Research On Chaos Property And Shadowing Property Of Dynamic Economic System

With the development of the society,the economic phenomenon is chang-ing from day to day.In order to control the future value of some economic indicators,researchers put forward various economic theories as well as analyt-ical methods and established various models.In fact,many current economic theory theories have limited ability to describe the real economy.The pos-sible reason for this situation lies in the imperfection of economic theory or analytical tools.With the development of natural science and social science,researchers found that some theories in natural science are also applicable to social science under certain conditions.For instance,chaos theory and track-ing theory are widely used and closely related in natural science.With the increasingly wide application of computer technology,economic system sim-ulation and numerical calculation have become one of the important topics,This dissertation is mainly based on the following two current situations:(1)In the computer simulation of economic systems,numerical errors always exists in the calculation of equilibrium trajectory described by the system(such orbit in dynamics is called pseudo orbit),but researchers want to make sure what they see on the computer screen is approximately the same as the genuine orbit of the system,thus the way to solve this contradiction is to shadow the classical notion which is closely related to the pseudo orbit and the genuine orbit of the system.However,there are few literature on the shadowing of economic system.(2)During the simulation of the economic system,researchers found that chaos still exists in the economic system.However,the research on the chaos of such economic system is limited to numerical simulation,and there is very little theoretical proof for the initial value sensitivity of economic systems(an important feature of chaos).This dissertation mainly studies the sensitivity of backward dynamic eco-nomic systems and the shadowing properties of economic system,so that some conclusions can be drawn.Those results are significant for analyzing the essence of economic systems.The introduction details are as follows:Firstly,three widely used economic models and the conditions under which they generate backward dynamics were introduced briefly.And an essential step was to generalize the definition of sensitivity and Li-Yorke sensitivity for dynamic systems and introduce the definition of P-sensitivity and P-Li-Yorke sensitivity for multi-valued forward dynamic systems,so that the gen-eralized definition can be applied to analyze economic models with backward dynamic.Particularly,as for the multi-valued forward dynamic systems,the connections between P-sensitivity(P-Li-Yorke sensitivity)of multi-valued forward dynamic systems and sensitivity(Li-Yorke sensitivity)of the induced single-valued dynamic systems were studied.In other words,they are equiva-lent respectively.Similarly,the relationship between sensitivity(Li-Yorke sen-sitivity)of single-valued forward dynamic system and sensitivity(Li-Yorke sen-sitivity)of the induced single-valued dynamic systems were studied.Secondly,the possible geometric forms of the dynamic trajectory of cap-ital equilibrium in several classical economic systems were classified and the various shadowing properties of different geometric forms were studied.It is meaningful to consider that the average error is very small in the numerical calculation of the model.Therefore,this dissertation also studied the rela-tionship between the asymptotic average shadowing property of the dynamic system function f and the asymptotic average shadowing property of the in-duced map g_?.In practice,sometimes it is only necessary to shadow certain segments of the system's orbit,thus the relationship among specification prop-erty,weak specification property,almost specification property,asymptotically average shadowing property and topological entropy were also studied.it turns out that weak specification property always implies the asymptotic average shadowing property,which generalized the result that the weak specification property implies the asymptotic average shadowing property for surjective dy-namical system.This dissertation also prove that there exists a large class of dynamical systems with both the almost specification property and zero topological entropy which do not coverage to some fixed points uniformly.Thirdly,according to the commonly used treatment of qualitative vari-ables in econometrics,the various levels or states of qualitative factors that cannot be accurately described by numerical values were described with sym-bols.Using the symbolic dynamic system theory and the iterative function system theory as basic tools,the research generalized the definition of two-sided limit shadowing property on iterated function system and introduces the definition of two-sided asymptotic shadowing property on iterated function sys-tem.The relationship among chain transitivity,totally transitivity,shadowing property and two-sided asymptotic shadowing property on iterated function system were studied.Finally,this dissertation studied a macroeconomic models with credit fric-tion.Particularly,the research discussed the case where there are only two heterogeneous items in the model and proved that the capital dynamics of the model are sensitive to initial values.