关于激变、分岔以及复杂网络描述的一些探索
摘要
本学位论文分为七章。第一章为绪论部分，第二章和第三章报道两个各有特色的非线性系统的研究。第四章、第五章和第六章分别报道了对三种具有中国特色的复杂性网络的研究。第七章提出了一个具有新意的统计参数并讨论了这种参数的意义。
论文在第二章中报导一种有特色的激变。这种激变是在一类分段连续力场作用下的受击转子模型中观察到的。描述系统的二维映象定义域中的函数不连续边界随离散时间发展振荡，从而使这个边界的向前象集构成一个承载混沌运动的胖分形。在控制参数的一个阈值下，一个椭圆周期轨道突然出现在此胖混沌奇异集中，使得迭代向它逃逸，胖混沌奇异集因此突然变为一个胖瞬态集。我们分析了系统的特点对激变后生存时间标度律的影响，解析出标度因子的值。这个具有显著特色的结论与数值结果很好地符合。
在本文第三章中讨论的系统是一个过电压保护电路。在这个电张弛振荡电路中，一个特征参数的连续改变可以产生一种所谓的“类耗散-不连续分岔”，或者称为“增强擦边”(enhanced grazing)分岔。这种分岔表征系统在一个特征参数的阈值从一个保守、连续的系统同时突变为一个类耗散、不连续的系统。所谓的“擦边”(grazing)分岔是上世纪八十年代在冲击振子模型中发现的，表征系统在一个特征参数的阈值从一个处处光滑的系统突变为一个不光滑的系统，突变的临界点对应于冲击碰撞的“擦边”情况，因此现在这种同时还伴随着类耗散性出现的“擦边”分岔也许还可以称为“增强擦边”(enhanced grazing)分岔。本章报道这种分岔可能产生一种保守、胖分形随机网向瞬态、瘦分形随机网的突然转变，从而导致原来保守、胖分形随机网上的迭代向一些保守椭圆岛的逃逸。这也可以看作是一种激变。这种激变的主要特征是这种特征分岔立即导致相平面上迭代禁区的出现，而且禁区的向前象集很可能仍是禁区，从而构成相平面上一个胖分形迭代空白区的主要部分。相平面被迭代空白区占据后所余下的允许迭代区域正是不连续边界象集构成的瞬态、瘦分形随机网。我们数值地说明了这个迭代空白区胖分形的分形指数随控制参数的变化显示指数关系。这应该是描述这种激变的最
扬州大学硕十学位论文
主要规律。除此之外，迭代在瞬态随机网中的平均生存时间随控制参数的变化仍
旧是激变的另一个主要规律。我们解析和数值地说明了这种变化显示幂律关系，
并且导出了幂律的标度指数。
第四章中报导了对中药方剂网络的研究。我们选取了将近1500付中药方剂，
从复杂网络角度研究了它们的统计性质，并通过分析，倾向于认为传统的平均道
路长、分支系数、顶点度分布律以及最近提出的格子系数不适于作为特征统计参
量。对具有“完全图团簇集合”这个拓扑特征的中药方剂网，建议使川“顶点团
簇度”进行最重要的统计描述。另外，假设中药方剂的成方过程可以用一个药材
的功能因子的分布函数来描述，从而通过与部分实际参数的拟合搜索这种分布函
数，建立一个再现中药方剂网络发展过程的模型。最后，通过与中国菜肴体系和
中国旅游线路体系的对比，认为顶点团簇度和这种建模力一法也许有较普遍的意义，
适用于更多的系统。
第五章我们从复杂网络角度研究了旅游线路体系的统计性质，并通过分析，
类似地倾向于认为传统的平均道路长、分支系数、顶点度分布律以及最近提出的
格子系数不适于作为特征统计参量。对同样具有“完全图团簇集合”这个拓扑特
征的旅游线路网，也建议使用“顶点团簇度”进行最重要的统计描述。我们建立
了一个与上述中药方剂网络发展过程的模型完全不同的再现旅游线路网络发展过
程的模型，所得到的模拟结果很好地与统计结果符合。最后，通过与中药方剂体
系和中国淮扬菜体系的对比，同样地认为顶点团簇度和这种建模方法也许有较普
遍的意义，适用于更多的系统。
第六章报道中国航空网的统计调研。定义飞机场为顶点，航线为边。共统计
了2002年、1998年及1997年这三个不同年度的中国航空网。我们还建立了一个
不停演化发展的自适应的动力学模型。数值模拟结果很好地与统计相吻合。
最后一章中我们定义了一种新的统计参数一“合作成功度”，它在不同类型
的具体网中表征不同意义。本文侧重讨论“合作成功度”在一类最典型的合作网
中的具体体现，及它在具体的三个典型合作网例子中的统计和模型结果。并以中
药网模型为例，研究改变建模法则而导致“平均合作成功度”的变化特征和意义。
This thesis is divided into seven chapters. Chapter one is an exordium; Chapter two and chapter three report the studies on two characteristic nonlinear systems; Chapter four, five and six report three kinds of Chinese characteristic complex networks. The last chapter, Chapter 7, suggests a novel statistical parameter and discusses its important meaning.
