非光滑动力系统周期解的分岔研究
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摘要
非光滑动力系统以其在实际工程领域中的普遍性和在诸多工业领域的重要性,吸引了众多科学工作者和工程技术人员的关注。近年来,非光滑动力系统已成为非线性动力学研究的热点之一。由于非光滑系统向量场的非光滑性,使得光滑系统中研究周期解稳定性和分岔的传统方法不再适用,需要从理论上探究一些分析非光滑系统周期解稳定性和分岔的新方法,因此在理论研究上具有很大的挑战性。本文的主要研究工作有以下几方面。
1.研究了高维非线性映射由Neimark-Sacker分岔产生的不变圈的计算。分析了不变圈的分岔条件,通过Fredholm择一方法分析了在计算不变圈过程中出现的一类方程解的存在性,再根据不变圈上映射到自身的不变性,通过分析振幅ε各阶项的系数,在一类高维映射中实现了不变圈的算法。
2.研究了碰撞系统中与碰撞面横截相交的周期解。通过零时间不连续映射方法推导出线性化矩阵的计算式。通过时间Poincaré映射对单自由度单边碰撞系统周期运动的倍化分岔现象做了分析,验证了理论的正确性。
3.研究了碰撞系统中与碰撞面相切的周期解(擦边轨道)附近的动力学行为。分析了碰撞系统存在擦边轨道的解析条件,详细地推导了零时间不连续映射和Poincaré截面不连续映射。引入参数空间,将未发生碰撞的周期轨道对应的光滑映射与前面得到的不连续映射复合得到两类擦边分岔的分段映射形式。然后,研究了单自由度单边碰撞系统的擦边分岔。利用擦边分岔条件找到了此系统的擦边分岔点,利用推导的擦边分岔的分段映射和直接数值仿真获得不连续分岔图,两者得到的分岔图在结构上是类似的。从分岔图发现,参数从一个不变的周期轨道区域到下一个不变的周期轨道区域的过渡点处都要发生一个跳跃而使得分岔不连续,通过相图做进一步分析,发现在跳跃处都要发生轨道的擦边现象。这与光滑系统典型的连续倍化分岔现象是不同的。
4.研究了碰撞振动系统通过取Poincaré截面得到的高维光滑映射的Hopf-pitchfork和Hopf-Hopf的余维二分岔。通过中心流形理论,将高维映射降阶为一个三维映射,再通过范式方法将降阶后的三维映射转化为范式映射。分析了三维范式映射在Hopf-pitchfork分岔点附近的参数开折。将上述结果应用于对一类三自由度双边碰撞振动系统周期运动的Hopf-pitchfork分岔分析并给出数值算例。通过映射的中心流形-范式方法,将高维映射降阶简化为四维的范式映射。理论分析了四维范式映射在Hopf-Hopf余维二分岔点附近的参数开折。通过对三自由度含间隙碰撞振动系统的Hopf-Hopf分岔行为进行数值仿真,验证了理论结果。
5.建立了n维分段光滑系统周期解横截穿越分界面这种情况下的时间Poincaré映射,使用零时间不连续映射方法推导了在分界面处的跳跃矩阵,进而给出了此类映射线性化矩阵的表达式。然后研究一类有一个分界面的单自由度分段线性系统周期运动始终横截穿越分界面的动力学行为。结合光滑系统的Floquet理论给出此非光滑系统发生倍化分岔的的条件。使用数值仿真进一步揭示了系统周期运动经倍化分岔通向混沌的现象。随后还研究了一类有两个分界面的两自由度分段线性非光滑系统周期运动始终横截穿越分界面的分岔现象和混沌行为。将得到的跳跃矩阵结合光滑系统的Floquet理论通过数值方法计算了周期运动发生Neimark-Sacker分岔和倍化分岔的分岔点,数值仿真表明系统存在Neimark-Sacker分岔,倍化分岔和亚谐分岔等连续分岔现象。
6.研究了分段光滑系统中擦边轨道附近的动力学行为。分析了分段光滑系统存在擦边轨道的条件,详细的推导了零时间不连续映射。结合不连续映射最后给出擦边分岔的分段映射形式。然后,具体研究了一个分界面的单自由度分段线性系统的擦边分岔。利用擦边分岔条件找到了此系统的擦边分岔点,再利用时间Poincaré分段映射给出了擦边分岔点附近的分岔图,通过数值方法得到了结构类似的分岔图并通过相图进一步揭示了擦边分岔的特点。数值结果表明随着参数的增大,该单自由度分段线性系统分岔过程为:一周期运动→倍化分岔→二周期运动→擦边分岔→二周期运动→擦边分岔→四周期运动→倍化分岔→混沌运动,随着参数减小,系统分岔过程为:一周期运动→倍化分岔→二周期运动→擦边分岔→三周期运动→擦边分岔→混沌运动。
7.研究了分段光滑系统周期运动的角点碰撞分岔。分析了角点附近轨线内外穿越分界面的两种几何结构以及角点碰撞轨线存在的条件。推导了零时间不连续映射并给出角点碰撞分岔的分段映射。然后研究了一类具有两个非光滑分界面的单自由度分段线性系统。数值结果表明该非光滑系统中存在周期运动的鞍结分岔和角点碰撞分岔现象。
8.分析了滑动分岔的几何结构以及滑动轨线存在的条件。介绍了四种可能的滑动分岔并分析了发生这四种滑动分岔的解析条件。然后研究了一类受简谐激励的单自由度干摩擦系统。通过数值仿真揭示了该系统发生的四种滑动分岔中的第1类滑动分岔和切换滑动分岔现象。通过选择合适的参数,揭示了此干摩擦系统周期运动特有的倍化分岔现象,此倍化分岔现象中伴随有擦边分岔的现象,是一种不同于光滑系统中倍化分岔的非光滑分岔现象。
