# A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Remarks

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

#### 2.1. Markov Chain $\xi $ is Markovian

#### 2.2. Markov Chain $\xi $ is Semi-Markovian

**Definition**

**3.**

**Theorem**

**1**

**Theorem**

**2**

- (1)
- there exists a solution ${B}_{s}={E}_{s}^{\left(2\right)}\left\{{X}_{0}\right\}+{W}_{s}>0$ to system of matrix Equations (12) and (13) under the condition ${E}_{s}^{\left(2\right)}\left\{{X}_{0}\right\}>0$,
- (2)
- the successive approximations$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {B}_{s}^{(j+1)}={E}_{s}^{\left(2\right)}\left\{{X}_{0}\right\}+\sum _{l=1}^{n}\sum _{k=1}^{\infty}{q}_{sl}\left(k\right){C}_{sl}{N}_{l}\left(k\right){B}_{l}^{\left(j\right)}{N}_{l}^{T}\left(k\right){C}_{sl}^{T},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {B}_{s}^{\left(0\right)}=0,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}s=1,\dots ,n,\phantom{\rule{4pt}{0ex}}j=0,1,2,\dots \phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$

## 3. Moment Equations for Difference Systems with Random Jumps

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Model Problems

#### 4.1. Threats to Security in Cyberspace Modelled by a System with Semi-Markov Parameters

- θ
_{1}— - the system operates in a threat-free environment, or threats appear with transition intensities ${q}_{1s},\phantom{\rule{0.166667em}{0ex}}s=1,2,3$, but do not represent damage with probability ${q}_{1}$;
- θ
_{2}— - the system operates in an environment where threats occur with transition intensities ${q}_{2s},\phantom{\rule{0.166667em}{0ex}}s=1,2,3$, but the programmer is ready to reflect it with probability ${q}_{2}$;
- θ
_{3}— - the system operates in an environment where threats occur with transition intensities ${q}_{2s},\phantom{\rule{0.166667em}{0ex}}s=1,2,3$, and the programmer is not ready to reflect it with probability ${q}_{3}$;

#### 4.2. Stability of Foreign Currency Exchange Market

- ${\theta}_{1}$—there is a currency crisis, $a\left({\xi}_{k}\right)={a}_{1}$,
- ${\theta}_{2}$—there is a stable foreign currency exchange market, $a\left({\xi}_{k}\right)={a}_{2}$,
- ${\theta}_{3}$—there is a market with currency restrictions, $a\left({\xi}_{k}\right)={a}_{3}$.

#### 4.3. Radiocarbon Dating Modelled by a System with Markov Parameters

- ${\xi}_{k}={\theta}_{1}$—the test sample (in terms of detecting age) is not affected by new pollution or the effects of radioactive substances,
- ${\xi}_{k}={\theta}_{2}$—the sample process is subject to some kind of influence,

