Bifurcation Dynamics Analysis And Application Of Coupled Discrete Map

The thesis devotes to the bifurcation analysis and its application on the synchronizability of a two-dimensional global coupled non-invertible discrete-time map.It elaborates on various of codimension-one and codimension-two bifurcations of this coupled system.The detailed bifurcation analysis and proofs are addressed.Several bifurcation software packages,e.g.,Ant4.669,Matcont M,are employed to verify numerically the presented theoretical results.The bifurcation analysis is also applied to the synchronizability of the coupled map.The main researches are given as below:1.A two-dimensional non-invertible coupled discrete-time map is constructed by using a symmetric coupling of conventional one-dimensional logistic maps.It is intrinsic symmetry due to the global coupled structure.In this part,the existence of fixed points together with their stability analysis of this coupled system is given.All possible bifurcations of this coupled system at each fixed point are listed.2.Based on bifurcation theory and center manifold theorem,the existence of codimension-one bifurcations of this two-dimensional coupled system is proved theoretically.For each codimension-one bifurcations,the critical normal form coefficients are calculated to check the non-degeneracy conditions and predict the bifurcation scenarios around the bifurcation points.The corresponding propositions are given.The theoretical results show that the coupled system undergoes such as,transcritical,pitchfork,symmetry-breaking,period-doubling and Neimark-Sacker bifurcations.To demonstrate the dynamics and to validate the theoretical results,the concerned phase diagrams are presented and the bifurcation curves of fixed points are plotted with the aid of the numerical continuation method.3.The part focuses on the codimension-two bifurcations of the coupled system.Firstly,two types of codimension-two bifurcation,namely,transcritical-flip bifurcation and pitchfork-flip bifurcation,are expounded initially.The existence of them is confirmed by the non-degenerate conditions.In addition,the evolution of two co-existence fixed points is given.And the transition sets of the fixed points are demarcated by the bifurcation curves clearly on the two-parameter bifurcation plane.The boundary curves of transition sets are completely in conformity with the continuations of codimension-one bifurcations.Secondly,the existence of strong resonance bifurcations,such as 1:2,1:3,and 1:4 is proved theoretically.The corresponding critical normal form coefficients are calculated to check the non-degeneracy conditions.All strong resonance bifurcations are detected and confirmed by numerical continuation method of fixed points of iterations on the two-parameter plane.Finally,the improved isoclines method is employed to the detection on both strong and weak resonances of coupled system.The cross points of Neimark-Sacker curve and the isoclines of different arguments coincide with the cusps of the Arnold tongues.More importantly and meaningful,the parametric loci of cusp points also can be specified explicitly.4.The bifurcation analysis is also applied to study the synchronizability of coupled system.The dynamics and bifurcation behaviors on the complete synchronized subspace of this two-dimensional coupled system are portraited.The tangential and transversal exponents are calculated numerically to illustrate the transverse/tangency stability of the periodic attractors embedded in the chaotic regions.The mechanisms of synchronizability,and strong and weak resonances generated by symmetrically coupled two-and three-dimensional Logistic maps are given by bifurcation analysis.