广义Lorenz映射的混沌行为及不变密度

Chaotic behavior of generalized Lorenz maps and its invariant density

用符号动力学证明了广义的、即具有2个或以上间断点的分段线性Lorenz映射以移位自同构为子系统,即系统是混沌的,并给出了拓扑熵的下界以及Lyapunov指数的上界与下界.讨论了广义Lorenz映射的不稳定周期轨道的周期及稠密性,给出了不稳定周期轨道的周期.用构造下界函数的方法论证了分段线性广义Lorenz映射在随机作用随机扰动下系统具有统计稳定性.

Proof is given to the conclusion that a subset of symbolic system is shift automorphic to a family of generalized and piecewise continuous Lorenz maps, which means that the generalized Lorenz map is chaotic. There are two or more points of discontinuity in the Lorenz maps which are linear on every sub-interval. The unstable periodic and dense orbits are discovered and hence periods are given. Lower bounds of the maps are given and both lower and upper bounds for the local Lyapunov exponent are also given. At last it is proved that, by constructing a lower bound function, the family of generalized Lorenz maps have invariant density under stochastic perturbations.