PERMUTATION GROUPS AND PERIODICITY OF SYSTEMS OF DIFFERENCE EQUATIONS

Here, stochastic asymptotic stability (SAS) of the zero solution of a stochastic delay differential equation (SDDE) of third order such as ... x (t) + f(ẋ(t))ẍ(t) + g(ẋ(t− τ)) + h(x(t− τ)) + σx(t)ω̇(t) = 0 is discussed. To arrive at the aim of this paper, a suitable Lyapunov functional (LF) is defined and then used to find conditions that guaranteeing the (SAS) of solutions. We give an example to verify the analysis made in this paper. 2010 Mathematics Subject Classification. 34K20, 34K50, 60H35


Introduction and preliminaries
The theory of permutation groups is important to diverse area of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.See for instance [1].In general, the theory of abstract groups plays an important part in present day mathematics and science.Groups arise in a bewildering number of apparently unconnected subjects.Thus they appear in algebra and analysis, in geometry and topology, in crystallography and quantum mechanics, in physics and chemistry, and even in biology.See [2] , [3].In this paper, as another application of the theory of permutation groups, we investigate the periodicity of systems of difference equations.Recently, there has been a great interest in difference equations, because they describe naturally many real-life problems in biology, ecology, genetics, psychology, sociology, and so forth.The authors in [4] investigated the periodicity of the system as well as the periodicity of the system x (1) n x n − 1 , . . .x For some related results about periodicity, we refer the reader to the interesting papers [5][6][7][8][9].Motivated by systems (1.1) and (1.2), we investigate the periodic character of the general system n−s ). (1.3) Here, π ∈ S k and f i is a certain function from a non-empty set into itself, i = 1, . . ., k.At the end of this paper, the periodicity of many systems of rational difference equations is investigated.It is well-known that for any permutation π ∈ S k , there is a natural number l such that the property π l = I holds, where I is the identity permutation and π l is the composition of π with itself l-times.The smallest l for which this property holds is called the order of π.
Definition 1.1.A system (1.3) is called periodic with period d if every solution (x n , . . ., x Here, I is the identity function.

Main results
Let π ∈ S k be a permutation of order l, X be any non-empty set and f i : X → X, i = 1, . . ., k.In this section we investigate the periodicity of the system in terms of the periodicity of each of the difference equations where g i is defined by We need the following lemma to prove our main result.
n , x n , . . ., x n ) is a solution of system (2.1), then it satisfies the following equations for every r ∈ Z ≥0 .
(ii) Assume that l is even.If a k-tuple (x n , . . ., x n ) is a solution of system (2.1), then it satisfies the following equations for every r ∈ Z ≥0 .
Proof.(i) Relation (2.3) is true at r = 0. Assume that relation (2.3) is true for a fixed r.We have (ii) can be shown similarly.
As a direct consequence, putting r = l − 1, we get the following result Corollary 2.2.
(ii) Assume that l is even.If a k-tuple (x n , . . ., x n ) is a solution of system (2.1), then x i n satisfies the equation for each i.
To arrive at our main result, we need to check the following Lemma Lemma 2.3.Let g : X → X.
(i) A solution of the equation is given by y (m+1)(s+1)l+r−s = g m+1 (y r−s ), r = 0, . . ., (s + 1)l, m ∈ Z ≥0 . ( (ii) Assume that l is even.A solution of the equation ), m upisodd. (2.8) Proof.(i) Relation (2.6) is true when m = 0. Assume it is true for a fixed m.Simple calculations show that (ii) Relation (2.8) is true at m = 0 and m = 1.Let m be a fixed natural number and the relation is true for every natural number less than m.For the case where m is even, we have That is relation (2.8) is true for m, when m is even.The case where m is odd can be treated similarly.Now, we are ready to prove our main results.
Theorem 2.4.A sufficient condition for the system (2.1) to be periodic with a period d is that each difference equation (2.5.i) ( resp.(2.6.i) when l is even ) is periodic, with a period n is a solution of (2.5.i) ( resp.(2.6.i) when l is even ).The periodicity of x (i) n with period d i , i = 1, . . ., k imply the periodicity of system (2.1) with period d = l.c.m(d 1 , . . ., d k ), the least common multiple of d 1 , . . ., d k .
Theorem 2.5.If f i is an involution on X for each i such that f i f j = f j f i , i, j ∈ Z k , then system (2.1) is periodic with period 2l(s + 1).
n ) be a solution of system (2.1).Then x i n is a solution of equation (2.5.i) for each i.By Lemma 2.3, this solution is given by The assumptions of the theorem imply that g 2 i is the identity function for each i.We deduce that each equation (2.5.i) is periodic with period 2l(s + 1).Indeed, we conclude that an involution on X and l is even, then system (2.1) is periodic with period l(s + 1).
Proof.The hypothesis implies that g i = f l = I, i ∈ Z k and equation (2.5.i) is periodic with period l(s + 1) for each i.
Theorem 2.7.Assume that f i is an idempotent on X for each i such that f i f j = f j f i , i, j ∈ Z k .Then system (2.1) is periodic with period l(s + 1).
n , . . ., x (k) n ) be a solution of system (2.1).Then x i n is a solution of equation (2.5.i) for each i.The assumptions of the theorem imply that g 2 i = g i for each i.By lemma 2.3, we have Consequently, system (2.1) is periodic with period l(s + 1).
Theorem 2.8.Let α and a be complex numbers.The system is periodic with period l(s + 1).
} is an involution for every i.By theorem 2.6, we insure that system (2.9) is periodic with period l(s + 1).
Proof.One can see that involution for every i.Applying theorem 2.5, we arrive at the periodicity of the system with period 4(s + 1).
3 Illustrative examples (i) The system is periodic with period 6(s + 1).Indeed, the permutation which corresponds to this system is π = (1 2 3).So, its order is 3.This implies that the system is periodic with period 6(s + 1).
(ii) The system is periodic with period 6(s + 1).The permutation which corresponds to this system is π = (1 3 2).Again its order is 3.This implies that the system is periodic with period 6(s + 1).
(iii) The system is periodic with period 4(s + 1).Indeed, The permutation which describes this system is π = (1 3 2 4) with order 4. Then the periodicity of the system is 4(s + 1).
(v) The system is periodic with period 4.

