多项式离散系统的分岔动力学

本书是第一本关于一维多项式非线性离散系统分岔动力学的专著。本书给出了多项式非线性离散系统分岔的一般条件，全面讨论了一维多项式离散系统中高阶奇异不动点的出现和切换分岔，系统分析了倍周期分岔和单调鞍点分岔所产生的周期-1到混沌的分岔树，提出了多项式离散系统的周期-2及全局倍周期重整化方法，并且首次确定了分岔树上周期-n不动点的出现机理和相应的倍周期重整化。读者将在书中看到非线性离散系统中妙趣横生的研究结果。 首先讨论了一维线性离散动力系统不动点的稳定性。 系统地讨论了二次、三次、四次多项式的离散动力系统不动点的稳定性及解的复杂性。 讨论了2m次多项式和2m+1次多项式的离散动力系统不动点的稳定性及分岔的复杂性。 展示了单调和振荡的出现和切换分岔，包括单调和振荡上下鞍点分岔、单调和振荡源分岔、单调和振荡汇分岔。 首次提出了此类多项式离散系统的周期-2及全局倍周期重整化方法。 首次确定了分岔树上周期-n不动点的出现机理。 给出了多项式离散系统通向混沌的解析表达式。

1 Quadratic Nonlinear Discrete Systems
1.1 Linear Discrete Systems
1.2 Forward Quadratic Discrete Systems
1.2.1 Period-1 Appearing Bifurcations
1.2.2 Period-1 Switching Bifurcations
1.3 Backward Quadratic Discrete Systems
1.3.1 Backward Period-1 Appearing Bifurcations
1.3.2 Backward Period-1 Switching Bifurcations
1.4 Forward Bifurcation Trees
1.4.1 Period-2 Appearing Bifurcations
1.4.2 Period-Doubling Renormalization
1.4.3 Period-n Appearing and Period-Doublization
1.4.4 Period-n Bifurcation Trees
1.5 Backward Bifurcation Trees
1.5.1 Backward Period-2 Quadratic Discrete Systems
1.5.2 Backward Period-Doubling Renormalization
1.5.3 Backward Period-n Appearing and Period-Doublization
1.5.4 Backward Period-n Bifurcation Trees
References
2 Cubic Nonlinear Discrete Systems
2.1 Period-1 Cubic Discrete Systems
2.2 Period-1 to Period-2 Bifurcation Trees
2.3 Higher-Order Period-1 Switching Bifurcations
2.4 Direct Cubic Polynomial Discrete Systems
2.5 Forward Cubic Discrete Systems
2.5.1 Period-Doubled Cubic Discrete Systems
2.5.2 Period-Doubling Renormalization
2.5.3 Period-n Appearing and Period-Doublization
2.5.4 Sampled Period-n Appearing Bifurcations
2.6 Backward Cubic Nonlinear Discrete Systems
2.6.1 Backward Period-2 Cubic Discrete Systems
2.6.2 Backward Period-Doubling Renormalization
2.6.3 Backward Period-n Appearing and Period-Doublization
Reference
3 Quartic Nonlinear Discrete Systems
3.1 Period-1 Appearing Bifurcations
3.2 Period-1 to Period-2 Bifurcation Trees
3.3 Higher-Order Period-1 Quartic Discrete Systems
3.4 Period-1 Switching Bifurcations
3.4.1 Simple Period-1 Switching Bifurcations
3.4.2 Higher-Order Period-1 Switching Bifurcations
3.5 Forward Quartic Discrete Systems
3.5.1 Period-2 Quartic Discrete Systems
3.5.2 Period-Doubling Renormalization
3.5.3 Period-n Appearing and Period-Doublization
3.6 Backward Quartic Discrete Systems
3.6.1 Backward Period-2 Quartic Discrete Systems
3.6.2 Backward Period-Doubling Renormalization
3.6.3 Backward Period-n Appearing and Period-Doublization
Reference
4 (2m)th-Degree Polynomial Discrete Systems
4.1 Global Stability and Bifurcations
4.2 Simple Fixed-Point Bifurcations
4.2.1 Appearing Bifurcations
4.2.2 Switching Bifurcations
4.2.3 Switching-Appearing Bifurcations
4.3 Higher-Order Fixed-Point Bifurcations
4.3.1 Appearing Bifurcations
4.3.2 Switching Bifurcations
4.3.3 Switching-Appearing Bifurcations
4.4 Forward Bifurcation Trees
4.4.1 Period-Doubled (2m)th-Degree Polynomial Discrete Systems
4.4.2 Renormalization and Period-Doubling
4.4.3 Period-n Appearing and Period-Doublization
References
5 (2m+1)th-Degree Polynomial Discrete Systems
5.1 Global Stability and Bifurcations
5.2 Simple Fixed-Point Bifurcations
5.2.1 Appearing Bifurcations
5.2.2 Switching Bifurcations
5.2.3 Switching-Appearing Bifurcations
5.3 Higher-Order Fixed-Point Bifurcations
5.3.1 Higher-Order Fixed-Point Bifurcations
5.3.2 Switching Bifurcations
5.3.3 Switching-Appearing Bifurcations
5.4 Forward Bifurcation Trees
5.4.1 Period-Doubled (2m+1)th-Degree Polynomial Systems
5.4.2 Renormalization and Period-Doubling
5.4.3 Period-n Appearing and Period-Doublization
References
Index