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第1章 圆映射描述的周期与准周期运动1

§1 The definition of circle maps1

Chapter 1 Periodic and quasiperiodic motion described by circle maps1

§1 圆映射的定义1

§2 可逆圆映射5

§2 Invertible circle maps5

§3 法里树的数论与法里序列的符号表示10

§3 Number theory of the Farey tree and symbolic representation of the Farey sequence10

§4 正弦圆映射参数平面上的相结构16

§4 Phase structure in parametric plan of the sine circle map16

§5 转数区间、阿诺尔德舌头、双稳和拓扑混沌19

§5 Rotation intervals，Arnol’d tongues，bistability and topological chaos19

第2章 圆映射的符号动力学23

§6 周期轨道的符号表示23

§6 Symbolic representation of periodic orbits23

Chapter 2 Symbolic dynamics of circle maps23

§7 Symbolic dynamics of Farey sequence26

§7 法里序列的符号动力学26

§8 Newborn orbits and theorem for topological degree29

§8 新生轨道与拓扑度定理29

§9 The * product of Farey sequence and M．S．S．sequence and the binary tree32

§9 法里序列与M．S．S．序列的*积及二元树32

第3章 由准周期向混沌过渡的标度律36

§10 标度行为的数值研究36

§10 Numerical research on the scaling behavior36

Chapter 3 Scaling law for transition from quasiperiod to chaos36

§11 临界准周期轨道的分形结构42

§11 Fractal structure of the critical quasiperiodic orbits42

§12 Renormalization group analysis46

§12 重正化群分析47

§13 临界线上转数阶梯标度性质的数值研究51

§13 Numerical research on the scaling properties of rotation number staircase at critcal line51

§14 Renormalization group analysis for devil’s staircase57

§14 关于魔梯的重正化研究57

§15 圆映射的一般标度性60

§15 Universal scaling in circle maps60

§16 超临界圆映射的一般分岔行为71

Chapter 4 Bifurcation behavior and scaling law of supercritical circle maps71

§16 General bifurcation behavior of the supercritical circle map71

第4章 超临界圆映射的分岔行为与标度律71

§17 The diffusion behavior described by supercritical circle map74

§17 w＝0超临界圆映射所描述的扩散行为74

§18 The similarity of the Arnol’d tongues near critical line78

§18 接近临界线处阿诺尔德舌头的相似性78

§19 Universal vector scaling of period-doubling bifurcation81

§19 倍周期分岔的矢量标度律81

§20 The similarity of the bifurcation in Arnol’d tongues for Fibonacci sequence88

§20 菲波那契序列阿诺尔德舌头中分岔的相似性88

§21 Chaotic measures and scaling for supercritical circle map90

§21 超临界圆映射的混沌测度及其标度律90

Chapter 5 Examples for circle maps in physical reality98

§22 Periodically forced damped pendulum，Josephson junction and charge-density wave systems98

§22 周期驱动阻尼单摆、约瑟夫逊结与电荷密度波系统98

第5章 实际问题中的圆映射举例98

§23 Periodically forced Rayleigh-Benard Convective fluid106

§23 周期强迫瑞利-贝纳德对流系统106

§24 Global universality in the Frenkel-Kontorova model109

§24 Frenkel-Kontorova模型中的整体普适性109

§25 Excitable oscillator under periodic perturbation and rhythm of heartbeat112

§25 可激振子的周期扰动与心脏的搏动节律112

§26 Phase locking and chaotic responses of a biological neuron under periodic stimulation119

§26 神经原对周期脉冲的锁相与混沌响应119

§27 Integrate-and-fire models and the transitiib from quasiperiodicity to nonchaotic complete phase locking125

§27 累积-释放模型与由准周期至完全锁相的非混沌转变125

§28 Complete phase locking and chaos in modulated relaxation oscillators129

§28 调制弛豫振子的完全锁相与混沌129

Chaper 6 Torus maps137

§29 Torus maps and resonance regions137

§29 环面映射及其共振区137

第6章 环面映射137

§30 Numerical experiments on N-frequency quasiperiodicity142

§30 关于环面映射的数值实验142

§31 Three frequency competing in a nonlinear electronic oscillator circuit149

§31 一个非线性电子振荡线路中的三频问题149

§32 Renormalization group analysis for three frequency quasiperod151

§32 三频问题的重正化群分析151

§33 The quasiperiodically forced circle map157

§33 准周期强迫圆映射157

§34 Identification and experrimental observation for a strange nonchaotic attractor163

§34 非混沌奇怪吸引子的鉴别与实验观测163

Refefences169

参考文献169