Research On The Stochastic Dynamics In Fractional Typical Economic Systems

Economic fluctuation is an important issue of macroeconomics and has been investigated by many scholars in recent years.It has guiding significance for making macroeconomic policy to have theoretical and application study on this issue.Business cycle and economic growth have always been the hot topics in the field of economics,and attracted more and more scholars' attention.Considering the memory property of economic variables in business cycle,this paper introduces fractional derivative to depict this features.At the same time,the time delay parameter is introduced to depict the inevitable time lag phenomenon in the real economy system.Then,the economic growth model with the disruption of ecosystem is built,and the efficiency of ecological environmental governance and the influence of random factors on economic growth is investigated.The main content of the paper is as follows:1.Stability and amplitude of an economy system with fractional derivative are studied comparing with classical Goodwin model.Firstly,the influence of the memory property of economic variables on the stability of the economy system is investigated.The result show that an economy system with fractional derivative cost more time to be the equilibrium state.Secondly,how fractional derivatives influence and transform the amplitude of the economic fluctuation is studied,and the results show that memory property of economic variables can lead to some different phenomena comparing with the model without considering the memory property of economic variables.2.A Goodwin type business cycle model with fractional-order time delay which describes the delay and memory property of macroeconomic regulation is investigated.Stochastic averaging method is applied to obtain the averaged stochastic differential equation.By solving Fokker-Planck-Kolmogorov equation,one can obtain the stationary probability function.Numerical simulations are done to verify the method.The effects of the fractional order,time delay,macroeconomic regulation and random excitation on the amplitude of economy system are investigated.The results show time delay,fractional order and intensity of random excitation can all magnify the amplitude and increase the volatility of the economy system.3.Multi-scales method is applied to obtain the primary resonance response of the business cycle model with fractional delay under the narrowband random excitation.The steady-state response in deterministic and stochastic cases.The validity of the method is verified by the simulation solutions.By analyzing two examples,the effect of the fractional derivative on the amplitude of the system and the stationary probability function is obtained.When the consumption function is nonlinear function,the change of the fractional derivative and time delay can both induce the stochastic jump.Also,the intensity of the random excitation can induce the P-bifurcation of the stationary probability function.4.A nonlinear model including capital accumulation,technological progress,resource consumption and the disruption of ecosystem to interpret the human forces is built to investigate the mechanism of human activities' forcing the earth's biosphere.The tipping point of the system with disruption of ecosystem is investigated.It shows economic growth is unsustainable with neglecting the disruption of ecosystem.And a collapse of the economic growth will happen.Then,two different kinds of environmental governance to control the tipping point are studied.And the tipping points of economic growth with two kinds of environmental governance are computed.The efficiencies of the environmental governances are investigated.The results indicate green production mode is just the crucial factor to sustain the economic growth.Finally,the effect of initial condition and random excitation on the steady-state of economic growth is investigated.The results show the initial condition of capital and technical skill can determine the steady-state of the economic growth.The stochastic shocks can change the value of the tipping points and disrupt the system away from the stable state.5.Considering the memory property,fractional derivative is added to the economic growth model with the disruption of ecosystem.The stability of the equilibrium state is investigated.Then the bifurcation properties of the parameters to sustain the economic growth are studied.The result show that the stability of the equilibrium state can be changed with considering the memory properties.The effect of random factors on the equilibrium state is also investigated,in indicates that the random shocks can disrupt the system away the stable state.