Figure 4.1.
Stability regions for equation (1.3) (red and green) and equation (3.3) (yellow), $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$
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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 |
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.
Keywords:
Delay differential neoclassical growth model, stability in probability,
zero and positive equilibrium points,
stochastic perturbations,
exponential mean square stability.
Mathematics Subject Classification:
34K20, 34K50, 65C20, 65C30.
Citation:
Leonid Shaikhet. Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete & Continuous Dynamical Systems - B,
2017, 22
(4)
: 1565-1573.
doi: 10.3934/dcdsb.2017075
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References:
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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. Google Scholar |
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| [5] | I. Gikhman and A. V. Skorokhod I., Stochastic Differential Equations, Springer. Berlin, 1972. Google Scholar |
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Matsumoto and F. Szidarovszky A., Delay differential neoclassical growth model, J. Econ. Behav. Organ., 78 (2011), 272-289. Google Scholar |
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Matsumoto and F. Szidarovszky A., Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5 (2013), 440-455. Google Scholar |
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J. Nicholson A., An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65. Google Scholar |
| [9] | Shaikhet L., Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer. London. Dordrecht. Heidelberg. New York, 2011. Google Scholar |
| [10] | Shaikhet L., Lyapunov functionals and Stability of Stochastic Functional Differential Equations, Springer. Dordrecht. Heidelberg. New York. London, 2013. Google Scholar |
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. Google Scholar |
| [2] |
Chen and W. Wang W., Global exponential stability for a delay differential neoclassical growth model, Adv. Differ. Equ., 2014 (2014), 325. Google Scholar |
| [3] |
Day R., Irregular growth cycles, Am. Econ. Rev., 72 (1982), 406-414. Google Scholar |
| [4] |
Day R., The Emergence of chaos from classical economic growth, Q. J. Econ., 98 (1983), 203-213. Google Scholar |
| [5] | I. Gikhman and A. V. Skorokhod I., Stochastic Differential Equations, Springer. Berlin, 1972. Google Scholar |
| [6] |
Matsumoto and F. Szidarovszky A., Delay differential neoclassical growth model, J. Econ. Behav. Organ., 78 (2011), 272-289. Google Scholar |
| [7] |
Matsumoto and F. Szidarovszky A., Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5 (2013), 440-455. Google Scholar |
| [8] |
J. Nicholson A., An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65. Google Scholar |
| [9] | Shaikhet L., Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer. London. Dordrecht. Heidelberg. New York, 2011. Google Scholar |
| [10] | Shaikhet L., Lyapunov functionals and Stability of Stochastic Functional Differential Equations, Springer. Dordrecht. Heidelberg. New York. London, 2013. Google Scholar |
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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