一維動態系統之歸返映射

Abstract

我們主要探討Schwarzian derivative為負值和有兩個極值點的一維離散型動態系統，並觀察它的疊代行為。同時，在這系統上定義歸返映射，運用歸返映射的許多重要性質與特徵，我們可以得到這個動態系統有拓撲轉移性（topology transitivity）、週期點的稠密性、及對初始條件的靈敏性（sensitively dependent to the initial conditions）等結果。然後，會舉一些三次多項式的例子來印證我們的結論。另一方面，我們也在具短暫混沌性質的神經網路上運用我們的理論，並舉一些例子來說明。

We investigate the iteration of maps of the interval which have negative Schwarzian derivative and two critical points. Using the characteristic of the first return map, we conclude the topological transitivity, dense periodic points, and sensitive dependence on initial conditions for the considered one-dimensional discrete-time dynamical systems. Some cubic polynomials are taken as examples to illustrate the results. We also attempt to apply the theory to the one-dimensional chaotic neural network.