Periodicity of a second-order switched difference system over integers

Abstract

In this paper, a second-order switched difference system which consists of two linear difference equations with a switching rule is proposed to study. Specifically, the periodicity of a particular case is addressed, deriving the appropriate rational values for parameter r which possess periodic integer solutions. By the transformation method, the particular second-order difference system is transformed into a first-order switched system. And, we prove that: (1) this system possesses periodic integer solutions of prime period two if and only if $r=-1/2$; (2) any rational r except for the integers arises periodic integer solutions of prime period three; (3) periodic integer solutions of prime period four exist if and only if $r=-1/2$; (4) this system possesses no periodic solutions of prime period five. We also prove that if $r>0$ and the system has periodic integer solutions of prime period $k⩾6$, then the only possible values of r are reciprocals of integers.

Introduction

A switched system is a dynamic system that consists of a finite number of subsystems and a logical rule that orchestrates switching between these subsystems. Traditional system theory is concerned with either continuous or discrete behavior, however, many dynamical systems encountered in practice include both types of dynamics. These problems could be modeled by switched systems or hybrid systems that are complex dynamic systems including continuous-time dynamics as well as discrete-time dynamics [1], [2], [3]. Many researchers in systems and control theory tend to regard hybrid systems as continuous systems with switching and place a greater emphasis on properties of the continuous state [4].

In recent years, studying the dynamics of difference equations is a hot topic, e.g., see [5], [6], [7], [8], [9] and the references therein. Especially, the periodic property of difference equations or systems had been widely investigated, such as [10], [11], [12], [13], [14], [15], [16], [17]. Mathematically, the subsystems of a switched system are often depicted by a collection of indexed differential or difference equations. A switched difference system refers to a switched system which consists of a family of discrete-time subsystems and a rule orchestrating the switching between the subsystems. The famous Collatz problem (also known as the $3x+1$ problem) [18] concerns the following mapping

$$g(n)=\left\{\begin{array}{c}3n+1\text{,}\mathrm{if}n\equiv 1(\text{mod}2)\hfill \\ n/2\text{,}\mathrm{if}n\equiv 0(\text{mod}2)\text{,}\hfill \end{array}\right.$$

where n is a natural number, i.e., a function taking odd integers n to $3n+1$ and even integers n to $n/2$. The Collatz Conjecture is that for any initial positive integer n, the repeated iterates of the above function, i.e., $g^k(n)$, eventually produces the periodic orbit $(1\text{,}4\text{,}2)$. This problem has been widely investigated in different places all over the world, however, it has turned out to be intractably hard to prove and is still unsolved by now in spite of massive try and effort. Paul Erdös ever commented regarding the intractability of the $3x+1$ Conjecture that Mathematics was not yet ripe for such problems.

Besides, much attention has been paid to the Collatz-like problems, which have simple forms but are also difficult to understand, such as [19], [20], [21], [22]. It was proved in [19] that each integer solution of the difference equation

$$a_n=\left\{\begin{array}{c}(a_{n-1}+a_{n-2})/2\text{,}\mathrm{if}{a}_{n-1}+a_{n-2}\equiv 0(\text{mod}2)\text{,}\hfill \\ a_{n-1}+a_{n-2}\text{,}\mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where the initial values $a_0\text{,}{a}_1$ are positive integers, was either stationary or unbounded. Ladas [20] gave the conjecture that all integer solutions to the difference equation

$$a_n=\left\{\begin{array}{c}(a_{n-1}+a_{n-2})/3\text{,}\mathrm{if}{a}_{n-1}+a_{n-2}\equiv 0(\text{mod}3)\text{,}\hfill \\ a_{n-1}+a_{n-2}\text{,}\mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

are unbounded except for certain obvious periodic solutions, such as $1\text{,}1\text{,}2\text{,}1\text{,}1\text{,}2\text{,}\dots$ and $7\text{,}14\text{,}7\text{,}7\text{,}14\text{,}7\text{,}\dots$. It was pointed out that for such kind of difference equations, any solution that is not eventually periodic must be unbounded, hence the only problem is to classify the periodic solutions.

However, this problem seems extremely difficult. For example, Greene and Niedzielski [21] studied a generalization of (1), (2) formulated by

$$a_n=\left\{\begin{array}{c}r(a_{n-1}+a_{n-2})\text{,}\mathrm{if}r(a_{n-1}+a_{n-2})\mathrm{is}\mathrm{an}\mathrm{integer}\text{,}\hfill \\ a_{n-1}+a_{n-2}\text{,}\mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where r is some fixed rational number. Finding that it was very hard to characterize its periodic solutions for a given specified r, they did some research by addressing a different and easier question that “Can we find other values of r which allow periodic solutions.”

Following this idea, He and Liu [22] studied the following third-order switched difference system

$$y_n=\left\{\begin{array}{c}r(y_{n-1}+y_{n-2}+y_{n-3})\text{,}\mathrm{if}r(y_{n-1}+y_{n-2}+y_{n-3})\in \mathbb{Z}\text{,}\hfill \\ y_{n-1}+y_{n-2}+y_{n-3}\text{,}\mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where r is some appropriate rational number, and proved that: (1) for any rational parameter r except for integers, system (4) possesses no periodic integer solutions of prime period two or three; (2) periodic integer solutions of prime period four exist if and only if $r=-1/3$ or $r=1/5$; (3) periodic integer solutions of prime period five exist if and only if $r=-7/15$.

