A general approach in the design of active controllers for nonlinear systems exhibiting chaos. International Journal of Bifurcation and Chaos.2000,10(1):165-178.
[163]Alexandra D. and Sinha S.C. Versal deformation and local bifurcation analysis of time-periodic nonlinear systems. Nonlinear Dynamics.2000,21:317-336.
[164]Alexandra D. and Sinha S.C. Bifurcation control of nonlinear systems with time-periodic coefficients. Journal of Dynamic Systems, Measurement, and Control. 2003,125:541-548.
[165]Sinha S.C., Redkar S. and Eric A. Butcher. Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. Journal of Sound and Vibration. 2005,284:985-1002.
[166]Sinha S.C., Gourdon E. and Zhang Y. Control of time-periodic system via symbolic computation with application to chaos control. Communications in Nonlinear Science and Numerical Simulation.2005,10:835-854.
[167]Hsu C.S. On approximating a general linear periodic system. Journal of Mathematical Analysis and Applicationss.1974,45; 234-251.
[168]Friendmann P., Hammond C.E. and Woo T.H. Efficient numerical treatment of periodic systems with application to stability problems. International Journal for Numerical Methods in Engineering.1977,11:1117-1136.
[169]Arneodo A., Coullet P.H. and Spiegel E.A. Cascade of period doublings of tori. Physics Letters.1983,94A(1):1-5.
[170]Yedavalli P.K. and Kolla S.R. Stability analysis of linear time-varying systems. International Journal of Systems SCI.1988,19(9):1853-1858.
[171]Lindtner E., Steindl A. and Troger H. Generic one-parameter bifurcations in the motion of a simple robot. Journal of Computational and Applied Mathematics.1989, 26:199-218.
[172]Guttalu R.S. and Flashner H. Stability analysis of periodic systems by truncated point mappings. Journal of Sound and Vibration.1996,189(1):33-54.
[173]Cai Zh.Q., Gu Y.X. and Zhong W.X. A new approach of computing Floquet transition matrix. Computers and Structures.2001,79:631-635.
[174]Yaghoobi H. and Abed E.H. Local feedback control of the Neimark-Sacker bifurcation. International Journal of Bifurcation and Chaos.2003,13(4):879-893.
[175]Venkatesh Y.V. On the stability of a nonlinear system with periodic coefficients. Proceedings of the IEEE.1974,30:278-279.
[176]Willems J.C. On the asymptotic stability of the null solution of linear differential equations with periodic coefficients. IEEE Transactions on Automatic Control. 1968,AC-13(1):65-72.
[177]Thompson J.M.T. and Ghaffari. Chaos after period-doubling bifurcations in the resonance of an impact oscillator. Physics Letters.1982,91 A(1):5-8.
[178]Thompson J.M.T. Complex dynamics of compliant off-shore structures. Proc. R. Lond.1983, A387:407-427.
[179]Kaneko K. Oscillation and doubling of torus. Progess of Theoretical Physics.1984, 72(2):202-215.
[180]Wu W.T., Griffin J.H. and Wickert J.A. Perturbation method for the Floquet eigenvalues and stsbility boundary of periodic linear systems. Journal of Sound and Vibration.1995,182(2):245-257.
[181]Liu Zh.L. Periodic solutions for nonlinear nth order ordinary differential equations. Journal of Mathematical Analysis and Applications.1996,204:46-64.
[182]Liu B. and Cheng S.S. Note on a nonlinear difference equation with periodic coefficients and constant delays. Appl. Math. Lett.1998,11(5):39-43.
[183]Shnol E.E. Stability of linear almost-Hamiltonian periodic systems. J. Appl. Maths. Mechs.1996,60(6):933-939.
[184]Bose T., Member S., Chen M.Q., Joo K.S. and Xu G.F. Stability of two-dimensional discrete systems with periodic coefficients. IEEE Transactions on Circuits and Systems-II:Analog and Digital Signal Processing.1998,45(7):839-847.
[185]Zevin A. A. Stability regions of linear canonical systems with periodic coefficients. J. Appl. Maths. Mechs.1998,62(1):37-42.
[186]Veovoda A.A, Khassoounekh V. On new equivalent mataix model for systems with periodic coefficients. Proceedings of the IEEE.2000,287:170-172.
[187]Mahmoud G.M., Bountis T. and Ahmed S.A. Stability analysis for systems of nonlinear Hill's equations. Physica A.2000,286:133-146.
[188]Natsiavas S., Theodossiades S. and Goudas I. Dynamic analysis of piecewise linear oscillators with time periodic coefficients. International Journal of non-linear mechanics.2000,35:53-68.
