The Periodic Orbits Of Continuous Connected Set-Valued Maps On The Interval And Y

Research of periodic orbits is an important content of dynamical systems. The periodic orbits of the continuous connected set-valued maps on the interval and Y are mainly studied in this thesis.In chapter3, the periodic orbits of the continuous connected set-valued maps on the interval are discussed. The relation between two periodic orbits whose periods are two adjacent odd is given. The main statement is as follows:Suppose that F:Iâ†'L(I) is a continuous connected set-valued map and n is an odd that is not less than5. If F has a n-periodic orbit O=(x1,x2,...)=(x1,x2,...,xn)°while has no (n-2)-periodic orbits, then the n-periodic orbit satisfies the following conditions(1) The periodic orbit O=(x1,x2,...,xn)°is a prime periodic orbit,(2) The periodic orbit O=(x1,x2,...,xn)°is a Stefan periodic orbit of F, that is, there is a point xi∈{x1,x2,...,xn} satisfying xi+n-2<…<xi+1<xi<xi+2...<xi+n-1or xi+n-1<…<xi+2<xi<xi+1…<xi+n-2.Suppose that F:Y-â†'L(Y) denotes a continuous connected set-valued map on Y,8:Yâ†'{1,2,3} denotes the endpoint number function of F on Y, Ns, Ng, Nr denote Sharkovskii ordering, green ordering and red ordering respectly, the point o∈Y denotes the branch point of Y and Per(F) denotes the set of periods of all periodic orbits of F.In chapter4, the periodic orbits of the continuous connected set-valued maps on Y are discussed. The main statements are as follows:(1)(a) Suppose that F:Yâ†'L(Y) is a continuous connected set-valued map. If the number of the connected components of δ-1(i)(i=1,2,3) is finite, then there exists three integers ns∈Ns, ng∈Ng and nr∈Nr such that Per(F)(?) S(ns)∪G(ng)∪R(nr).(b) If ns∈Ns, ng∈Ng and nr∈Nr, then there exists a continuous connected set-valued map F:Y-â†'L(Y) such that Per(F)=S(ns)∪G(ng)∪R?(nr).(2) Suppose that F:Yâ†'L(Y) is a continuous connected set-valued map, and the number of the continuous connected components of δ-1(i)(i=1,2,3) is finite. If F contains five prime periodic orbits P2,P3, P4, P5, P7whose periods are2,3,4,5,7respectively such that o (?) Pi(i=2,3,4,5,7) and Pi∩Pj=φfor i≠j and i, j∈{2,3,4,5,7}, then Per(F)=N.