As the non-smooth dynamical systems are becoming more and more important in engineering fields,they draw attention of many scientific workers and engineers.The non-smooth dynamical systems have been one of the hot spot of non-linear dynamics study in recent years.Due to non-smooth characteristic of vector field of non-smooth dynamical systems the conventional methods in analyzing stability and bifurcation of periodic solution of smooth systems don't apply,and the new methods need be developed to analyze the dynamic behaviors of periodic solution of non-smooth systems,which is a big challenge for theoretical research.The main respects of the research are followings:
1.The computation method of invariant circles for high dimensional maps is addressed.The bifurcating conditions of invariant circles are analyzed.A necessary condition for the existence of a solution of a kind of equations,which rises in computing invariant circle,is presented through Fredholm method,then the variable of phase angle is expanded as a Fourier series and the expressions of computing invariant circles are given by identification of the Fourier coefficients. The computation method of invariant circles for mapping is realized in a three-dimensional map finally.
2.The transversal periodic solutions of impacting systems are addressed.The linearized matrix is derived by means of zero-time discontinuity mapping.The stroboscopic Poincarémap is established to analyze the period-doubling bifurcation of periodic motions in the single-degree-of-freedom impacting system. The theoretical result is verified.
3.The local dynamics behavior of grazing orbit of impacting systems is investigated.The conditions for the existence of grazing orbit are analyzed.The Zero-time discontinuity mapping and the Poincaré-section discontinuity mapping are deduced in detail.The parameter space is included to construct a compound piecewise smooth map by composing previous discontinuity mappings with a Poincarémap for smooth periodic solution.The grazing bifurcation of periodic motions in the single-degree-of-freedom impacting system is investigated.The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through previous piecewise smooth map and numerical calculation,they are similar structurally.A saltation arises to cause the discontinuity bifurcation when the parameter is a transitional point from one periodic orbit area to the other one and the grazing phenomenon happens at the transitional point.This bifurcation phenomenon is different from continuity period-doubling bifurcation of smooth systems.
