摘要
该文研究一簇Ｌｏｒｅｎｚ映射Ｓａ：［０，１］＋［０．１］（０＜ａ＜１） 从拓扑的角度考虑了Ｓａ的混沌行为．证明了：Ｓａ有稠密轨道；Ｓａ的周期的集合ＰＰ（Ｓａ）＝ ｛１，ｍ＋１，ｍ＋２，…｝，其中ｍ为使广ａｍ＜１－ａ成立的最小正整数；Ｓａ的拓扑熵ｈ（Ｓａ）＞０；几乎所 有（关于Ｌｅｂｅｓｇｕｅ测度）的点ｘ的Ｌｙａｐｕｎｏｖ指数λ（Ｓａ，ｘ）＝λａ＞０． 从统计的角度讨论了Ｓａ的稳定性．我们用下界函数方法证明了Ｓａ是统计稳定的，并且 为Ｓａ的唯一绝对连续（关于Ｌｅｂｅｓｇｕｅ测度）不变概率测度．同 时，不变密度ｇａ在参数扰动和随机作用的随要扰动下是稳定的．
The property of a family of Lorenz maps Sa[0,1]→［0,1](0 < a < 1 ) is studied. The chaotic behavior is considered from the topological point of view. It is proved that Sa has dense orbit and infinite periodic orbits. The topological entropy of Sa is positive and the local Lyapunov exponent is positive for almost points in [0,1]. The sta- tistical stability of Sa is considered from the statistical point of view. It is proved that Sa is statistical stable and Sa admits a unique absolute continuous invariant measure ga with re- spect to Lebesgue measure. ga is stable under parameter perturbation and randomly applied stochastic perturbations.
出处
《数学物理学报（A辑）》
CSCD
北大核心
2001年第4期559-569,共11页
Acta Mathematica Scientia
基金
国家自然科学基金资助项目(19847005
69874039)
