Logistic Map on the Ring of Multisets and Its Application in Economic Models

In this paper, we extend complex polynomial dynamics to a set of multisets endowed with some ring operations (the metric ring of multisets associated with supersymmetric polynomials of infinitely many variables). Some new properties of the ring of multisets are established and a homomorphism to a function ring is constructed. Using complex homomorphisms on the ring of multisets, we proposed a method of investigations of polynomial dynamics over this ring by reducing them to a finite number of scalarvalued polynomial dynamics. An estimation of the number of such scalar-valued polynomial dynamics is established. As an important example, we considered an analogue of the logistic map, defined on a subring of multisets consisting of positive numbers in the interval [0, 1]. Some possible application to study the natural market development process in a competitive environment is proposed. In particular, it is shown that using the multiset approach, we can have a model that takes into account credit debt and reinvestments. Some numerical examples of logistic maps for different growth rate multiset [r] are considered. Note that the growth rate [r] may contain both “positive” and “negative” components and the examples demonstrate the influences of these components on the dynamics.


Introduction
The difference equation t n+1 = rt n (1 − t n ) is known as the logistic map, where n ∈ Z + is the discrete time, r ∈ [0, 4] is the growth rate and the sequence (t n ) ∞ n=0 , 0 < t n < 1 describes the dynamic of the "population" value t n at any time n. The logistic map was considered first by R. May [1] in 1976 for a discrete-demographic model of biological populations. But this equation is well applicable in Economics, Sociology, Cryptography, Physics and etc. There are a lot of publications in different areas of sciences with using dynamic models based on chaotic properties of the logistic map. For example, economic models which involve cycles and chaos were considered in [2]. Applications of the logistic map for investigations of exogenous shocks in economic models were studied in [3]. In [4] was introduced a logistic map with memory and studied its applications in economic models. Some other generalizations of the logistic map can be found in [5]. Complex behavior of two combined logistic models was investigated in [6]. Dynamics of the logistic difference equation on fuzzy numbers were studied in [7,8].
It is well known (see e. g. [9,6]) that if r ∈ [0, 1), the population will eventually die, if r ∈ [1,3], the population will approach r−1 r . If 3 < r ≤ 1 + √ 6, from almost all initial conditions, the population will have permanent oscillations between two values (the bifurcation points). Practically, the logistic map becomes chaotic when r ≥ 3.5699. The number of bifurcation points rapidly grows, when r approaches 3.85. If 3.85 < r < 4, the population has a chaotic behaviour with periodic windows in the interval [0, 1]. If r = 4, t n has a chaotic behavior on the whole interval [0, 1]; the dynamics has lost near-all its determinism and the population evolves as a random number generator.
In this paper, we consider the logistic map defined on the ring of multisets, which was introduced in [10]. The ring of multisets can be considered as a natural domain of supersymmetric polynomials of infinitely many variables. Supersymmetric polynomials of infinitely many variables give us some kind of generalization of symmetric polynomials. Symmetric polynomials and analytic functions on infinite dimensional Banach spaces were studied in [11,12,13,14,15,16]. In particular, there were considered some semiring algebraic operations on the spectra of algebras of symmetric polynomials which can 425 be applied to a set of multisets. Supersymmetric polynomials allow to extend this operations to a ring operations on a larger set [10]. Some further investigations in this direction can be found in [17]. Algebras of analytic functions, generated by a sequence of polynomials on a Banach space, in the general case were investigated in [18]. For the general information about algebraic theory of symmetric functions we refer the reader to [19].
In Section 2 we introduce the ring of multisets and consider its basic properties. In Section 3 we investigate general properties of polynomial dynamics on the ring of multisets. In Section 4 we consider the logistic map dynamic on the ring of multisets and its possible application to economical models.

Ring of Multisets
Let us recall that a multiset is an unordered finite collection of nonzero numbers which can have multiple instances for each of its elements. Let us denote by M + 0 the set of all finite multisets. That is, if x ∈ M + 0 , then we can write x = (x 1 , x 2 , . . . , x m ) for some m, where x i are, in the general case, nonzero complex numbers. Since we are taking into account only nonzero numbers, we can write ( and It is easy to check that these operations are associative, commutative and we have the distributive law. So M + 0 (•, ) is a semiring (see [20]). To get a ring, we need to have the invertibility of operation "•". Let us consider the Cartesian product We can extend the introduced operations to M + 0 × M + 0 by the following way: let u = (y|x) and u = (y |x ). Then Let us define the following relation of equivalence on M + 0 × M + 0 : u ∼ u if and only if there exists a ∈ M + 0 such that u = u • (a|a) or u = u • (a|a). In particular, (a|a) ∼ 0 = (0|0) for every a ∈ M + 0 .
We denote by M 0 the quotient set M + 0 × M + 0 / ∼ and by [u] the class of equivalence containing u. From [10] we know that the operations are well defined on M 0 and (M 0 , +, ·) is a commutative ring with the unit I = (0|1).
From [10] we have that and Note that polynomials T k form the algebra of supersymmetric polynomials [10] and are difference of symmetric polynomials The completion of M 0 with respect to ρ is denoted by M.
The operation of M 0 and complex homomorphisms T k , k = 1, 2, . . . are continuous with respect to the metric ρ and can be continuously extended to the completion M (see [10]). On the other hand, there are discontinuous complex homomorphisms of M 0 which can not be extended to M. For example, let where we assume that 0 0 = 0. Then So T 0 is well defined on M 0 and simple calculations show that T 0 is additive and multiplicative. But the completion M of M 0 contains elements with infinitely many "coordinates" like u = (0|1/2, 1/4, . . . , 1/2 n , . . .), where T 0 is not defined. We need, also, describe some homomorphisms and subrings of M 0 . For every k ∈ N the map where u = (y 1 , . . . , y j |x 1 , . . . , x m ), is a ring homomorphism from M 0 to M 0 . We will use also notation [u] (k) := Φ k ([u]). If U is a subset of C which is closed with respect to the multiplication in C, then we have that the following map  Proof. Note first that the sum on the right side consists only of a finite numbers of terms. So, it is well defined for every [u] ∈ M 0 . Let u be as in (2), then

