考虑记忆性质与时间滞后效应的非线性经济周期模型分析
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摘要
考虑非线性经济周期模型中经济变量存在记忆性质与时间滞后现象,研究随机周期作用激励下Goodwin模型的随机响应,以此研究记忆性质与时间滞后现象对经济周期波动的具体影响。通过随机多尺度方法得到了模型的确定性与随机情形下的稳态响应。结果发现:当考虑非线性投资函数时,经济变量的时间记忆性质和时间滞后现象均可以导致经济波动方式的改变;当考虑非线性消费函数时,经济变量的时间记忆性质与时间滞后现象均可以诱导出经济周期波动的随机跳跃现象,即引发经济系统的突变。同时,随机周期作用也可以诱发系统出现稳态概率密度函数的分岔现象出现,说明外部随机周期作用可以诱发经济系统的突变现象产生。
In this paper,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 responses in deterministic and stochastic cases are studied.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 investigated.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.
In this paper,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 responses in deterministic and stochastic cases are studied.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 investigated.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.
引文
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[24]Hattaf K,Riad D,Yousfi N.A Generalized Business Cycle Model with Delays in Gross Product and Capital Stock[J].Chaos Solitons&Fractals,2017,98.
[2]李佼瑞.概述随机非线性动力系统在经济周期研究中的运用[J].统计与信息论坛,2012,27(4).
[3]Goodwin R M.The Nonlinear Accelerator and the Persistence of Business Cycles[J].Econometrica,1951,19(1).
[4]Strotz R H,Mcanulty J C,Naines J B.Goodwin's Nonlinear Theory of the Business Cycle:An Electro-Analog Solution[J].Econometrica,1953,21(3).
[5]Lorenz H W,Nusse H E.Chaotic Attractors,Chaotic Saddles,and Fractal Basin Boundaries:Goodwin's Nonlinear Accelerator Model Reconsidered[J].Chaos Solitons&Fractals,2002,13(5).
[6]Li Jiaorui,Feng Changshui.First-passage Failure of a Business Cycle Model under Time-delayed Feedback Control and Wide-band Random Excitation[J].Physica A:Statistical Mechanics&Its Applications,2010,389(24).
[7]Li Jiaorui,Ren Zhengzheng,Wang Zuoren.Response of Nonlinear Random Business Cycle Model with Time Delay State Feedback[J].Physica A:Statistical Mechanics&Its Applications,2008,387(23).
[8]Li Shuang,Li Qian,Li Jiaorui,et al.Chaos Prediction and Control of Goodwin’s Nonlinear Accelerator Model[J].Nonlinear Analysis:Real World Applications,2011,12(4).
[9]Rong Haiwu,Wang Xiangdong,Xu Wei,et al.Resonant Response of a Non-linear Vibro-impact System to Combined Deterministic Harmonic and Random Excitations[J].International Journal of Non-Linear Mechanics,2010,45(5).
[10]Xu Yong,Li Yongge,Liu Di,et al.Responses of Duffing Oscillator with Fractional Damping and Random Phase[J].Nonlinear Dynamics,2013,74(3).
[11]Xu Yong,Li Yongge,Liu Di.Response of Fractional Oscillators with Viscoelastic Term under Random Excitation[J].Journal of Computational&Nonlinear Dynamics,2014,9(3).
[12]Xu Yong,Li Yongge,Liu Di.A Method to Stochastic Dynamical Systems with Strong Nonlinearity and Fractional Damping[J].Nonlinear Dynamics,2016,83(4).
[13]Lin Zifei,Li Jiaorui,Li Shuang.On a Business Cycle Model with Fractional Derivative under Narrow-band Random Excitation[J].Chaos,Solitons and Fractals,2016,87.
[14]Machado J T,Duarte F B.Fractional Dynamics in Financial Indices[J].International Journal of Bifurcation&Chaos,2012,22(10).
[15]Pan I,Das S,Das S.Multi-objective Active Control Policy Design for Commensurate and Incommensurate Fractional Order Chaotic Financial Systems[J].Applied Mathematical Modelling,2015,39(2).
[16]Tsuzuki E.A New Keynesian Model with Delay:Monetary Policy Lag and Determinacy of Equilibrium[J].Economic Analysis&Policy,2014,44(3).
[17]Maria C D,Smulders S,Werf E V D.Absolute Abundance and Relative Scarcity:Environmental Policy with Implementation Lags[J].Ecological Economics,2012,74(1).
[18]Niu Jiangchuan,ShenYongjun,Yang Shaopu,et al.Analysis of Duffing Oscillator with Time-delayed Fractional-order PIDController[J].International Journal of Non-Linear Mechanics,2017,92.
[19]Gori L,Guerrini L,Sodini M.Disequilibrium Dynamics in a Keynesian Model with Time Delays[J].Communications in Nonlinear Science&Numerical Simulation,2018,58.
[20]Vliet K V D,Reindorp M J,Fransoo J C.The Price of Reverse Factoring:Financing Rates vs Payment Delays[J].European Journal of Operational Research,2015,242(3).
[21]Matsumoto A,Szidarovszky F.Continuous Hicksian Trade Cycle Model with Consumption and Investment Time Delays[J].Journal of Economic Behavior&Organization,2010,75(1).
[22]Lin Zifei,Xu Wei,Li Jiaorui,et al.Study on the Business Cycle Model with Fractional-Order Time Delay under Random Excitation[J].Entropy,2017,19(7).
[23]Matsumoto A,Szidarovszky F.Nonlinear Multiplier-accelerator Model with Investment and Consumption Delays[J].Structural Change&Economic Dynamics,2015,33.
[24]Hattaf K,Riad D,Yousfi N.A Generalized Business Cycle Model with Delays in Gross Product and Capital Stock[J].Chaos Solitons&Fractals,2017,98.
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