On the Global Character of the System of Piecewise Linear Difference Equations and

Abstract

We consider the system in the title where the initial condition We show that the system has exactly two prime period-5 solutions and a unique equilibrium point . We also show that every solution of the system is eventually one of the two prime period-5 solutions or else the unique equilibrium point.

1. Introduction

In this paper, we consider the system of piecewise linear difference equations

(1.1)

where the initial condition . We show that every solution of System (1.1) is eventually either one of two prime period-5 solutions or else the unique equilibrium point .

System (1.1) was motivated by Devaney's Gingerbread man map [1, 2]

(1.2)

or its equivalent system of piecewise linear difference equations [3, 4]

(1.3)

We believe that the methods and techniques used in this paper will be useful in discovering the global character of solutions of similar systems, including the Gingerbread man map.

2. The Global Behavior of the Solutions of System (1.1)

System (1.1) has the equilibrium point given by

(2.1)

System (1.1) has two prime period-5 solutions,

(2.2)

Set

(2.3)

Theorem 2.1.

Let . Then there exists an integer such that the solution is eventually either the prime period-5 solution , the prime period-5 solution , or else the unique equilibrium point .

The proof is a direct consequence of the following lemmas.

Lemma 2.2.

Suppose there exists an integer such that and . Then , and so is the equilibrium solution.

Proof.

Note that

(2.4)

and so the proof is complete.

Lemma 2.3.

Suppose there exists an integer such that and . Then , and so is .

Proof.

We have

(2.5)

and so the proof is complete.

Lemma 2.4.

Suppose there exists an integer such that and . Then the following statements are true.

• (1) .

• (2) If , then is .

• (3) If , then .

Proof.

We have and . Then

(2.6)

and so statement (1) is true.

If , then . That is, and so statement (2) is true.

If , then , and so statement (3) is true.

Lemma 2.5.

Suppose there exists an integer such that and . Then the following statements are true.

• (1) .

• (2) If , then .

• (3) If , then is .

Proof.

We have and . Then

(2.7)

and so statement (1) is true.

Now if , then . Thus , and so statement (2) is true.

Lastly, if , then . Thus ; that is, and so statement (3) is true.

Lemma 2.6.

Suppose there exists an integer such that and . Then the following statements are true.

• (1) If , then is .

• (2) If , then is .

• (3) If , then and .

Proof.

First consider the case and . Then

(2.8)

and so statement (1) is true.

Next consider the case and . Then

(2.9)

If , then and so . That is, and so statement (2) is true.

If , then . Thus , and so statement (3) is true.

Lemma 2.7.

Suppose there exists an integer such that and . Then the following statements are true.

• (1) .

• (2) If , then .

• (3) If , then is .

Proof.

Let and . Then

(2.10)

and so statement (1) is true.

If , then . Thus , and so statement (2) is true.

If , then and . That is, and so is and the proof is complete.

We now give the proof of Theorem 2.1 when is in .

Lemma 2.8.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then is eventually the equilibrium solution.

• (2) If , then the solution is .

• (3) If , then the solution is eventually .

Proof.

• (1) We will first show that statement (1) is true. Suppose ; for each , let

  

  (2.11)

Observe that

(2.12)

Thus there exists a unique integer such that .

We first consider the case ; that is, . By statements (1) and (3) of Lemma 2.4, and . Clearly , and so

(2.13)

Now and , and so by Lemma 2.2, is the equilibrium solution.

Without loss of generality, we may assume .

For each integer such that , let be the following statement:

(2.14)

Claim 1.

is true for .

The proof Claim 1 will be by induction on . We will first show that is true.

Recall that and . Then by statements (1) and (3) of Lemma 2.4, we have and .

Note that,

(2.15)

and so is true. Thus if , then we have shown that for , is true. It remains to consider the case . So assume that . Let be an integer such that and suppose is true. We will show that is true.

Since is true, we know

(2.16)

It is easy to verify that for ,

(2.17)

Thus by statements (1) and (3) of Lemma 2.4,

(2.18)

Recall that .

In particular,

(2.19)

and so is true. Thus the proof of the claim is complete. That is, is true for . Specifically, is true, and so

(2.20)

In particular,

(2.21)

That is, , and so by case , is the equilibrium solution, and the proof of statement (1) is complete.