My thesis reports a kind of characteristic crisis in Chapter two. The crisis is observed in a kicked rotor subjected to a piecewise continuous force field. The discontinuity border in the definition range of a two-dimensional mapping, which describes the system, oscillates as the discrete time progresses so that the forward images of the border form a fat fractal. By choosing particular parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter an elliptic periodic orbit suddenly appears inside the fat strange set so that the iterations on the set escape to the orbit. The fat chaotic attractor thus suddenly transfers to a fat transient set. The effect of the feature of the crisis on the rule of the lifetime in the transient set is analyzed. It shows that the dependence of the lifetime on the control parameter follows a universal scaling
law suggested by Grebogy, Ott and Yorke (Phys. Rev. Lett. 57, 1284 (1986)). This unique conclusion is in a good agreement with the numerical results.
In Chapter three, the system investigated is an electronic circuit with over-voltage protection. A "quasi-dissipative and discontinuous" bifurcation, or a so-called "enhanced grazing" bifurcation can be produced in it by varying a characteristic parameter. The bifurcation signifies that the system suddenly changes from a conservative, continuous one to a quasi-dissipative, discontinuous one at a threshold of a parameter value. Grazing bifurcation was discovered firstly in 1980's in an impact oscillator. It signifies that the system suddenly changes from an everywhere smooth one to a piecewise smooth one at threshold of a parameter value. The critical point of the bifurcation corresponds to the "grazing" situation of the impact. Thus the bifurcation
discussed here may be addressed as a "enhanced grazing" one since the quasi-dissipative transition is added. This chapter reports that the bifurcation can produce a sudden transition from a conservative, fat fractal stochastic web to a transient, thin stochastic web so that the iterations on the original web escape to some conservative elliptic islands. This phenomenon can also be viewed as a crisis. Its main feature can be described like this: The bifurcation induces at once the appearance of a forbidden gap for iterations on the phase plane, the forward image set of the gap is probably the forbidden region as well, and therefore forms the main part of a fat fractal "iteration-absent" region on the phase space. The remaining parts of the phase space where iterations are permitted are just the transient, thin fractal stochastic web formed by the images of the discontinuous borderlines. We have numerically shown that the fractal exponent of the fat fractal iteration-absent region displays an exponential dependence on the control parameter. This should be the principal rule describing the crisis. In addition, another main rule is still the dependence of the averaged lifetime of the iterations inside the transient stochastic web on the control parameter. We analytically and numerically show that the dependence obeys a universal power law and deduce the scaling exponent.