Motivated by the above papers, this paper proposes to study the following second-order switched difference system

$$z_n=\left\{\begin{array}{c}r(b_1{z}_{n-1}+b_2{z}_{n-2})\text{,}\mathrm{if}r(b_1{z}_{n-1}+b_2{z}_{n-2})\in \mathbb{Z}\text{,}\hfill \\ c_1{z}_{n-1}+c_2{z}_{n-2}\text{,}\mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where $b_1\text{,}{b}_2\text{,}{c}_1\text{,}{c}_2$ are integer coefficients, r is a rational parameter and the initial values $z_{-2}\text{,}{z}_{-1}$ are integers.

Clearly, system (5) is another generalization of (1), (2), (3). It is easy to observe that zero is a trivial integer equilibrium of system (5). First, we are interested in that whether system (5) possesses any other nonzero integer equilibria or not. To answer this question, we proceed by the following three cases:

• if $b_1+b_2=0$, then system (5) has the unique zero equilibrium;

• if $b_1+b_2\ne 0$ and $r=1/(b_1+b_2)$, then any integer is an equilibrium;

• if $b_1+b_2\ne 0$ and $r\ne 1/(b_1+b_2)$, then zero is the unique equilibrium.

Note that system (5) consists of two linear difference equations. In particular, when r is an integer, system (5) reduces to the following linear difference equation

$$z_n=r(b_1{z}_{n-1}+b_2{z}_{n-2})\text{,}n\in {\mathbb{N}}_0\text{.}$$

In this case, explicit solutions of Eq. (6) can be derived by the linear difference equation theory, thus in the following it suffices to consider the cases $r\in \mathbb{Q}⧹\mathbb{Z}$, the set of rational numbers except for integers.

Next, we will state an equivalent description of system (5). Our idea is to transform system (5) into a first-order switched system.

Denote $\mathbf{x}={(x_1\text{,}{x}_2)}^T\in \mathbb{Z}\times \mathbb{Z}$, and define a vector function $\mathbf{f}:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\times \mathbb{Z}$ such that

$$\mathbf{f}(\mathbf{x})=\left\{\begin{array}{cc}\mathbf{Dx}\text{,}\hfill & \mathrm{if}\mathbf{Dx}\in \mathbb{Z}\times \mathbb{Z}\text{,}\hfill \\ \mathbf{Ex}\text{,}\hfill & \mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where $\mathbf{D}=\left(\begin{array}{cc}{\mathit{rb}}_1\hfill & {\mathit{rb}}_2\hfill \\ 1\hfill & 0\hfill \end{array}\right)\text{,}\mathbf{E}=\left(\begin{array}{cc}{c}_1\hfill & c_2\hfill \\ 1\hfill & 0\hfill \end{array}\right)\text{,}r$ is a rational parameter, and $b_1\text{,}{b}_2\text{,}{c}_1\text{,}{c}_2$ are integers.

By the above function $\mathbf{f}$, system (5) can be rewritten as a first-order switched system.

Denote ${\mathbf{x}}_n={(x_n^{(1)}\text{,}{x}_n^{(2)})}^T\in \mathbb{Z}\times \mathbb{Z}\text{,}n\in {\mathbb{N}}_0$, then ${\mathbf{x}}_n={\mathbf{f}}^n({\mathbf{x}}_0)\text{,}n\in \mathbb{N}$, thus we get the following first-order system

$${\mathbf{x}}_{n+1}=\mathbf{f}({\mathbf{x}}_n)=\left\{\begin{array}{cc}{\mathbf{Dx}}_n\text{,}\hfill & \mathrm{if}{\mathbf{Dx}}_n\in \mathbb{Z}\times \mathbb{Z}\text{,}\hfill \\ {\mathbf{Ex}}_n\text{,}\hfill & \mathrm{otherwise}\text{.}\hfill \end{array}\right.$$

Let ${\{z_n\}}_{n⩾2}$ be a solution of (5), and denote ${\mathbf{x}}_n={(z_{n-1}\text{,}{z}_{n-2})}^T\text{,}n\in {\mathbb{N}}_0$, then $\{{\mathbf{x}}_n\}$ is a solution of (7).

This paper is devoted to studying of a special case of system (5) with $b_1=-b_2=c_1=c_2=1$, i.e. the following second-order switched difference system

$$z_n=\left\{\begin{array}{cc}\hfill & r(z_{n-1}-z_{n-2})\text{,}\mathrm{if}r(z_{n-1}-z_{n-2})\in \mathbb{Z}\text{,}\hfill \\ \hfill & z_{n-1}+z_{n-2}\text{,}\mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where $r\in \mathbb{Q}⧹\mathbb{Z}$, and the initial values $z_{-1}\text{,}{z}_{-2}$ are integers.

Obviously, system (8) corresponds to the following first-order switched system

$${\mathbf{x}}_{n+1}=\mathbf{f}({\mathbf{x}}_n)=\left\{\begin{array}{cc}{\mathbf{Ax}}_n\text{,}\hfill & \mathrm{if}{\mathbf{Ax}}_n\in \mathbb{Z}\times \mathbb{Z}\text{,}\hfill \\ {\mathbf{Bx}}_n\text{,}\hfill & \mathrm{otherwise}\text{,}\hfill \end{array}\right.$$

where $\mathbf{A}=\left(\begin{array}{cc}\hfill r\hfill & \hfill -r\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)\text{,}\mathbf{B}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$.