4.Hopf-pitchfork bifurcation and Hopf-Hopf bifurcation of high dimensional smooth maps obtained in impacting systems are investigated.The high dimensional map is reduced to a three-dimensional map by the center manifold theorem.The reduced map is further transformed into its normal form by theory of normal forms.The two parameter unfoldings of the map near the point of Hopf-pitchfork bifurcation is investigated theoretically.The obtained result is applied to analyzing the Hopf-pitchfork bifurcation of periodic motion in a three-degree-of freedom vibro-impact system and is verified by numerical work. The high dimensional map is reduced to a four-dimensional normal form map by the center manifold theorem and theory of normal forms.The two-parameter unfoldings of local dynamical behavior,near the point of Hopf-Hopf bifurcation, is investigated theoretically.Hopf-Hopf bifurcation in a three-degree-of freedom vibro-impact system is further investigated by means of numerical simulations. Numerical simulation results indicate that the vibro-impact system presents interesting and complicated dynamical behavior.
5.With regard to transversal periodic solutions of piecewise smooth systems, the stroboscopic Poincarémap is established.The saltation matrix is deduced by zero-time discontinuity mapping method and then the linearized matrix of the map is given.The dynamics behaviors of transversal periodic motions of a single-degree-of-freedom piecewise linear system is studied.The period-doubling bifurcation of periodic motions of the piecewise linear system is investigated by the saltation matrix and the Floquet theory.The period-doubling bifurcations and chaotic behaviors in the non-smooth system are further investigated by means of numerical simulations.The bifurcation and chaos of transversal periodic motions of a two-degree-of-freedom non-smooth system with piecewise-linearity is investigated.The saltation matrix is given out at the switching boundaries and the Neimark-Sacker bifurcation point and period-doubling bifurcation point of periodic motions of the system are investigated by the numerical calculation. Numerical results demonstrate the existence of Neimark-Sacker bifurcation, period-doubling bifurcation and subharmonic bifurcation in the non-smooth system.
6.The local dynamics behaviors of grazing orbit of piecewise smooth systems are investigated.The conditions for the existence of grazing orbit are analyzed.Zero-time discontinuity mapping is deduced in detail.A compound piecewise smooth map is given by composing the discontinuity mapping with a Poincarémap for no intersection periodic solution.The grazing bifurcation of periodic motions in the single-degree-of-freedom piecewise linear system is investigated.The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through the previous piecewise smooth map and numerical calculation,they are similar structurally.Then the grazing behavior is further analyzed by phase portraits.The numerical results indicate that the piecewise linear non-smooth system presents period 1 motion→period-doubling bifurcation→period 2 motion→grazing bifurcation→period 2 motion→grazing bifurcation→period 4 motion→period-doubling bifurcation→chaos with the increase of parameter and period 1 motion→period-doubling bifurcation→period 2 motion→grazing bifurcation→period 3 motion→grazing bifurcation→chaos with the decrease of parameter.
7.The corner-collision bifurcation of piecewise smooth systems is addressed. The external and internal corner collisions geometry and the conditions for the existence of corner collision orbit are investigated.The Zero-time discontinuity mapping is deduced in detail and the compound piecewise smooth map of corner-collision bifurcation is given.The single-degree-of-freedom piecewise linear system with two non-smooth switching boundaries is studies.The numerical results indicate that the piecewise linear system presents saddle-node bifurcation and comer-collision bifurcation.
8.The sliding orbit geometry and the conditions for the existence of sliding orbit are investigated.Four types of sliding bifurcations and their bifurcation conditions are introduced.The single-degree-of-freedom dry friction system is considered.The crossing-sliding bifurcation and switching-sliding bifurcation are shown by numerical simulations.The numerical results also indicate that the dry friction system presents the special period-doubling bifurcation accompanied by grazing bifurcation.