We reported our research about the Chinese traditional prescription formulation web in Chapter four. We selected about 1500 Chinese traditional prescription formulations and performed statistical investigation on them from complex network's viewpoint. After some analyzing, we believe that traditional statistical parameters, like average path length, clustering coefficient, node degree distribution, and t
论文在第二章中报导一种有特色的激变。这种激变是在一类分段连续力场作用下的受击转子模型中观察到的。描述系统的二维映象定义域中的函数不连续边界随离散时间发展振荡，从而使这个边界的向前象集构成一个承载混沌运动的胖分形。在控制参数的一个阈值下，一个椭圆周期轨道突然出现在此胖混沌奇异集中，使得迭代向它逃逸，胖混沌奇异集因此突然变为一个胖瞬态集。我们分析了系统的特点对激变后生存时间标度律的影响，解析出标度因子的值。这个具有显著特色的结论与数值结果很好地符合。
在本文第三章中讨论的系统是一个过电压保护电路。在这个电张弛振荡电路中，一个特征参数的连续改变可以产生一种所谓的“类耗散-不连续分岔”，或者称为“增强擦边”(enhanced grazing)分岔。这种分岔表征系统在一个特征参数的阈值从一个保守、连续的系统同时突变为一个类耗散、不连续的系统。所谓的“擦边”(grazing)分岔是上世纪八十年代在冲击振子模型中发现的，表征系统在一个特征参数的阈值从一个处处光滑的系统突变为一个不光滑的系统，突变的临界点对应于冲击碰撞的“擦边”情况，因此现在这种同时还伴随着类耗散性出现的“擦边”分岔也许还可以称为“增强擦边”(enhanced grazing)分岔。本章报道这种分岔可能产生一种保守、胖分形随机网向瞬态、瘦分形随机网的突然转变，从而导致原来保守、胖分形随机网上的迭代向一些保守椭圆岛的逃逸。这也可以看作是一种激变。这种激变的主要特征是这种特征分岔立即导致相平面上迭代禁区的出现，而且禁区的向前象集很可能仍是禁区，从而构成相平面上一个胖分形迭代空白区的主要部分。相平面被迭代空白区占据后所余下的允许迭代区域正是不连续边界象集构成的瞬态、瘦分形随机网。我们数值地说明了这个迭代空白区胖分形的分形指数随控制参数的变化显示指数关系。这应该是描述这种激变的最
扬州大学硕十学位论文
主要规律。除此之外，迭代在瞬态随机网中的平均生存时间随控制参数的变化仍
旧是激变的另一个主要规律。我们解析和数值地说明了这种变化显示幂律关系，
并且导出了幂律的标度指数。
第四章中报导了对中药方剂网络的研究。我们选取了将近1500付中药方剂，
从复杂网络角度研究了它们的统计性质，并通过分析，倾向于认为传统的平均道
路长、分支系数、顶点度分布律以及最近提出的格子系数不适于作为特征统计参
量。对具有“完全图团簇集合”这个拓扑特征的中药方剂网，建议使川“顶点团
簇度”进行最重要的统计描述。另外，假设中药方剂的成方过程可以用一个药材
的功能因子的分布函数来描述，从而通过与部分实际参数的拟合搜索这种分布函
数，建立一个再现中药方剂网络发展过程的模型。最后，通过与中国菜肴体系和
中国旅游线路体系的对比，认为顶点团簇度和这种建模力一法也许有较普遍的意义，
适用于更多的系统。
第五章我们从复杂网络角度研究了旅游线路体系的统计性质，并通过分析，
类似地倾向于认为传统的平均道路长、分支系数、顶点度分布律以及最近提出的
格子系数不适于作为特征统计参量。对同样具有“完全图团簇集合”这个拓扑特
征的旅游线路网，也建议使用“顶点团簇度”进行最重要的统计描述。我们建立
了一个与上述中药方剂网络发展过程的模型完全不同的再现旅游线路网络发展过
程的模型，所得到的模拟结果很好地与统计结果符合。最后，通过与中药方剂体
系和中国淮扬菜体系的对比，同样地认为顶点团簇度和这种建模方法也许有较普
遍的意义，适用于更多的系统。
第六章报道中国航空网的统计调研。定义飞机场为顶点，航线为边。共统计
了2002年、1998年及1997年这三个不同年度的中国航空网。我们还建立了一个
不停演化发展的自适应的动力学模型。数值模拟结果很好地与统计相吻合。
最后一章中我们定义了一种新的统计参数一“合作成功度”，它在不同类型
的具体网中表征不同意义。本文侧重讨论“合作成功度”在一类最典型的合作网
中的具体体现，及它在具体的三个典型合作网例子中的统计和模型结果。并以中
药网模型为例，研究改变建模法则而导致“平均合作成功度”的变化特征和意义。
This thesis is divided into seven chapters. Chapter one is an exordium; Chapter two and chapter three report the studies on two characteristic nonlinear systems; Chapter four, five and six report three kinds of Chinese characteristic complex networks. The last chapter, Chapter 7, suggests a novel statistical parameter and discusses its important meaning.
