# 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

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**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