The Non-linear Analysis And The Dynamics Research For Non-linear Economical Period Systems

This paper uses the method of multiple scales and Chebyshev orthogonal polynomi-al approximation to transform a system of nonlinear economic fluctuations, and focuses on system dynamic analyses through the stability theories of ordinary differential equa-tions as well as bifurcation theory, then finds the factors affecting the economic system operation.1.This paper summarizes firstly the development of business cycle theory (RBC) and the current research overseas and introduces the research purpose and significance. Then elaborates a series of theories such as the stability, Hopf bifurcation theory, bifurcation control method, multiple scales method and Chebyshev polynomial approximation.2.The second part is a brief description of different economic fluctuations theories. This paper obtains the Samuelson-Hicks model by improving the Samuelson’s multiplier acceleration principle, and transfers the discrete system into continuous system using the calculus of variation. Then we transform the external excitation system of spontaneous function into an autonomous system by applying multiple scales method. Through the Hopf bifurcation analysis to the stability of this system, combined with numerical simu-lation, the factors affecting the stable operation of the economy are achieved. After that, we work on the hysteresis phenomenon of economic policy by adding the time delay con-trol, at last, achieves a preliminary understanding of the time delay effects to the Hopf bifurcation system.3.In the third part of this paper, considering the marginal propensity to save (MPS) change randomly with the economic conditions in the dynamic economies currently, we add the random saving rate. Then by the Chebyshev orthogonal polynomial approxima-tion, we gain a deterministic system with equivalent to the original system, and analyzes the stability and bifurcation accordingly in the end.