Conley Index Theory Of Interconnected Networks And Its Applications To Propagation Dynamics

The study of the theory about differential dynamical systems and its applications in other fields are two important contents in applied mathematics.This doctoral Disserta-tion considers the theory and applications of dynamical systems,mainly in following three aspects:the establishment and analysis of an infectious disease model on a directed in-terconnected network,the generalization of winding number theory in a planar dynamical system,and the use of Conley index theory to study the dynamical structure between two attractors.The main research contents of the dissertation are listed as follows:First of all,there are increasing indications that many real networks interact with each other,and sometimes the spread of disease on the network is directed or asymmetrical.Based on these two facts,an epidemic model in a directed interconnected network is established.At the same time,this model is also a generalization of the epidemic model in the undirected network,single-layer network,and bipartite network.By calculation,we obtain that the basic reproduction number R₀,which is one of the most concerning parameters in epidemiology and also obtain the basic reproduction number of some special networks.Through theoretical analysis,we explain the reason that coupling leads to the increase of basic reproduction number,and also give a necessary condition for disease outbreak caused by coupling.Besides,we give the necessary and sufficient conditions for a disease outbreak in the whole network,and the necessary conditions for disease outbreak only in one single subnetwork.Interestingly,the latter can only occur in the directed interconnected networks,but not in the undirected interconnected networks.Finally,through numerical simulation,we also find that the independence of a joint degree greatly reduces the influence of heterogeneity of degree on the prevalence.Next,researches usually think that the winding number of a simply closed curve and the index of an equilibrium can only be defined on a planar dynamical system,but in this dissertation,we generalize this index theory to an n-dimensional system.For a closed,oriented?n-1?-dimensional hypersurface S,we define the winding number S as the number of the image f?S?of S under the vector field winding around the origin,namely,w?S?=¹/??_n????_s^*??₀?.Some properties of the winding number of a hypersurface are given,including discreteness,Non-triviality,and homotopy invariance.As a result,we can define the winding number of a point,which is defined as the winding number of a hypersurface that contains no other equilibria except this point,and its additivity and continuity are proved as well.We also contrasted the winding number with other topological theories and find that the Brouwer degree of continuous mappings can be defined directly in terms of the winding number.We also find that the winding number and Conley index theory have many similar characteristics,for example,both of them are insensitive to dimensions and have the same research ideas.Finally,some applications corresponding to the properties of winding number are given.Finally,the Conley index was used to study the dynamical structure of two attractors.The background of this study is that bistable phenomena are very common in the field of life-related sciences.In the range from the microscopic gene expression to the macroscopic species competition,biological models with the bistable structure are ubiquitous.In this dissertation,the stable state is regarded as an attractor in a dynamical system,and two types of bistable structures are studied:one is bistable with two point-attractors,the other is bistable with one point-attractor and one cycle attractor.Our approach is that we can obtain the detailed dynamical behavior of a bistable structure by gradually increasing the preconditions,and then relaxing the preconditions to arrive at a more general conclusion about bistable structures.We find that under certain conditions,no matter which kind of bistable structure,there is always an invariant set between the two attractors,and there also exist connecting orbits from the invariant set to the two attractors,respectively.Also,in each type of bistable structures,there is a separatrix or cycle separatrix,which divides the area concerned into two sub-areas,and almost all the orbits in each sub-area tend to the corresponding attractor.Finally,a competitive model in ecology is used as an example to explain how the results are applied,and four other biological models are given to show the commonness of bistable structures.