[189]Andrey V.S. Lie-and chronologico-algebraic tools for studying stability of time-varying systems. Systems & Control Letters.2001,43:59-76.
[190]Alexandra Z. and Pinsky M. Stability criteria for linear systems with partly uncertain periodic coefficients. Society for Industrial and Applied Mathematics.2003,42(4): 1185-1197.
[191]Montagnier P., Christopher C.P. and Spiteri R.J. Real Floquet factors of linear time-periodic systems. Systems & Control Letters.2003,50:251-262.
[192]Tyshchenko V.Y. Equivalence of Riccati equations with periodic coefficients. Translated from Differentsial'nye Uravneniya.2003,39(4):565-567.
[193]Hong J.L. and Liu Y. A novel numerical approach to simulating nonlinear Schrodinger equations with varying coefficients. Applied Mathematics Letters.2003, 16:759-765.
[194]Yuan R. On the second-order differential equation with piecewise constant argument and almost periodic coefficients. Nonlinear Analysis.2003,52:1411-1440.
[195]Yenicerioglu F A and Talcinbas S. On the stability of the second-order delay differential equations with variable coefficients. Applied Mathematics and Computation.2004,152:667-673.
[196]Zevin A.A. A new approach to the theory of the stability of linear canonical systems of differential equations with periodic coefficients. J. Appl. Maths. Mechs.2004, 68(2):183-198.
[197]Ma H.T., Deshmukh V, Butcher E and Averina V. Delayed state feedback and chaos control for time-periodic systems via a symbolic approach. Communications in Nonlinear Science and Numerical Simulation.2005,10:479-497.
[198]Gaishun I.V. Stability of linear Hamiltonian systems in total differentials with periodic coefficients. Translated from Differentsial'nye Uravneniya.2005,41(1): 33-40.
[199]Gergely R. Neimark-Sacker bifurcation for periodic delay differential equations. Nonlinear Analysis.2005,60:1025-1044.
[200]Liu Y.J., Ge W.G. and Gui Zh.J. Three positive periodic solutions of nonlinear differential equations with periodic coefficients. Analysis and Applications.2005, 3(2):145-155.
[201]冯兆生,张培军,张后扬.关于周期系数Abel方程的周期解问题.北方交通大学学报.1998,22(2)：9-11.
[202]徐鉴,陈予恕.非自治时滞反馈控制系统的周期解分岔和混沌.力学学报.2003,35(4)：443-451.
[203]黎雄.周期系数的高维Riccati方程的周期解.数学进展.1999,28(4)：313-322.
[204]徐君祥,尤建功.关于几乎周期系数的线性微分方程的可约化性.数学年刊.1996,17A(5)：607-616.
[205]冯立新,马富明.解一类具有周期系数的Helmholtz方程的Galerkin谱方法.吉林大学学报(理学版).2003,41(3)：253-258.
[206]Li X.M. Oscillation of neutral differential-difference equations with positive and negative periodic coefficients. Chinese Quarterly Journal of Mathematics.2000, 15(2):17-22.
[207]Li X.M. and Huang L.H. Oscillation and linearized oscillations for difference equations with positive and negative periodic coefficients. Mathematica Applicata. 2001,14(4):26-30.
[208]Huang Zh.K. and Chen F.D. Global asymptotic stanility of the type II functional response of three-species model with almost periodic coefficients. Journal of Mathetical Study.2003,36(2):124-132.
[209]王天成.线性周期系数系统的稳定性.烟台师范学院学报(自然科学版)1999,15(4)：246-249.
[210]陈茂海.具有周期系数线性微分方程的性质.工科数学.2001,17(1)：94-96.
[211]杨秀香.周期系数非线性差分系统的稳定性.渭南师范学院学报.2000,15(5)：33-34.
[212]徐俊峰,刘名生.一类周期线性微分方程的复振荡.华南师范大学学报(自然 科学版).2004,3：11-15.
[213]朱萌纾.一类双周期系数Riccati方程的单值群的性质.南京师大学报(自然科学版).2002,25(3)：44-48.
[214]汤光宋,熊炎.具有周期系数的几类线性微分系统的周期解.南都学坛.1990,3：46-53.
[215]李雪臣,王鸿燕.具周期系数时滞差分方程的全局渐近稳定性.扬州大学学报(自然科学版).2005,8(1)：14-17.
[216]张海燕,唐友刚,陈芳启.非线性Mathieu方程的局部分岔和在余维2退化点的Hopf分岔.机械强度.2007,29(5)：717-721.
[217]洪灵,徐建学.非线性强迫Mathieu方程的激变和瞬态混沌.力学学报.2001,33(3)：423-429.
[218]Troger H. and Steindl A. Nonlinear stability and bifurcation theory. Springer-verlag, 1991.