Bifurcation Analysis Of Two Discrete Dynamical Systems

We consider the discrete FHN system and discrete BVP model as the research objects.Based on the qualitative theory and stability theory of discrete dynamical system,local bifurcation theory,the bifurcations of the fixed points are strictly analyzed.We control the flip bifurcation and Neimark-Sacker bifurcation by using the linear state feedback method and the hybrid control method respectively.And the corresponding numerical simulations not only certify the theoretical analysis results,but also verbalize more dynamic behaviors.This paper is comprised of four chapters:In the first chapter,we chiefly generalize the background of discrete dynamical systems,the development of the discrete FHN system and the discrete BVP model.And the basic theories are presented,including local bifurcation theory of the fixed points,bifurcation control methods,and the Marotto’s theorem.In the second chapter,the discrete FHN systems are exhaustively investigated as follows: We discuss the existence,stability,and types of the fixed points;The sufficient conditions for the existence of the period-2 orbit are rendered;Using center manifold theory and local bifurcation theory,we demonstrate the existence of pitchfork bifurcation,fold bifurcation,flip bifurcation,and Neimark-Sacker bifurcation;A linear state feedback controller is designed to accurately control the bifurcation value of flip bifurcation and Neimark-Sacker bifurcation,which doesn’t change the position of the bifurcation point;What’s more,We strictly proves that the chaos in the sense of Marotto exists under certain parameter conditions by applying the Marotto’s theorem;Meanwhile,bifurcation diagrams and phase diagrams obtained by numerical simulations verify the accuracy of theoretical analysis,and phase diagrams exhibit some complex dynamics such as high-period orbits.In the third chapter,the bifurcations in the discrete BVP model are discussed.We explore the types of the fixed point,the sufficient parameter conditions that the fixed point undergo the flip bifurcation and Neimark-Sacker bifurcation.These bifurcations are controlled to delay by adopting the hybrid control strategy which adjust the controllable parameters.In the fourth chapter,we conclude the whole thesis,and project the crucial research contents and directions of future work.