Research On Dynamic Properties Of The Cyclical Mapping

The dynamic system is widely used in economics and occupies an im-portant position. The antitriangular map gives the mathematical description of two operators in the same field competing in the market. However, when the operators are more than two, antitriangular map can not express the competitive phenomenon among them. To solve the problem, the dissertation gives the concept of the cyclical mapping. It can provide guidance in decision-making for the operators. The main contents of the dissertation are as follows:Firstly, for the cyclical mapping, the relationship between the periodic sets, the ω-limit sets, the recurrent sets, non-wandering sets, strong recurrent sets, almost periodic sets, chain recurrent sets of the cyclical mapping F and the product of those sets of cyclical compound mapping of f, h, g are studied. It is proved that the isolate periodic point of F also is the isolated recurrent point.Secondly, in this dissertation, the periods are linked to the periodic sets of the cyclical mapping. The equivalent conditions for the periodic sets to be closed are derived.Thirdly, for a certain special mapping, the closure of the periodic set equals to the closure of the recurrent set. For the open cuboid, if there is no periodic point in the corresponding interval, then for any x, the points of the trajectory/(x) which lie in the interval form a strictly monotonic sequence. For the non-wandering sets, the dissertation derives a conclusion that if (x,x,x) belongs to the non-wandering set of F, then x belongs to the non-wandering set of f which dose not hold for the general cyclical mapping.Finally, an example is constructed to illustrate that the recurrent set of F may be properly included in the product of the recurrent sets of cyclical compound mapping of f, h, g; A counter-example is constructed to illustrate that the the non-wandering set of F is not included in the product of the non-wandering sets of cyclical compound mapping of f, h, g.