My thesis reports a kind of characteristic crisis in Chapter two. The crisis is observed in a kicked rotor subjected to a piecewise continuous force field. The discontinuity border in the definition range of a two-dimensional mapping, which describes the system, oscillates as the discrete time progresses so that the forward images of the border form a fat fractal. By choosing particular parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter an elliptic periodic orbit suddenly appears inside the fat strange set so that the iterations on the set escape to the orbit. The fat chaotic attractor thus suddenly transfers to a fat transient set. The effect of the feature of the crisis on the rule of the lifetime in the transient set is analyzed. It shows that the dependence of the lifetime on the control parameter follows a universal scaling
law suggested by Grebogy, Ott and Yorke (Phys. Rev. Lett. 57, 1284 (1986)). This unique conclusion is in a good agreement with the numerical results.
In Chapter three, the system investigated is an electronic circuit with over-voltage protection. A "quasi-dissipative and discontinuous" bifurcation, or a so-called "enhanced grazing" bifurcation can be produced in it by varying a characteristic parameter. The bifurcation signifies that the system suddenly changes from a conservative, continuous one to a quasi-dissipative, discontinuous one at a threshold of a parameter value. Grazing bifurcation was discovered firstly in 1980's in an impact oscillator. It signifies that the system suddenly changes from an everywhere smooth one to a piecewise smooth one at threshold of a parameter value. The critical point of the bifurcation corresponds to the "grazing" situation of the impact. Thus the bifurcation
discussed here may be addressed as a "enhanced grazing" one since the quasi-dissipative transition is added. This chapter reports that the bifurcation can produce a sudden transition from a conservative, fat fractal stochastic web to a transient, thin stochastic web so that the iterations on the original web escape to some conservative elliptic islands. This phenomenon can also be viewed as a crisis. Its main feature can be described like this: The bifurcation induces at once the appearance of a forbidden gap for iterations on the phase plane, the forward image set of the gap is probably the forbidden region as well, and therefore forms the main part of a fat fractal "iteration-absent" region on the phase space. The remaining parts of the phase space where iterations are permitted are just the transient, thin fractal stochastic web formed by the images of the discontinuous borderlines. We have numerically shown that the fractal exponent of the fat fractal iteration-absent region displays an exponential dependence on the control parameter. This should be the principal rule describing the crisis. In addition, another main rule is still the dependence of the averaged lifetime of the iterations inside the transient stochastic web on the control parameter. We analytically and numerically show that the dependence obeys a universal power law and deduce the scaling exponent.
We reported our research about the Chinese traditional prescription formulation web in Chapter four. We selected about 1500 Chinese traditional prescription formulations and performed statistical investigation on them from complex network's viewpoint. After some analyzing, we believe that traditional statistical parameters, like average path length, clustering coefficient, node degree distribution, and t
引文
[1] Grebogi C, Ott E, and Yorke J A 1982 Phys. Rev. Lett. 48 1507; 1983 Physica (Amsterdam) 7D 181
[2] Grebogi C, Ott E, and Yorke J A 1986 Phys. Rev. Lett. 57 1284; 1987 Phys. Rev. A, 36 5365
[3] Grebogi C, Ott E, and Yorke J A 1983 Phys. Rev. Lett. 50 935; 1985 Ergod. Theor. Dynam. Sys. 5 341.
