Border Collision Bifurcation And Chaos Control Of Piecewise Map

Border collision bifurcation and chaos are the typical phenomena of the piecewise map systems, which always have very complicated dynamic behavior and have been received extensive attention by many scholars. In this thesis, border collision bifurcation of two classes of the piecewise maps and the bifurcation control and chaos control of a piecewise map are taken into consideration. The main results are given as follows:(1) For the piecewise map that consists of a left linear function and a right nonlinear function, the border collision bifurcation of the system is studied. And the condition for border collision bifurcation of period n+1solutions occurring is deduced. The global bifurcation diagrams of the system are depicted in the three-dimensional parameter space. When the constant term of left linear function is given as b, the sufficient condition for fold bifurcation of the system can be obtained. Then, numerical simulations are carried out for the system with two and three bifurcation parameters, respectively. The simulation results show that there exist the period adding sequence and superposition sequence in the considered system. (2) A class of piecewise maps owning left and right nonlinear functions are investigated. Firstly, when the constant of the right nonlinear function has been selected as bifurcation parameter, the conditions for border collision bifurcation and flip bifurcation of the system are deduced. The simulation results show that there is the phenomenon of period adding sequence in the system. Secondly, when the constant of the left nonlinear function has been selected as bifurcation parameter, simulation results show that there exists only chaos phenomenon in the system.(3) The parameter adjustment and state feedback control method is employed for controlling the bifurcation of a class of piecewise maps. Firstly, bifurcation phenomenon of the system is verified by the numerical simulation. Secondly, an appropriate controller is designed for controlling the bifurcation of the system. The results of numerical simulation show that this control method can extend to the piecewise map.(4) The chaos of a class of piecewise maps is suppressed to the periodic orbits by using the existing methods as well as the newly proposed approachs in the thesis. First of all, numerical simulations show that there exists chaotic phenomenon in the system. Then, the suitable prediction feedback controller, delayed feedback controller, the nonlinear feedback controller and two proposed new controllers, namely, the prediction delayed controller and delayed nonlinear controller are applied to this map, and the sufficient conditions for stability of the controlled system are obtained. Finally, with the range of the controller and the iteration times of the system converging to stable periodic orbit for reference, numerical simulation discloses that both prediction delay feedback method and delay feedback control method have their own advantages while delay nonlinear feedback method is better than delay feedback control method for the chaos control of the map discussed.