**2. Notation and preliminaries**

Now we introduce the main concepts which we will use below. Let *w t*ð Þ, *t*∈ ½ Þ 0, ∞ be a Wiener process or a (standard) Brownian motion, i. e.

$$\bullet \ w(0) = 0,$$


A *ν*ð Þ *t*, *A* is called a Poisson random measure or standard Poisson measure (PM) if it is non-negative integer random variable with the Poisson distribution *ν*ð Þ� *t*, *A Poi t*ð Þ Πð Þ *A* , and it has the properties of measure:


$$\mathbf{P}\_t(\#\!A=k) = \frac{\left(t\Pi(A)\right)^k}{k!} \exp\left\{-t\Pi(A)\right\}.$$

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

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

Take a vector *γ* ∈ Θ with values in *<sup>n</sup>*<sup>0</sup> . Denote by *ν*ð Þ Δ*t*, Δ*γ* the PM on 0, ½ �� *T <sup>n</sup>*<sup>0</sup> 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 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 Equation [12]:

$$d\mathbf{x}(t) = A(t)dt + B(t)d\mathbf{w}(t) + \int\_{R\_{\mathcal{I}}} \mathbf{g}(t, \boldsymbol{\gamma}, \mathbf{x})\nu(dt, d\boldsymbol{\gamma}),\tag{1}$$

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 *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> ≕*R<sup>γ</sup>* , while **w**ð Þ*t* is an *m*-dimensional 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 perturbations, which named below as a Jump-diffusion Itô equations system (GSDES).

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.
