Modeling And Dynamics Learning Of Several Complex Systems

From the composition of the entire universe to all areas of human social activity,and then to the small atomic structure,complex systems are everywhere.Complexity science,which originated in the 1970s and focuses on complex systems,represents a new level of human understanding of the world.The breakthrough and innovation of complexity science in research methodology,in a sense,can be said to be a reformation in methodology or way of thinking.With the development of complexity science research,complex systems theory has been sufficiently improved,and substantial progress has been made in the quantitative understanding of complex systems by combining basic theories with computer simulations.In a complex system,there may be complex interactions among its components,allowing the system to exhibit a variety of complex dynamic behaviors.Based on complex systems theory,we study the modeling and dynamics of several complex systems related to economic,social,and physical fields to reveal the inherent characteristics and evolutionary laws of the systems.This paper is divided into six chapters.In the first chapter,Introduction,we introduce the development and significance of complex systems and complexity science,elucidate the importance of dynamics and introduce several important categories of complex systems—economics and markets,human society and physical systems.We also introduce some of the mathematical techniques involved in the study of complex systems,including nonlinear dynamics and chaos,complex networks,game theory,and neural networks.The second to fourth chapters are the main part and innovation of this paper.In the fifth chapter,we first summarize what we have done,and on the basis,make further research plans and prospects.First,we combine economic games and chaos theory to study the complex dynamics of the contest game model with bounded rationality.Contest theory is widely used in many different fields such as society and economics.At each step of the contest,the agent simultaneously decides how much effort to make to compete for an object,and the probability of the agent winning the object is related to his/her effort.Different from the classical Cournot and Stackelberg models,the natural asymmetric nonlinear demand characteristic allows the contest system to exhibit highly complex dynamic behavior.For example,there exist two different routes to chaos for the system:the period-doubling/flip bifurcation which leads to periodic cycles and chaos,and the Neimark-Sacker bifurcation which derives an attractive invariant closed curve.There are significant differences between the two routes in the economic perspective.Through the complex dynamic study of the contest model with bounded rationality,we analyze the characteristics of economic system and the evolution laws of economic dynamics,reveal the causes of instability in social and economic markets,and provide short-term predictions and long-term control methods for unstable markets.Second,we study the coevolution of game dynamics and network topologies to reveal potential mechanisms for promoting cooperation in social systems.Cooperation plays an important role in the development of human civilization and society.How cooperation emerges in human society,however,is both an evolutionary puzzle and a practical one that has real implications for social harmony.Recently,scientists have revealed multiple mechanisms for promoting cooperation,one of which is population structure,as it enables localized reciprocity.But this explanation assumes static social interactions,whereas human interactions are often dynamic.To this end,we propose a social reward and punishment mechanism,combined with network reciprocity,to achieve the co-evolution of individual behavior and network structure.The results show that rewards and punishments are not simply against each other;on the contrary,they are interdependent in promoting cooperation.Neither a single reward measure nor a pure punishment mechanism can extensively promote cooperation.Instead,a combination of appropriate punishment and reward mechanisms can greatly promote the emergence of cooperation and maintain social order and development.In addition,under this combination mechanism,the heterogeneity of social relations and individual influence occurs spontaneously,which support the persistence of cooperative behavior.In a way,we provide unique insights into the emergence of structural heterogeneity in networks from the perspective of evolutionary game theory.Finally,we use neural network as a tool to realize the automatic modeling of complex dynamic system under the data drive,so that we can quickly identify a batch of soliton solutions.Because the solitons can maintain their shapes during propagation and collisions,they are attractive for both theoretical studies and practical applications,such as using the soliton as a carrier of information to achieve ultra-long-distance and ultra-large capacity communication.The numerical determination of solitary states of nonlinear partial differential equations is crucial in various research fields such as Bose-Einstein condensates(BECs),and nonlinear optics.Traditional methods,such as Newton’s method and the variational method,can figure out one soliton at a time by solving the stationary equation based on strong prior information of soliton structure.We develop a machine learning system called the complex-valued neural operator instead of traditional differential equations to model dynamical systems,and combined with an energy-constrained optimization search algorithm,we can directly identify a batch of soliton solutions within a certain energy range.We concretely apply our approach to the one-dimensional BEC systems with both homogeneous and inhomogeneous nonlinearities.In terms of computational complexity,calculations based on complex-valued neural operators are carried out more efficient than those based on traditional partial differential equation solvers.This advantage also facilitates the stability analysis of solitons.Our work is a new attempt at numerical identification of solitons and provides a new perspective for the numerical study of solitons in nonlinear quantum systems,which is promisingly useful in the study of such nonlinear quantum systems as the inhomogeneous BEC and nonlinear optical systems.