Hopf Bifurcation And Applications Of Economic System Based On Complexity Theory

Along with the development and progress of society, the economic and financial system we facedbecome more and more complex, and the root of the situation of leading to this complexity is internalnonlinearity of economic system, which prompted scholar have the research on the laws of thenonlinear behind the complex economic phenomena, and then reveals the essence of economicphenomenon, so as to guide practical economic activity.Based on the research work of scholars,we use the theory and method of management science,economics and nonlinear dynamic system for the duties as follows:This paper summarized the research progress of nonlinear economic theory, including the twofields of chaos and fractal t heory, enumerated the important nonlinear model in the field of economy,then this paper introduces the application of the two big classical nonlinear theories: chaos, fractal inthe field of economic.Based on the research work of scholars at home and abroad. we ha ve a research on a series ofnonlinear ch aracteristics u nder th e co ndition of different param eter com bination of econom ic andfinancial system, including: stable node, saddle points, bifurcation, Hopf bifurcation. And then westudy the parameters interdepende nce a nd the changed ci rcumstances u nder t he con dition ofcombination of a b c, and conduct the global complexity analysis. Focus on the research ofeconomic stag nation ri gid, stab ility o r not stab le growth, i nflation, an d eco nomic depression oreconomic situation out of control, etc, which can lead to significant social unrest and correspond to thevariation of parameter.This paper studies a series of Hopf bifurcation of complex economic system and the conditions ofparameter evolution under the condition of high resilience, the stability of the period orbits before Hopfbifurcation appears, and successively research the system parameters’ near value of topologicalstructure evolution, and according to Taken’s estimate, have the research on the complexity of internalevolution of the system under the condition of all kinds of cases above, Apply recovery plan (RP) andthe coherent recovery plan (CRP) method, combining the ApEn algorithm complexity analysis, havethe research on time series complexity and different characteristics of the time-series data recovery planand the coherent recovery plan, research the Lorenz system, stable focal point, saddle points,bifurcation and Hopf bifurcation point, time-series data recovery plan and the coherent recovery plan,and studies the complexity of its corresponding timing data, and got five new recovery plan and thecoherent recovery plan。This paper have the research on inner complexity of a class o f financial system, through numericalsimulation, this paper found two paths which causes the system into chaos: The Ruelle-Takens path tothe chaos, and high order equilibrium point bifurcation meet period-doubling bifurcation leads to chaos, which produce the singular chaotic attractor (SNA) in this process; Then used the delay parameterτas a bifurcation parameter to study the influence of the delay to the financial system;Derived the mathematical expression of the economic system’s lyapunov index,with the change ofJacobain matrix, and have a research on the situation of the real part changes with the parameters a, b,c, fu ther study th e system en ter into th e bifurcation, chao tic w ay under th e co ndition of d ifferentparameter combination,which lay the foundation for chaotic control of the system.The result of this paper will have further promote to thorough analysis of the economic and financialsystem’s internal operation condition, puts forward two new paths of the system into chaos, and providea basis for the government to make policy while controlling the economic system, and it has a practicalvalue.