**Undergraduate #70**

**Discipline:**Mathematics and Statistics

**Subcategory:**

**Modesta Trejo**

**- Texas Southern University**

**Co-Author(s):**Willie Taylor, Rodrick Holmes, and Jaida Rice, Texas Southern University

In this research we develop to construct periodic difference equations of orders two and three. This is accomplished using so-called I-functions, i.e., functions that are both symmetric and isovertible. After the construction is completed additional equations are derived using the aggregate operators max and min. One of the interesting results shows that periods depend on the largest initial value.

I. Introduction

Let where denotes the set of real numbers and consider the difference equation

(1)

where and our study will focus on the cases where and even though the method can be used for larger . In particular we construct certain functions so that (1) will have periodic solutions and then employ the maximum operator to generate additional periodic difference equations. Recall that a solution of (1) is said to be eventually periodic with prime period if there is some such that for all and for . Equation (1) is said to be globally periodic of period if every solution is p-periodic regardless of the initial terms .

Definition: A function is said to be symmetric if it is invariant under any permutation of

Definition: A function is said to be isovertible if it is an invertible function in each of its independent variables, i.e., , … are invertible functions.

Example 1: The functions and are both symmetric and isovertible in two variables.

Example 2: The functions , and are both symmetric and isovertible in three variables.

The functions listed in examples 1 and 2 will be called principal in two and three variables. Generally, a function will be called an if it is both symmetric and isovertible.

Letting and , the set will be referred to as the fundamental class of in two variables. Similarly, letting and , the set is the fundamental set of of three variables.

II. Main Results

The following two theorems are based on results found in [1] and forms a basis for the construction of periodic difference equations of orders 2 and 3.

Theorem A: Suppose is a positive integer so that and divides . Then (1) is periodic with period if

,

For , (1) takes on the following forms

For , (1) has the following form

For third order equations we get the following companion theorem

Theorem B: suppose is a positive integer such that >3 and divides 3. Then (1) is periodic with period p if

,

For , (1) has one of the forms

For , (1) becomes

or

III. Constructions based on Theorem A and recommendations for future study

Using the right-hand side of equations for and the maximum operator many additional periodic difference equations can be derived.

Example: Using and with we construct the maxi linear difference equation

Using the fact that is 3-periodic and is 4-periodic we found the following to be true.

Theorem: If is a solution of with initial values where and are positive integers, then is periodic with period where . In a similar fashion we construct another example

Example: Using and with we get the equation

Taking and will generate a 7-cycle following the method used in constructing and we can construct up to 15 different periodic difference equations. Moreover by expanding to triples on the right-hand side we get an additional ten different periodic difference equations.

Example: Using and with we get

By taking and , we get an eventual 7-cycle . Note that using a triple we lost periodicity but we still have eventual periodicity.

Clear this work serves as only a beginning to unlock the mysteries of periodic difference equations.

Using Theorem B together with the construction methods discussed above enables the construction of third periodic difference equations. Some results of these equations are forthcoming in a separate work.

References

1. K. T. Al-Dosary, Periodicity and Transformation of Difference Equations, Int. J. Math. Anal., Vol. 4, 2010(12) 565-576

2. Jaida Rice, Modesta Trejo, Some Results on Periodic Maxilinear Difference Equations, preprint, summer 2016.

**Faculty Advisor: **Willie Taylor,
taylor_we@tsu.edu

**Role:** Constructed periodic difference equations of order two and three