3 Dynamics of Complex Polynomials on M 0
Let q : C → C be a polynomial of a complex variable t. Then the dynamic of q with the initial condition t 0 ∈ C is the sequence where ϕ(p) is a polynomial on C defined by . , x k ). We denote by card (x) = k the cardinality of x, card (y) = m and card (u) = k + m. From [10] we know that any element [u] ∈ M is completely defined by the number sequence (T n (u)) ∞ n=1 . On the other hand, if [u] ∈ M 0 , that is a representative u ∈ [u] is finite, then [u] can be defined by a finite sequence (T n (u)) N n=1 , where the number N should depend of the cardinality of u. Proof.
According to [10,Corollary 3] any element u = (y|x) = (y 1 , . . . , y m |x 1 , . . . , x k ) can be represented as a rational function Moreover, if u = (y |x ) = (y 1 , . . . , y s |x 1 , . . . , x r ), then and . (6) From (4) and (5) we can see that the representative u of [u] is irreducible if and only if the fraction in (4) is irreducible. Clearly that the irreducible fraction is unique. 2 Note that from the representation [u] → w(u)(t) it follows that a polynomial dynamic on M 0 is equivalent to a dynamics defined on rational functions (4) with "addition" (5) and "multiplication" (6). Proof.
for all j = 1, 2, . . . , m + k. It is well known that symmetric polynomials F 1 , . . . , F m+k form an algebraic basis in the algebra of symmetric polynomials on C m+k . Since F j are symmetric, each F j , j > m + k can be represented as an algebraic combination of F 1 , . . . , F m+k . So F j (y • x ) = F j (x • y ) for all j ∈ N. But it implies that T j ([u]) = T j ([u ]) for all j ∈ N. Hence [u] = [u ]. The converse implication is trivial. 2 We can use, also, the alternative power map (1), to introduce a polynomial dynamics on the ring of multisets M 0 .
"negative" part is responsible to return some part of the profit to costumers. It may be considered as a tax which returns some money to support the capacity of the market. We suppose that T k ([z]) ≥ 0 for every k. If all components of [r] are less than 1, then T k ([r]) tends to zero as k → ∞.
Example 4.2 The following example shows that for some special choice of [r] we can guaranty that only some finite number of functions T k will be essential for the behavior of [u n ] what is natural for economic systems. Let Then T 1 ([r]) = 5/2 and for almost all initial conditions In particular, it is equal to zero, if m = j and s = l.

Remarks
(i) Note that equation (8) is not a unique way to extend the logistic map (7) to multisets. If we rewrite (7) by t n+1 = rt n − rt 2 n , then we can consider dynamic where the power function It is easy to see that [u] (2) = [u] 2 in general. So the dynamic (9) is not equivalent to (8) and is interesting in the mathematical sense. However, the simple example [u 0 ] = (0|1/2), [r] = (0|1, 1) shows that T 1 ([u 3 ]) < 0 and so (9) does not describe a reasonable model of development.
(ii) The dynamic of logistic map (8) can be defined also on the quotient ring M ∆ 0 /M ∆ε 0 for some small positive ε. Practically it means that all transactions x i and y j are automatic vanishing if they are less than ε.

Conclusion
This paper is an invitation to investigations of polynomial dynamics on multisets and their applications in economic models. As an example, we considered the logistic map dynamic on a ring of multisets. We can see that the logistic map on multisets is a generalization of the classical scalar logistic map and the multiset logistic map dynamic contains the scalar dynamic. Thus, the multiset dynamic is applicable to practical problems where one can use classical dynamic systems. However, the proposed approach allows as to use the logistic map dynamic for modeling the natural market development process which takes into account credit debt and reinvestments. We show that in this case, the growth rate [r] may contain both "positive" and "negative" components and demonstrate examples of the influences of these components on the dynamic.