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lo, C.F. Stochastic Nonlinear Gompertz Model of Tumour Growth. In Proceedings of the World Congress on Engineering, London, UK, 1–3 July 2009. [Google Scholar]
- Dzhalladova, I.; Růžičková, M.; Štoudková Růžičková, V. Stability of the zero solution of nonlinear differential equations under the influence of white noise. Adv. Differ. Equ.
**2015**, 2015, 143. [Google Scholar] [CrossRef][Green Version] - Geezt, V.M.; Klebanova, T.S.; Chernyak, O.I.; Ivanov, V.V. Models and Methods of Socio-Economic Forecasting; INZHEK: Kharkov, Ukraine, 2005. (In Russian) [Google Scholar]
- Diblík, J.; Dzhalladova, I.; Michalková, M.; Růžičková, M. Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty. Abstr. Appl. Anal.
**2013**, 2013, 172847. [Google Scholar] [CrossRef] - Diblík, J.; Dzhalladova, I.; Růžičková, M. Stabilization of company’s income modelled by a system of discrete stochastic equations. Adv. Differ. Equ.
**2014**, 2014, 289. [Google Scholar] [CrossRef] - Dzhalladova, I.; Michalková, M.; Růžičková, M. Model of stabilizing of the interest rate on deposits banking system using by moment equations. Tatra Mt. Math. Publ.
**2013**, 54, 45–59. [Google Scholar] [CrossRef] - Klebanova, T.S.; Rayevneva, E.V.; Strizhenko, K.A. Mathematical Models of Transformational Economy; INZHEK: Kharkov, Ukraine, 2006. (In Russian) [Google Scholar]
- Lavinski, G.V.; Pshenishniuk, O.S.; Ustenko, S.V.; Shadarov, O.D. Modeling of Economy Dynamics; ATIKA: Kiev, Ukraine, 2006. (In Russian) [Google Scholar]
- Růžičková, M.; Dzhalladova, I. Dynamic system with random structure for modeling security and risk management in cyberspace. Opuscula Math.
**2019**, 39, 3–37. [Google Scholar] [CrossRef] - Picchini, U.; Ditlevsen, S.; De Gaetano, A.; Lansky, P. Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal. Neural Comput.
**2008**, 20, 2696–2714. [Google Scholar] [CrossRef] [PubMed] - Bect, J.; Baili, H.; Fleury, G. Fokker-Planck-Kolmogorov equation for stochastic differential equations with boundary hitting resets. arXiv
**2005**, arXiv:math/0504583. [Google Scholar] - Mao, W.; Mao, X. Approximate solutions of hybrid stochastic pantograph equations with Levy jumps. Abstr. Appl. Anal.
**2013**, 2013, 718627. [Google Scholar] [CrossRef] - Lu, Z.; Yang, T.; Hu, Y. Convergence rate of numerical solutions for nonlinear stochastic equations with Markovian switching. Abstr. Appl. Anal.
**2013**, 2013, 420648. [Google Scholar] [CrossRef] - Kloeden, P.; Platen, E. Numerical Solution of Stochastic Differential Equations; Springer: Berlin, Germany, 1992. [Google Scholar]
- Martínez-García, M.; Gordon, T. A multiplicative human steering control model. In Proceedings of the 2017 IEEE International Conference on Systems, Man, and Cybernetics (SMC), Banff, AB, Canada, 5–8 October 2017. [Google Scholar] [CrossRef]
- Martínez-García, M.; Gordon, T. A new model of human steering using far-point error perception and multiplicative control. In Proceedings of the 2018 IEEE International Conference on Systems, Man, and Cybernetics (SMC), Miyazaki, Japan, 7–10 October 2018. [Google Scholar] [CrossRef]
- Valeev, K.G.; Dzhalladova, I. Derivation of Moment Equations for Solutions of a System of Differential Equations Dependent on a Semi-Markov Process. Ukrainian Math. J.
**2002**, 54, 1906–1911. [Google Scholar] [CrossRef] - Růžičková, M.; Dzhalladova, I.; Laitochová, J.; Diblík, J. Solution to a stochastic pursuit model using moment equations. Discrete Contin. Dyn. Syst. B
**2018**, 23, 473–485. [Google Scholar] [CrossRef] - Diblík, J.; Dzhalladova, I.; Michalková, M.; Růžičková, M. Modeling of applied problems by stochastic systems and their analysis using the moment equations. Adv. Differ. Equ.
**2013**, 2013, 152. [Google Scholar] [CrossRef][Green Version] - Diblík, J.; Dzhalladova, I.; Růžičková, M. The Stability of Nonlinear Differential Systems with Random Parameters. Abstr. Appl. Anal.
**2012**, 2012, 924107. [Google Scholar] [CrossRef] - Dzhalladova, I.; Růžičková, M. Stabilization of solutions of the system of linear differential equations with semi-Markov coefficients and random transformations of solutions. J. Num. Appl. Math.
**2010**, 2, 20–34. [Google Scholar] - Růžičková, M.; Dzhalladova, I. The optimization of solutions of the dynamic systems with random structure. Abstr. Appl. Anal.
**2011**, 2011, 486714. [Google Scholar] [CrossRef] - Gikhman, I.I.; Skorokhod, A.V. Controlled Stochastic Processes; Springer: Berlin, Germany, 1979. [Google Scholar]

**Figure 1.**The actual boundaries of the foreign exchange market stability area as well as the information system stability area if $a=b=d=e=k=l=\frac{1}{3}$ and $c=1$.

**Figure 2.**The area of stability of the decay of matter into isotopes is constructed in the plane with parameters $\lambda $ and $\nu $ that express the transition probabilities.

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**MDPI and ACS Style**

Diblík, J.; Dzhalladova, I.; Růžičková, M.
A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena. *Symmetry* **2019**, *11*, 1338.
https://doi.org/10.3390/sym11111338

**AMA Style**

Diblík J, Dzhalladova I, Růžičková M.
A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena. *Symmetry*. 2019; 11(11):1338.
https://doi.org/10.3390/sym11111338

**Chicago/Turabian Style**

Diblík, Josef, Irada Dzhalladova, and Miroslava Růžičková.
2019. "A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena" *Symmetry* 11, no. 11: 1338.
https://doi.org/10.3390/sym11111338