非线性离散系统中的分岔与稳定性

本书讨论非线性离散动力系统中的分岔和稳定性，包括单调和振荡稳定性理论与单调和振荡分岔理论。本书不仅给出了周期-1的不动点在某特定特征向量方向上的稳定性与分岔理论、并讨论其不动点的高阶奇异性，而且描述了一般非线性离散系统具有重特征根的正则形式、并介绍了具有无限不动点的离散动力系统。作者发现了不动点的局部分析理论不能解释或获得倍周期不动点解的稳定性和分岔。本书分五章介绍非线性离散系统的近期新发展，以便读者更好地、系统地理解与学习非线性离散系统的分岔与稳定性。首先讨论了非线性离散动力系统不动点的稳定性一般理论与稳定性分类。系统地给出了在不动点的特征向量方向上的高阶奇异稳定性与分岔理论。系统地讨论一维离散系统不动点的单调和振荡稳定性与分岔的局部理论。展示了一维离散系统的单调和振荡上下鞍点分岔、单调和振荡源分岔以及单调和振荡汇分岔。基于非线性离散系统阴阳（Yin-Yang）理论，讨论了具有前向和后向映射的非线性离散动力系统的动力学及其不动点的存在理论与分岔。给出了一般非线性离散系统具有重特征根的正则形式。讨论了具有无限不动点的离散动力系统理论。

罗朝俊教授系国际非线性动力学及力学知名专家，目前为美国南伊利诺伊大学爱德华分校突出贡献研究型讲席教授。

1 Local Stability and Bifurcations
1.1 Discrete Systems
1.2 Fixed—Points and Stability
1.3 Stability Switching
1.3.1 Monotonic and Oscillatory Stability
1.3.2 Spiral Stability
1.4 Local Bifurcations
1.4.1 Monotonic Bifurcations
1.4.2 One—Step Oscillatory Bifurcations
1.4.3 Two—Step Oscillatory Bifurcations
1.4.4 Neimark Bifurcations
References
2 LOW-Dimensional Discrete Systems
2.1 One—Dimensional Discrete Systems
2.2 Monotonic and Oscillatory Stability
2.3 Bifurcations
2.3.1 Monotonic Bifurcations
2.3.2 One.Step 0scillatory Bifurcations
2.3.3 Two.Step Oscillatory Bifurcations
2.4 Sample Bifurcation Analysis
2.4.1 Saddle—Node Appearing Bifurcations
2.4.2 Saddle—Node Switching Bifurcations
2.4 -3 Third.Order Monotonic Sink Pitchfork Bifurcations
2.4.4 Third一0rder Monotonic Source Pitchfork Bifurcations
2.5 TwO—Dimensional Discrete Systems
2.5.1 Stability and Singularity
2.5.2 An Example
References
3 Global Stability of 1.D Discrete Systems
3.1 Discrete System Classifications
3.2 Fixed—Point Global StabilitV
3-3 One Fixed—Point Systems
3-3.1 Monotonic Stability
3-3.2 Oscillatory Stability
3.4 Two.Fixed.Point Systems
3.4.1 Monotonic Two—Fixed.Point Systems
3.4.2 Oscillatory Two—Fixed—Point Systems
3.5 Thtee—Fixed—Point Systems
3.5.1 Monotonic Three—Fixed.Point Systems
3.5.2 Oscillatory Three—Fixed—Point Systems
References
4 Forward and Backward Discrete Systems
4.1 Yin—Yang Theory Of Discrete Systems
4.2 Forward Discrete Systems
4.3 Backward Discrete Systems
Reference
5 Infinite-Fixed-Point Discrete Systems
5.1 Fixed.Point Computations
5.2 Normal Forms of Discrete Systems
5.3 Infnite—Fixed—Point Systems
5.3.1 0ne—Infinite—Fixed.Point Systems
5.3.2 TWO.Infinite—Fixed.Point Systems
5.3.3 Higher—Order Infinite—Fixed.Point Systems
5.4 Infinite.Fixed—Point—Network Systems
5.4.1 An Infinite—Fixed—Point.Network System
5.4.2 A Spiral Infinite.Fixed.Point System
5.4.3 A Linear Infinite—Fixed.Point System
5.4.4 A Hyperbolic Infinite—Fixed.Point System
References
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