1.研究了高维非线性映射由Neimark-Sacker分岔产生的不变圈的计算。分析了不变圈的分岔条件,通过Fredholm择一方法分析了在计算不变圈过程中出现的一类方程解的存在性,再根据不变圈上映射到自身的不变性,通过分析振幅ε各阶项的系数,在一类高维映射中实现了不变圈的算法。
2.研究了碰撞系统中与碰撞面横截相交的周期解。通过零时间不连续映射方法推导出线性化矩阵的计算式。通过时间Poincaré映射对单自由度单边碰撞系统周期运动的倍化分岔现象做了分析,验证了理论的正确性。
3.研究了碰撞系统中与碰撞面相切的周期解(擦边轨道)附近的动力学行为。分析了碰撞系统存在擦边轨道的解析条件,详细地推导了零时间不连续映射和Poincaré截面不连续映射。引入参数空间,将未发生碰撞的周期轨道对应的光滑映射与前面得到的不连续映射复合得到两类擦边分岔的分段映射形式。然后,研究了单自由度单边碰撞系统的擦边分岔。利用擦边分岔条件找到了此系统的擦边分岔点,利用推导的擦边分岔的分段映射和直接数值仿真获得不连续分岔图,两者得到的分岔图在结构上是类似的。从分岔图发现,参数从一个不变的周期轨道区域到下一个不变的周期轨道区域的过渡点处都要发生一个跳跃而使得分岔不连续,通过相图做进一步分析,发现在跳跃处都要发生轨道的擦边现象。这与光滑系统典型的连续倍化分岔现象是不同的。
4.研究了碰撞振动系统通过取Poincaré截面得到的高维光滑映射的Hopf-pitchfork和Hopf-Hopf的余维二分岔。通过中心流形理论,将高维映射降阶为一个三维映射,再通过范式方法将降阶后的三维映射转化为范式映射。分析了三维范式映射在Hopf-pitchfork分岔点附近的参数开折。将上述结果应用于对一类三自由度双边碰撞振动系统周期运动的Hopf-pitchfork分岔分析并给出数值算例。通过映射的中心流形-范式方法,将高维映射降阶简化为四维的范式映射。理论分析了四维范式映射在Hopf-Hopf余维二分岔点附近的参数开折。通过对三自由度含间隙碰撞振动系统的Hopf-Hopf分岔行为进行数值仿真,验证了理论结果。
5.建立了n维分段光滑系统周期解横截穿越分界面这种情况下的时间Poincaré映射,使用零时间不连续映射方法推导了在分界面处的跳跃矩阵,进而给出了此类映射线性化矩阵的表达式。然后研究一类有一个分界面的单自由度分段线性系统周期运动始终横截穿越分界面的动力学行为。结合光滑系统的Floquet理论给出此非光滑系统发生倍化分岔的的条件。使用数值仿真进一步揭示了系统周期运动经倍化分岔通向混沌的现象。随后还研究了一类有两个分界面的两自由度分段线性非光滑系统周期运动始终横截穿越分界面的分岔现象和混沌行为。将得到的跳跃矩阵结合光滑系统的Floquet理论通过数值方法计算了周期运动发生Neimark-Sacker分岔和倍化分岔的分岔点,数值仿真表明系统存在Neimark-Sacker分岔,倍化分岔和亚谐分岔等连续分岔现象。
6.研究了分段光滑系统中擦边轨道附近的动力学行为。分析了分段光滑系统存在擦边轨道的条件,详细的推导了零时间不连续映射。结合不连续映射最后给出擦边分岔的分段映射形式。然后,具体研究了一个分界面的单自由度分段线性系统的擦边分岔。利用擦边分岔条件找到了此系统的擦边分岔点,再利用时间Poincaré分段映射给出了擦边分岔点附近的分岔图,通过数值方法得到了结构类似的分岔图并通过相图进一步揭示了擦边分岔的特点。数值结果表明随着参数的增大,该单自由度分段线性系统分岔过程为:一周期运动→倍化分岔→二周期运动→擦边分岔→二周期运动→擦边分岔→四周期运动→倍化分岔→混沌运动,随着参数减小,系统分岔过程为:一周期运动→倍化分岔→二周期运动→擦边分岔→三周期运动→擦边分岔→混沌运动。
7.研究了分段光滑系统周期运动的角点碰撞分岔。分析了角点附近轨线内外穿越分界面的两种几何结构以及角点碰撞轨线存在的条件。推导了零时间不连续映射并给出角点碰撞分岔的分段映射。然后研究了一类具有两个非光滑分界面的单自由度分段线性系统。数值结果表明该非光滑系统中存在周期运动的鞍结分岔和角点碰撞分岔现象。
8.分析了滑动分岔的几何结构以及滑动轨线存在的条件。介绍了四种可能的滑动分岔并分析了发生这四种滑动分岔的解析条件。然后研究了一类受简谐激励的单自由度干摩擦系统。通过数值仿真揭示了该系统发生的四种滑动分岔中的第1类滑动分岔和切换滑动分岔现象。通过选择合适的参数,揭示了此干摩擦系统周期运动特有的倍化分岔现象,此倍化分岔现象中伴随有擦边分岔的现象,是一种不同于光滑系统中倍化分岔的非光滑分岔现象。
