| This thesis mainly studies border-collision bifurcation in a class of piecewise mapping with two discontinuous points and period-doubling bifurcation control in another class of piecewise mapping with only one discontinuous point. The research contents and results are:(1) For a class of three-stage map with two discontinuous points, it is divided into two two-stage maps with only one discontinuous point respectively and border-collision bifurcation is discussed, then bifurcation conditions of period k+1 solutions and period k+2 solutions with the first complexity region are derived respectively, and the global bifurcation diagram of this map is drawn on the parameter plane. Secondly, based on above analysis, the second and third complexity periodic domains of this map are given out respectively, and there exist periodic-adding sequences and period stacking sequences in the system known by the numerical simulation. Eventually, even loops such as the period 2k+2 solutions and the period 2k+4 solutions, etc., and the corresponding global bifurcation diagram of three-stage map are discussed entirely. The bifurcation condition of second complexity periodic domain of periodic solution is obtained. Simulation results show that period-adding sequences and period stacking sequences also exist in this system.(2) For the period-doubling bifurcation control in a class of piecewise mapping with only one discontinuous point, parameters adjustment and linear controller are designed in the system. Numerical simulations show that the period-doubling bifurcation of system can be delayed through both methods, and the chaotic behavior also can be controlled effectively. However, parameters adjustment controller can control period-doubling bifurcation only in a small range, which means this method is not significant. In contrast, the linear controller consisting of coarse and fine tuning can control the period-doubling bifurcation and chaos of the system effectively. |
PDF Full Text Request