Constructive periodic orbits in Lorenz maps systems

Based on symbolic dynamics, the admissibility conditions are improved and thus the concepts of monotone basic 1-strings and 0-strings are defined. The basic strings generate all the possible basic strings of any periodic series thus the concepts improve the algorithm of finding periods under the admissibility conditions. It provides a satisfactory and necessary condition for existence of consecutive periodic orbits of Lorenz maps, that is, there exist two co-prime periods under some conditions, which overcomes the restriction of continuity of the functions in the Sarkovskii's theorem on consecutive periods, and the Lorenz maps are not restricted within piecewise linear ones. A corresponding algorithm and results of consecutive periods by symbolic series are given for some examples. The algorithm is of high efficiency and can be extended to other dynamic systems after some corresponding variation.