**1. Introduction**

Models for actual dynamical processes are based on some restrictions. These restrictions are represented as a conservation law.

The conservation law states that a particular measurable property of an isolated dynamical system does not change as the system evolves over time.

Actual dynamical systems are open, and they are subject to strong external disturbances that violate the laws of conservation for the given system.

Conventionally, deterministic dynamical systems have an invariant function. Doobko<sup>1</sup> V. in [1] proved that stochastic dynamical systems have an invariant function as well. For dynamical system which are described using a system of stochastic differential Itô equations, a first integral – or an invariant function, exists with probability 1 [2–10].

When we know only a conservation law for a dynamical system, and equations which describing this system are unknown, the invariant functions are a good tool for determination of these equations.

<sup>1</sup> Different variant of transliteration of the name: Dubko

Our method differs for other (see, for example, [11]) preliminary in the fact that we construct a system of differential equation with the given first integral under arbitrary initial conditions. Besides, this algorithm is realized as software and it allows us to choose a set of functions for simulation. Moreover, we can construct both a system of stochastic differential equations and a system of deterministic ones. ~*ν*ð Þ¼ *t*, *A ν*ð Þ� *t*, *A* **E**½ � *ν*ð Þ *t*, *A* is called a centered Poisson measure (CPM). Let **<sup>w</sup>**ðÞ¼ *<sup>t</sup>* ð Þ *<sup>w</sup>*1ð Þ*<sup>t</sup>* , … , *wm*ð Þ*<sup>t</sup>* <sup>∗</sup> be an *<sup>m</sup>*-dimensional Wiener process, such that

modeling independent random variables on disjoint intervals and sets. The Wiener processes *wk*ð Þ*t* , *k* ¼ 1, … , *m*, and the Poisson measure *ν*ð Þ ½ � 0, *T* , *A* defined

> ð *Rγ*

. Denote by *ν*ð Þ Δ*t*, Δ*γ* the PM on 0, ½ �� *T*

*g t*ð Þ , *γ*, **x** *ν*ð Þ *dt*, *dγ* , (1)

≕*R<sup>γ</sup>* , while **w**ð Þ*t* is an *m*-dimen-

½ � **v**ðÞ�*t d***w**ð Þ*t* , (2)

the one-dimensional Wiener processes *wk*ð Þ*t* for *k* ¼ 1, … , *m* is mutually

*Invariants for a Dynamical System with Strong Random Perturbations*

on the specified space are F*t*-measurable and independent of one another. Consider a random process *x t*ð Þ with values in **<sup>R</sup>***<sup>n</sup>*, *<sup>n</sup>*≥2, defined by the

where *A t*ðÞ¼ f g *<sup>a</sup>*1ð Þ*<sup>t</sup>* , … , *an*ð Þ*<sup>t</sup>* <sup>∗</sup> , *B t*ðÞ¼ *<sup>b</sup> <sup>j</sup>*,*<sup>k</sup>*ð Þ*<sup>t</sup>* � � is ð Þ *<sup>n</sup>* � *<sup>k</sup>* - matrix, and

perturbations, which named below as a Jump-diffusion Itô equations system

**3. An existence of an invariant function (with Prob.1) for stochastic**

Consider the diffusion Itô equation in **R**<sup>3</sup> with orthogonal random action with

*b* ∣**v**ð Þ*t* ∣

V. Doobko in [1] showed that the system (2) have an invariant function called a

*u t*ð Þ¼ , **<sup>v</sup>** exp 2f g *<sup>μ</sup><sup>t</sup>* j j **<sup>v</sup>**ð Þ <sup>0</sup> <sup>2</sup> � *<sup>b</sup>*<sup>2</sup>

lim*<sup>t</sup>*!<sup>∞</sup> j j **<sup>v</sup>**ð Þ*<sup>t</sup>* <sup>2</sup> <sup>¼</sup> *<sup>b</sup>*<sup>2</sup>

i.e. process ∣**v**ð Þ*t* ∣ is a nonrandom function and the random process **v**ð Þ*t* itself is

*μ* p .

, and *wi*ð Þ*t* , *i* ¼ 1, 2, 3 are independent Wiener processes.

!

*μ* ,

*μ*

*:*

**dynamical system under strong perturbations**

*d***v**ðÞ¼� *t μ***v**ð Þ*t dt* þ

This equation is a specific form of the Langevin equation.

respect to the vector of the solution

, **w** ∈ **R**<sup>3</sup>

This, in particular, implies that

generated in a sphere of constant radius *<sup>b</sup>*ffiffi

where **v**∈ **R**<sup>3</sup>

**63**

first integral of this system:

sional Wiener process. In general the coefficients *A t*ð Þ, *B t*ð Þ, and *g t*ð Þ , *γ* are random functions depending also on **x**ð Þ*t* . Since the restrictions on these coefficients relate explicitly only to the variables *t* and *γ*, we use precisely this notation for the coefficients of (1) instead of writing *A t*ð Þ , **x**ð Þ*t* , ð Þ *t*, **x**ð Þ*t* , and *g t*ð Þ , **x**ð Þ*t* , *γ* .

A system (1) is the stochastic differential Itô equation with Wiener and Poisson

We will consider the dynamical system described using ordinary deterministic differential equations (ODE) system and ordinary stochastic differential Itô equations (SDE) system of different types, taking into account the fact that **x**∈ **R***<sup>n</sup>*, *n* ≥2.

*dx t*ðÞ¼ *A t*ð Þ*dt* þ *B t*ð Þ*d***w**ðÞþ*t*

Take a vector *γ* ∈ Θ with values in *<sup>n</sup>*<sup>0</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.96235*

*g t*ð Þ¼ , *<sup>γ</sup> <sup>g</sup>*1ð Þ *<sup>t</sup>*, *<sup>γ</sup>* , … , *gn*ð Þ *<sup>t</sup>*, *<sup>γ</sup>* � �<sup>∗</sup> <sup>∈</sup> *<sup>n</sup>*, and *<sup>γ</sup>* <sup>∈</sup> *<sup>n</sup>*<sup>0</sup>

independent.

Equation [12]:

(GSDES).

*<sup>n</sup>*<sup>0</sup>

The goal of this chapter is representation of modern approach to describe of dynamical systems having a set of invariant functions.

This chapter is structured as follows. Firstly, we show that the invariant functions for stochastic systems exist. Then, the generalized Itô – Wentzell formula is represented. It is a differentiated rule for Jump-diffusion function under variables which solves the Jump-diffusion equations system. This rule is basic for the necessary and sufficient conditions for the stochastic first integral (or invariant function with probability 1) for the Jump-diffusion equations system. The next step is the construction of the differential equations system using the given invariant functions. It can be applied for stochastic and nonstochastic cases. The concept of PCP1 (Programmed control with Prob. 1) for stochastic dynamical systems is introduced. Finally, we show an application of the stochastic invariant theory for a transit from deterministic model with invariant to the same stochastic model. Several examples of application of this theory are given and confirmed by results of numerical calculations.
