2 Discontinuous dynamical systems Let $^{R}$, $^{N}$and $^{Z}$be the sets of all real numbers, natural numbers and integers, respectively. Consider the set D ∈ R n such that D = ∪ D$_{i}$, where D$_{i}$, i = 1 , 2 , . . . , k, components of D, are disjoint open connected subsets of R $^{n}$. To describe the surface of discontinuity, we present a two times continuously differentiable function Φ : D r → R $^{n}$. The set can be defined as Γ = Φ − $^{1}$(0) and is a closed subset of ¯ D, where ¯ D is the closure of D. Denote ∂ Γ as the boundary of Γ . One can easily see that Γ = ∪ k i $_{=1}$Γ$_{i}$ , where Γ$_{i}$ are parts of the surface of discontinuity in the components of D. Denote ˜ Γ = J (Γ) , ˜ Φ( x ) = Φ( J − $^{1}$( x )) . Denote an r − neighborhood of D in R n for a fixed r > 0 as D $^{r}$. Let Γ r be the r − neighborhood of Γ in R $^{n}$, for a fixed r > 0 and define functions J : Γ r → D r and ˜ J : ˜ Γ r → D $^{r}$, such that, J (Γ) , ˜ J ( ˜ Γ) ⊂ D. Assume that a function f ( x ) : D r → R n is continuously differentiable in D $^{r}$. Set the gradient vector of Φ as ∇ Φ( x ) . The following definitions will be utilized in the remaining part of the paper. Let x ( t − ) be the left limit position of the trajectory and x ( t +) be the right limit of the position of the trajectory at the moment t. Define Δ x ( t ) := x ( t +) − x ( t − ) as the jump operator for a function x ( t ) such that x ( t ) ∈ Γ and t is a moment of discontinuity (discontinuity moment). In other words, the discontinuity moment t is the moment when the trajectory meets the surface of discontinuity Γ . The function I ( x ) will be used in the following part of the paper which is defined as I ( x ) := J ( x ) − x, for x ∈ Γ . The following assumptions are needed throughout this paper. One can verify that ˜ Γ = { x ∈ D | ˜ Φ( x ) = 0 } and ˜ J ( x ) = x on ˜ Γ since of ( C 2) . Condition ( C 1) implies that for every x$_{0}$ ∈ Γ , there exist a number j and a function φ$_{x}$$_{0}$ ( x$_{1}$$^{, . . . , x$_{j}$}$− $_{1}$, x$_{j}$$_{+1}$ , . . . , x$_{n}$ ) such that Γ is the graph of the function x$_{j}$ = φ$_{x}$$_{0}$ ( x$_{1}$$^{, . . . , x$_{j}$}$− $_{1}$, x$_{j}$$_{+1}$ , . . . , x$_{n}$ ) in a neighborhood of x$_{0}$. Same is true for every x$_{0}$ ∈ ˜ Γ . Moreover, ∇ ˜ Φ( x ) = 0 , for all x ∈ ˜ Γ , can be verified by using the condition ( C 2) . The conditions ( C 2) , ( C 6) , ( C 7) , imply that the equality ˜ J ( x ) = x, is true for all x ∈ ∂ ˜ Γ . Let A be an interval in $^{Z}$. We say that the strictly ordered set θ = { θ$_{i}$ } , i ∈ A , is a B − sequence [1] if one of the following alternatives holds: ( i ) θ =  , ( ii ) θ is a nonempty and finite set, ( iii ) θ is an infinite set such that | θ$_{i}$ | → ∞ as i → ∞ . In what follows, θ is assumed to be a B − sequence . (C1) ∇ Φ( x ) = 0 for all x ∈ Γ , (C2) J ∈ C $^{1}$(Γ $^{r}$) and det [ ∂J ( x ) ∂x ] = 0 , for all x ∈ Γ r \ ∂ Γ , (C3) Γ ⋂ ˜ Γ ⊆ ∂ Γ ∩ ˜ ∂ Γ , (C4) 〈∇ Φ( x ) , f ( x ) 〉 ̸ = 0 if x ∈ Γ \ ∂ Γ , (C5) 〈∇ ˜ Φ( x ) , f ( x ) 〉 ̸ = 0 if x ∈ ˜ Γ \ ∂ ˜ Γ , (C6) J ( x ) = x for all x ∈ ∂ Γ , (C7) ˜ J ( x ) = x for all x ∈ ∂ ˜ Γ . 3