Numerical Study of the Effect of Stochastic Disturbances on the Behavior of Solutions of Some Differential Equations
Abstract
Nowadays interest of the deterministic differential system of Lorentz equations is still primarily due to the problem of gas and fluid turbulence. Despite numerous existing systems for calculating turbulent flows, new modifications of already known models are constantly being investigated.
In this paper we consider the effect of stochastic additive perturbations on the Lorentz convective turbulence model. To implement this and subsequent interpretation of the results obtained, a numerical simulation of the Lorentz system perturbed by adding a stochastic differential to its right side is carried out using the programming capabilities of the MATLAB programming environment.
References
2.Sparrow C. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Applied Mathematical Sciences, vol. 41. Springer, New York, NY; 1982. (In Eng.) DOI: https://doi.org/10.1007/978-1-4612-5767-7
3.Sparrow C. An introduction to the Lorenz equations. IEEE Transactions on Circuits and Systems. 1983; 30(8):533-542. (In Eng.) DOI: https://doi.org/10.1109/TCS.1983.1085400
4.Leonov G.A., Kuznetsov N.V.
5.Boichenko V.A., Leonov G.A., Reitmann V. Dimension Theory for Ordinary Differential Equations.Teubner-Texte zur Mathematik, vol. 141. Teubner, Stuttgart; 2005. (In Eng.)
6.Leonov G.A. Criteria for the existence of homoclinic orbits of systems Lu and Chen. Doklady Mathematics. 2013; 87(2):220-223. (In Eng.) DOI: https://doi.org/10.1134/S1064562413020300
7.Algaba A., Domínguez-Moreno M.C., Merino M. et al. Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems. Nonlinear Dynamics. 2015; 79(2):885-902. (In Eng.) DOI: https://doi.org/10.1007/s11071-014-1709-2
8.Chen G. The Chen system revisited. Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applications & Algorithms. 2013; 20:691-696. Available at: https://www.ee.cityu.edu.hk/~gchen/pdf/Chen_Sys_Revited_2013.pdf (accessed 04.02.2021). (In Eng.)
9.Chen Y., Yang Q. The nonequivalence and dimension formula for attractors of Lorenz-type systems. International Journal of Bifurcation and Chaos. 2013; 23(12):1350200. (In Eng.) DOI: https://doi.org/10.1142/S0218127413502003
10.Rozanov Yu.A. Probability Theory, Random Processes and Mathematical Statistics. Mathematics and Its Applications, vol. 344. Springer, Dordrecht; 1995. (In Eng.) DOI: https://doi.org/10.1007/978-94-011-0449-4
11.Kannan D., Wu D.T. A numerical study of the additive functionals of solutions of stochastic differential equations.Dynamic Systems and Applications. 1993; 2(1-4):291-310. (In Eng.)
12.Kushner H.J., Dupuis P.G. Numerical Methods for Stochastic Control Problems in Continuous Time. Applications of Mathematics, vol. 24. Springer, New York, NY; 1992. (In Eng.) DOI: https://doi.org/10.1007/978-1-4684-0441-8
13.Kulchitskiy O.Yu., Kuznetsov D.F. Numerical Methods of Modeling Control Systems Described by Stochastic Differential Equations. Journal of Automation and Information Sciences. 1999; 31(1-3):47-61. (In Eng.) DOI: https://doi.org/10.1615/JAutomatInfScien.v31.i1-3.70
14.Kuznetsov D.F. New Representations of Explicit One-Step Numerical Methods for Jump-Diffusion Stochastic Differential Equations. Computational Mathematics and Mathematical Physics. 2001; 41(6):874-888. (In Eng.)
15.Kuznetsov D.F. Development and Application of the Fourier Method for the Numerical Solution of Ito Stochastic Differential Equations. Computational Mathematics and Mathematical Physics. 2018; 58(7):1058-1070. (In Eng.) DOI: https://doi.org/10.1134/S0965542518070096
16.Nakonechnyi A.G., Polyvyanaya Yu.V. Minimax Adjusters for Stochastic Linear Differential Equations with Branching Structures. Journal of Automation and Information Sciences. 1999; 31(11):31-39. (In Eng.) DOI: https://doi.org/10.1615/JAutomatInfScien.v31.i11.60
17.Arato M. Linear Stochastic Systems with Constant Coefficients. A Statistical Approach. Lecture Notes in Control and Information Sciences, vol. 45. Springer-Verlag Berlin Heidelberg; 1982. (In Eng.) DOI: https://doi.org/10.1007/BFb0043631
18.Kloeden P.E., Platen E. Numerical Solution of Stochastic Differential Equations. Applications of Mathematics, vol. 23. Springer, Berlin, Heidelberg; 1992. (In Eng.) DOI: https://doi.org/10.1007/978-3-662-12616-5
19.Platen E. An introduction to numerical methods for stochastic differential equations. Acta Numerica. 1999; 8:197-246. (In Eng.) DOI: https://doi.org/10.1017/S0962492900002920
20.Akhtari B. Numerical solution of stochastic state-dependent delay differential equations: convergence and stability. Advances in Difference Equations. 2019; 2019:396. (In Eng.) DOI: https://doi.org/10.1186/s13662-019-2323-x
21.Baker C., Buckwar E. Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations. LMS Journal of Computation and Mathematics. 2000; 3:315-335. (In Eng.) DOI: https://doi.org/10.1112/S1461157000000322
22.Särkkä S., Solin A. Applied Stochastic Differential Equations. Cambridge: Cambridge University Press; 2019. (In Eng.) DOI: https://doi.org/10.1017/9781108186735
23.Mao X. Stochastic Differential Equations and Applications. 2nd ed. Woodhead Publ.; 2008. (In Eng.)
24.Mauthner S. Step size control in the numerical solution of stochastic differential equations. Journal of Computational and Applied Mathematics. 1998; 100(1):93-109. (In Eng.) DOI: https://doi.org/10.1016/S0377-0427(98)00139-3
25.Rümelin W. Numerical Treatment of Stochastic Differential Equations. SIAM Journal on Numerical Analysis. 1982; 19(3):604-613. Available at: https://www.jstor.org/stable/2156972 (accessed 04.02.2021). (In Eng.)
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