[4] Qu S X, Christiansen B and He D R 1995 Acta Phys. Sin. 44 841 (in Chinese) [屈世显、何大韧 1995 物理学报 44 841]
[5] Qu S X, Christiansen B and He D R 1995 Phys. Lett. 201A 413
[6] Ding X L, Wu S G, Yin Y C, He D R, 1999 Chinese Phys. Lett. 16 167
[7] Wang J, Ding X L, Hu B, Wang B H, Mao J S and He D R 2001 Phys. Rev. E 64 026202
[8] Wang J, Ding X L, Wang B H, and He D R Chin. Phys. Lett. 2001 18 13
[9] Wang X M, Wang Y M, D.-R. He et al. 2002 Eur. Phys. J. 19D 119
[10] Christian M 1996 Inter J. Bifur. and Chaos 6 893
[11] Wang B H 1995 Weak chaos and quasi-regular patterns, (Shanghai: Shanghai Scientific and Technological Education Publishing House) (in Chinese) [汪秉宏 2001 弱混沌与准规则斑图(上海：上海科技教育出版社)]
[12] Jiang Y M, Lu Y Q and He D R to appear on 2004 Acta Phys. Sin. 53 (2) (in Chinese)[姜玉梅、陆云清、何大韧 将发表于2004物理学报53(2)]
[13] Grebogi C, McDonald S W, Ott E and Yorke J A 1985 Phys. Lett. 110A 1
[14] Umberger D K and Farmer J D 1985 Phys. Rev. Lett. 55 661
[15] Li H Q and Wang F Q 1993 Theory of Fractal and its application in Molecular Science, (Beijing: Science Publishing House) (in Chinese) [李后强、汪富泉 1993 分形理论及其在分子科学中的应用(北京：科学出版社)]
[16] Falconer K (Translated by Zeng W Q and Liu S Y) 1991 Fractal Geometry,(Shenyang: Northeast University Publishing House) (in Chinese) [Falconer K著, 曾文曲、刘世耀译 1991 分形几何(沈阳：东北大学出版社)]
[17] T. Tél, Transient Chaos, in Directions in Chaos, Vol. Ⅲ, edited by Hao B L, (Singapore: World Scentific, 1990).
[18] Chao X G 2003 Thisis in Yangzhou University, (in Chinese) [巢小刚 2003 扬州大
[19] Jiang y m and He D R ,Stochastic web crisis, European Physical Journal D
[2] Grebogi C, Ott E, and Yorke J A 1986 Phys. Rev. Lett. 57 1284; 1987 Phys. Rev. A, 36 5365
[3] Grebogi C, Ott E, and Yorke J A 1983 Phys. Rev. Lett. 50 935; 1985 Ergod. Theor. Dynam. Sys. 5 341.
[4] Qu S X, Christiansen B and He D R 1995 Acta Phys. Sin. 44 841 (in Chinese) [屈世显、何大韧 1995 物理学报 44 841]
[5] Qu S X, Christiansen B and He D R 1995 Phys. Lett. 201A 413
[6] Ding X L, Wu S G, Yin Y C, He D R, 1999 Chinese Phys. Lett. 16 167
[7] Wang J, Ding X L, Hu B, Wang B H, Mao J S and He D R 2001 Phys. Rev. E 64 026202
[8] Wang J, Ding X L, Wang B H, and He D R Chin. Phys. Lett. 2001 18 13
[9] Wang X M, Wang Y M, D.-R. He et al. 2002 Eur. Phys. J. 19D 119
[10] Christian M 1996 Inter J. Bifur. and Chaos 6 893
[11] Wang B H 1995 Weak chaos and quasi-regular patterns, (Shanghai: Shanghai Scientific and Technological Education Publishing House) (in Chinese) [汪秉宏 2001 弱混沌与准规则斑图(上海：上海科技教育出版社)]
[12] Jiang Y M, Lu Y Q and He D R to appear on 2004 Acta Phys. Sin. 53 (2) (in Chinese)[姜玉梅、陆云清、何大韧 将发表于2004物理学报53(2)]
[13] Grebogi C, McDonald S W, Ott E and Yorke J A 1985 Phys. Lett. 110A 1
[14] Umberger D K and Farmer J D 1985 Phys. Rev. Lett. 55 661
[15] Li H Q and Wang F Q 1993 Theory of Fractal and its application in Molecular Science, (Beijing: Science Publishing House) (in Chinese) [李后强、汪富泉 1993 分形理论及其在分子科学中的应用(北京：科学出版社)]
[16] Falconer K (Translated by Zeng W Q and Liu S Y) 1991 Fractal Geometry,(Shenyang: Northeast University Publishing House) (in Chinese) [Falconer K著, 曾文曲、刘世耀译 1991 分形几何(沈阳：东北大学出版社)]
[17] T. Tél, Transient Chaos, in Directions in Chaos, Vol. Ⅲ, edited by Hao B L, (Singapore: World Scentific, 1990).
[18] Chao X G 2003 Thisis in Yangzhou University, (in Chinese) [巢小刚 2003 扬州大
[19] Jiang y m and He D R ,Stochastic web crisis, European Physical Journal D