As the non-smooth dynamical systems are becoming more and more important in engineering fields,they draw attention of many scientific workers and engineers.The non-smooth dynamical systems have been one of the hot spot of non-linear dynamics study in recent years.Due to non-smooth characteristic of vector field of non-smooth dynamical systems the conventional methods in analyzing stability and bifurcation of periodic solution of smooth systems don't apply,and the new methods need be developed to analyze the dynamic behaviors of periodic solution of non-smooth systems,which is a big challenge for theoretical research.The main respects of the research are followings:
1.The computation method of invariant circles for high dimensional maps is addressed.The bifurcating conditions of invariant circles are analyzed.A necessary condition for the existence of a solution of a kind of equations,which rises in computing invariant circle,is presented through Fredholm method,then the variable of phase angle is expanded as a Fourier series and the expressions of computing invariant circles are given by identification of the Fourier coefficients. The computation method of invariant circles for mapping is realized in a three-dimensional map finally.
2.The transversal periodic solutions of impacting systems are addressed.The linearized matrix is derived by means of zero-time discontinuity mapping.The stroboscopic Poincarémap is established to analyze the period-doubling bifurcation of periodic motions in the single-degree-of-freedom impacting system. The theoretical result is verified.
3.The local dynamics behavior of grazing orbit of impacting systems is investigated.The conditions for the existence of grazing orbit are analyzed.The Zero-time discontinuity mapping and the Poincaré-section discontinuity mapping are deduced in detail.The parameter space is included to construct a compound piecewise smooth map by composing previous discontinuity mappings with a Poincarémap for smooth periodic solution.The grazing bifurcation of periodic motions in the single-degree-of-freedom impacting system is investigated.The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through previous piecewise smooth map and numerical calculation,they are similar structurally.A saltation arises to cause the discontinuity bifurcation when the parameter is a transitional point from one periodic orbit area to the other one and the grazing phenomenon happens at the transitional point.This bifurcation phenomenon is different from continuity period-doubling bifurcation of smooth systems.
