June  2017, 22(4): 1565-1573. doi: 10.3934/dcdsb.2017075

Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model

School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel

Received  May 2016 Revised  October 2016 Published  February 2017

The known nonlinear delay differential neoclassical growth model is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the zero or a positive equilibrium. Sufficient conditions for stability in probability of the positive equilibrium and for exponential mean square stability of the zero equilibrium are obtained. Numerical calculations and figures illustrate the obtained stability regions and behavior of stable and unstable solutions of the considered model. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with the order of nonlinearity higher than one.

Citation: Leonid Shaikhet. Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1565-1573. doi: 10.3934/dcdsb.2017075
References:
[1]

Bradul and L. Shaikhet N., Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dyn. Nat. Soc., 2007 (2007), 92959-92983. 

[2]

Chen and W. Wang W., Global exponential stability for a delay differential neoclassical growth model, Adv. Differ. Equ., 2014 (2014), 325. 

[3]

Day R., Irregular growth cycles, Am. Econ. Rev., 72 (1982), 406-414. 

[4]

Day R., The Emergence of chaos from classical economic growth, Q. J. Econ., 98 (1983), 203-213. 

[5] I. Gikhman and A. V. Skorokhod I., Stochastic Differential Equations, Springer. Berlin, 1972. 
[6]

Matsumoto and F. Szidarovszky A., Delay differential neoclassical growth model, J. Econ. Behav. Organ., 78 (2011), 272-289. 

[7]

Matsumoto and F. Szidarovszky A., Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5 (2013), 440-455. 

[8]

J. Nicholson A., An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65. 

[9] Shaikhet L., Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer. London. Dordrecht. Heidelberg. New York, 2011. 
[10] Shaikhet L., Lyapunov functionals and Stability of Stochastic Functional Differential Equations, Springer. Dordrecht. Heidelberg. New York. London, 2013. 

show all references

References:
[1]

Bradul and L. Shaikhet N., Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dyn. Nat. Soc., 2007 (2007), 92959-92983. 

[2]

Chen and W. Wang W., Global exponential stability for a delay differential neoclassical growth model, Adv. Differ. Equ., 2014 (2014), 325. 

[3]

Day R., Irregular growth cycles, Am. Econ. Rev., 72 (1982), 406-414. 

[4]

Day R., The Emergence of chaos from classical economic growth, Q. J. Econ., 98 (1983), 203-213. 

[5] I. Gikhman and A. V. Skorokhod I., Stochastic Differential Equations, Springer. Berlin, 1972. 
[6]

Matsumoto and F. Szidarovszky A., Delay differential neoclassical growth model, J. Econ. Behav. Organ., 78 (2011), 272-289. 

[7]

Matsumoto and F. Szidarovszky A., Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5 (2013), 440-455. 

[8]

J. Nicholson A., An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65. 

[9] Shaikhet L., Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer. London. Dordrecht. Heidelberg. New York, 2011. 
[10] Shaikhet L., Lyapunov functionals and Stability of Stochastic Functional Differential Equations, Springer. Dordrecht. Heidelberg. New York. London, 2013. 
Figure 4.1.  Stability regions for equation (1.3) (red and green) and equation (3.3) (yellow), $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$
Figure 4.2.  Stability regions for equation (1.3) (red and green) and equation (3.3) (yellow), $\gamma=2$ , $b=2$ , $h=0.02$ , $p=20$
Figure 4.3.  Trajectories of solution of equation (1.3) in unstable equilibrium for different initial functions: $x_0=1.19$ (green), $x_0=1.1805$ (red), $x_0=1.17$ (yellow), $A(700,300)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x_1.*=1.1805$
Figure 4.4.  Trajectories of solution of equation (1.3) in stable equilibrium for different initial functions: $x_0=4.6$ (green), $x_0=1.65$ (red), $x_0=1.1$ (yellow), $A(700,300)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x_2.*=3.1270$
Figure 4.5.  Trajectories of solution of equation (1.3) in unstable equilibrium for different initial functions: $x_0=0.6536$ (yellow) and $x_0=4.5215$ (red), $B(900,200)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x_1.*=0.6527$ and $x_2.*=4.5215$
Figure 4.6.  Trajectories of solution of equation (3.3) in stable zero equilibrium for different initial functions: $b_0=0.8$ (green), $b_0=1.55$ (red), $b_0=100$ (yellow), $C(600,400)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x.*=0$
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