Introduction
A stochastic differential equation (SDE) involves a variable which indicates random white noise calculated as the derivative of Brownian motion or the Wiener process.However, other types of random behaviors are possible, for example,jump processes.There are two dominating versions of stochastic calculus (SC), the Itô (SC) and the Stratonovich (SC).Both of these (SC)have advantages and disadvantages(see, for example, [1]) and conveniently, one can readily convert an Itô (SDE) to an equivalent Stratonovich (SDE) and back again.
The theory of (SDEs) has attracted considerable attention of many scholars in the last years.Therefore, it has extensively been studied in the relevant literature and there exists a large number of books, which provides full details of the background of probability theory and (SC), for example, see the books of [1], [2], [3], [4], [5], [6] and the references of these sources.
Since then the number of contributions to statistics, numerics and control theory of (SDEs) have been rapidly increasing.It is well known that stochastic models (SMs) have important roles in science and industry, where many authors can encounter with (SDEs).
Therefore, (SMs) can have many important roles in various scientific areas such as biology, economy, medicine, engineering and so on (see, [7], [8], [9]).By this time, numerous kind of (LFs) have been used as basic tools to study qualitative problems in many deterministic/(SDEs) and (DDEs).
In addition, (SDDEs) represent a relatively new field of the qualitative theory of (DEs) and (DDEs).The significance of (SDDEs) has become more evident in recent years due to a great variety of their applications in modeling real world phenomena.Unfortunately, it is in general not possible to give or obtain analytic expressions for solutions of (SDEs) and (SDDEs).Therefore, most of the papers are interested in to characterize the qualitatively behaviors of solutions (QBSs) without solving (SDEs) and (SDDEs).To the best of our knowledge,up to now in the relative literature, the Lyapunov's theory is the most powerful tool for qualitative analysis of (SDEs)and (SDDEs) of higher order without solving that equations.
With respect to our observation, up to this moment, the corresponding problem for the stability (S) and boundedness (B) of solutions of higher order (SDDEs) have been studied in the literature by only a few researches, see, for instance, [28], [29], [30], [31], [32] and [33].Therefore, it is worth discussing the (QBSs)of (SDDEs).In 2015, the authors of [30] discussed the (AS) of the zero solution of (SDDEs) such as ...
In 2017, the author of [32] studied (S) and (B) of solutions of the (SDDEs) in the following form ...
It should be noted that throughout the references of this papers, (LFs) have been used as a basic tool to investigate the (QBSs) mathematical modes therein.
Here, we take into consideration the (SDDE) of third order ...
(SDDE) (1.1) can be written as where σ and τ are two positive constants; f ( ẋ), g( ẋ) and h(x) are continuous functions, g(0 is a m-dimensional standard Brownian motion or Wiener process defined on the probability space, a stochastic process representing the noise. In this work, we aim to find new conditions for the (SAS) of the zero solution of(SDDE) (1.1) by defining an appropriate (LF).
Remark 1.1.We have the following observations: (1) If σ = 0 in (1.1), then we have a general (DDE) of third-order , which has been investigated extensively in relevant literature.
Motivated by references of this paper and the papers and books can found in the literature, we aim to do a contribution to the literature by obtaining a new result on the (SAS) of solutions of a new (SDDE) model, which has not been discussed in the literature yet.This fact is the novelty and originality of this article.

Preliminary results
Let ω(t) = (ω 1 (t), . . ., ω m (t)) be an m-dimensional Brownian motion.Let us consider an n-dimensional (SDE) with initial condition x(0) = x 0 ∈ R n .We suppose that the functions F : R + × R n → R n and G : R + × R n → R n×m are continuous and satisfy the Lipschitz condition (see, for example, [22], [26]).Hence, the existence and uniqueness of solutions of (SDE) (2.1) are guaranteedon t ≥ 0, say x(t; x 0 ).In addition, we assume that f (t, 0) = 0 and g(t, 0) = 0. We will show family of non-negative and differentiable functions V (t, x t ) by C 1,2 (R + × R n ; R + ).That is, those functions once and twice continuously differentiable in t and x, respectively.
Let us define an operator L on C 1,2 (R + × R n ; R + ) by where Furthermore, Further, let K be the family of all nondecreasing and continuous functions ϑ : R + → R + with ϑ(0) = 0 and ϑ(r) > 0, if r > 0.
Theorem 2.2.( [2], [4]) Assume that there exist Then the zero solution of the (SDE) is stochastically asymptotic stable in the large.

Main result
We now state the main (S) result for (SDDE) (1.1).
Then the zero solution of (SDDE) (1.1) is stochastically asymptotic stable if , Proof of Theorem 3.1.Define the (LF) V (•) = V (t, x t , y t , z t ) as where x t = x(t + s), s ≤ 0 and λ and δ are positive constants, and we determine them later.Next, we have to show that V (t, x t , y t , z t ) satisfies the assumptions of Theorem 2.2.Moreover, by applying Itô formula in (3.1) using system (1.2), we find that  .