4.Hopf-pitchfork bifurcation and Hopf-Hopf bifurcation of high dimensional smooth maps obtained in impacting systems are investigated.The high dimensional map is reduced to a three-dimensional map by the center manifold theorem.The reduced map is further transformed into its normal form by theory of normal forms.The two parameter unfoldings of the map near the point of Hopf-pitchfork bifurcation is investigated theoretically.The obtained result is applied to analyzing the Hopf-pitchfork bifurcation of periodic motion in a three-degree-of freedom vibro-impact system and is verified by numerical work. The high dimensional map is reduced to a four-dimensional normal form map by the center manifold theorem and theory of normal forms.The two-parameter unfoldings of local dynamical behavior,near the point of Hopf-Hopf bifurcation, is investigated theoretically.Hopf-Hopf bifurcation in a three-degree-of freedom vibro-impact system is further investigated by means of numerical simulations. Numerical simulation results indicate that the vibro-impact system presents interesting and complicated dynamical behavior.
5.With regard to transversal periodic solutions of piecewise smooth systems, the stroboscopic Poincarémap is established.The saltation matrix is deduced by zero-time discontinuity mapping method and then the linearized matrix of the map is given.The dynamics behaviors of transversal periodic motions of a single-degree-of-freedom piecewise linear system is studied.The period-doubling bifurcation of periodic motions of the piecewise linear system is investigated by the saltation matrix and the Floquet theory.The period-doubling bifurcations and chaotic behaviors in the non-smooth system are further investigated by means of numerical simulations.The bifurcation and chaos of transversal periodic motions of a two-degree-of-freedom non-smooth system with piecewise-linearity is investigated.The saltation matrix is given out at the switching boundaries and the Neimark-Sacker bifurcation point and period-doubling bifurcation point of periodic motions of the system are investigated by the numerical calculation. Numerical results demonstrate the existence of Neimark-Sacker bifurcation, period-doubling bifurcation and subharmonic bifurcation in the non-smooth system.
6.The local dynamics behaviors of grazing orbit of piecewise smooth systems are investigated.The conditions for the existence of grazing orbit are analyzed.Zero-time discontinuity mapping is deduced in detail.A compound piecewise smooth map is given by composing the discontinuity mapping with a Poincarémap for no intersection periodic solution.The grazing bifurcation of periodic motions in the single-degree-of-freedom piecewise linear system is investigated.The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through the previous piecewise smooth map and numerical calculation,they are similar structurally.Then the grazing behavior is further analyzed by phase portraits.The numerical results indicate that the piecewise linear non-smooth system presents period 1 motion→period-doubling bifurcation→period 2 motion→grazing bifurcation→period 2 motion→grazing bifurcation→period 4 motion→period-doubling bifurcation→chaos with the increase of parameter and period 1 motion→period-doubling bifurcation→period 2 motion→grazing bifurcation→period 3 motion→grazing bifurcation→chaos with the decrease of parameter.
7.The corner-collision bifurcation of piecewise smooth systems is addressed. The external and internal corner collisions geometry and the conditions for the existence of corner collision orbit are investigated.The Zero-time discontinuity mapping is deduced in detail and the compound piecewise smooth map of corner-collision bifurcation is given.The single-degree-of-freedom piecewise linear system with two non-smooth switching boundaries is studies.The numerical results indicate that the piecewise linear system presents saddle-node bifurcation and comer-collision bifurcation.
8.The sliding orbit geometry and the conditions for the existence of sliding orbit are investigated.Four types of sliding bifurcations and their bifurcation conditions are introduced.The single-degree-of-freedom dry friction system is considered.The crossing-sliding bifurcation and switching-sliding bifurcation are shown by numerical simulations.The numerical results also indicate that the dry friction system presents the special period-doubling bifurcation accompanied by grazing bifurcation.
引文
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[29]Peterka F.Bifurcation and transition phenomena in an impact oscillator.Chaos,Solitons and Fractals,1996,7(10):1635-1647.
[30]舒仲周,谢建华.冲击动力学的发展与展望.一般力学(动力学、振动与控制)最新进展(黄文虎等主编).科学出版社,1994,:198-204.
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[46]曹登庆,舒仲周.存在间隙的多自由度系统的周期运动与Robust稳定性.力学学报.1997,29(1):74-83.
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