Limit Cycle For A Class Of Multi-scale Problems Of Triangular Networks

Complex networks widely exist in many scientific and engineering fields,such as food chain networks,neural networks,and intelligent cluster networks.They are one of the important topics that have attracted much attention in the current era of big data and artificial intelligence.In the study of practical problems,people often use certain laws or properties to reduce the modeling of specific complex networks,triangular networks and multi-scale systems are the most important and common reduction models.Geometric singular perturbation theory is one of the important methods in the study of multi-scale systems,its essence is to analyze the fast and slow variables corresponding to the corresponding system,and then make a comprehensive analysis according to the local properties,the global dynamic behavior of the original system is obtained.The blow-up method expands the application range of the geometric singular perturbation theory,and can further study the dynamic behavior near the non-hyperbolic point,which changes the difficulty of the normal non-hyperbolic manifold analysis.In recent decades,based on the above two methods,people have made fruitful achievements in the study of multi-scale systems,especially the relaxation oscillation,duck solution,and the existence of other limit cycles of the planar system.This paper mainly studies the existence of limit cycles for a class of triangular network multi-scale problems.We use the geometric singular perturbation theory and the blow-up method to perform local analysis on nonhyperbolic points,and obtain the dynamic behavior near non-hyperbolic points.Finally,by means of Poincare mapping,we obtain the global analysis of the system,and prove that there is a periodic orbit in the system.This paper is divided into three chapters.The first chapter is the introduction,which briefly introduces the research background of triangular network,geometric singular perturbation and blow-up method.The second chapter introduces the preparation knowledge of fast and slow systems,Fenichel theorem and blow-up method.The third chapter is the local analysis and global analysis of complex network systems,local analysis is to study the dynamic behavior near non-hyperbolic points,and global analysis is to prove the existence of limit cycles of the system.