• (2) We will next show that statement (2) is true. Suppose . Note that . Thus the solution is .

• (3) Finally, we will show that statement (3) is true. Suppose .

First consider . By statement (2) of Lemma 2.4, the solution is .

Next consider the case . For each , let

(2.22)

Observe that

(2.23)

Thus there exists a unique integer such that .

Note that the statement which we stated and proved in the proof of statement (1) of this lemma still holds. Specifically is true, and so

(2.24)

Recall that for .

In particular,

(2.25)

By statement (2) of Lemma 2.4, the solution is .

We now give the proof of Theorem 2.1 when is in .

Lemma 2.9.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then is eventually the equilibrium solution.

• (2) If , then the solution is .

• (3) If , then the solution is eventually .

Proof.

• (1) We will first show that statement (1) is true. So suppose .

Case 1.

Suppose . Then

(2.26)

In particular, and , and so by Lemma 2.2, is the equilibrium solution.

Case 2.

Suppose . By statements (1) and (2) of Lemma 2.5, and . Then

(2.27)

Thus and , and so by Lemma 2.2, is the equilibrium solution.

Case 3.

Suppose . By statements (1) and (2) of Lemma 2.5, and . Note that and so by statement (1) of Lemma 2.8, is eventually equilibrium solution.

• (2) We will next show that statement (2) is true. Suppose . By direct calculations we have . So the solution is .

• (3) Finally, we will show that statement (3) is true. Suppose and .

Case 1.

Suppose . By statements (1) and (2) of Lemma 2.5, we have and . Note that and so by statement (3) of Lemma 2.8, the solution is eventually .

Case 2.

Suppose . By statement (3) of Lemma 2.5, the solution is .

We now give the proof of Theorem 2.1 when is in .

Lemma 2.10.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then is eventually the equilibrium solution.

• (2) If , then the solution is .

• (3) If , then the solution is eventually .

Proof.

• (1) We will first show that statement (1) is true. So suppose and . By statement (3) of Lemma 2.6, and . In particular, and so by statement (1) of Lemma 2.8 and statement (1) of Lemma 2.9, is eventually the equilibrium solution.

• (2) We will next show that statement (2) is true. Suppose . By direct calculations we have . Thus the solution is .

• (3) Finally, we will show statement (3) is true.

First consider the case . By statement (3) of Lemma 2.6, and . Now, and so by statement (3) of Lemma 2.8, the solution is eventually .

Next consider the case . Then by statements (1) and (2) of Lemma 2.6, if then is , and if then is .

We next give the proof of Theorem 2.1 when is in .

Lemma 2.11.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then is eventually the equilibrium solution.

• (2) If , then the solution is .

• (3) If , then the solution is eventually .

Proof.

• (1) We will first prove statement (1) is true. Suppose .

First consider the case . Then

(2.28)

In particular, and and so by Lemma 2.2, is the equilibrium solution.

Next consider the case . By statements (1) and (2) of Lemma 2.7, and . In particular, and so by statement (1) of Lemma 2.8 and statement (1) of Lemma 2.9, is eventually the equilibrium solution.

• (2) We will next show that statement (2) is true. Suppose . By direct calculations, we have . That is, is .

• (3) Lastly, we will show that statement (3) is true. Suppose .

First consider the case . By statements (1) and (2) of Lemma 2.7, and . In particular, and so by statement (3) of Lemma 2.8, the solution is eventually .

Next consider the case . By statement (3) of Lemma 2.7, the solution is .

We next give the proof of Theorem 2.1 when is in .

Lemma 2.12.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then the solution is .

• (2) If , then there exists an integer such that .

Proof.

Suppose and .

Then

(2.29)

Case 1.

Suppose . Then, in particular, and . Thus

(2.30)

and so statement (1) is true.

Case 2.

Suppose . Then, in particular, .

Subcase 1.

Suppose .

Then . It follows by a straight forward computation, which will be omitted, that . Hence .

Subcase 2.

Suppose .

Then . It follows by a straight forward computation, which will be omitted, that . Hence , and the proof is complete.

We next give the proof of Theorem 2.1 when is in .

Lemma 2.13.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then the solution is the equilibrium solution.

• (2) If , then .

Proof.

By assumption, we have and .

If , then

(2.31)

Hence is the equilibrium solution and statement (1) is true.

If , then it follows by a straight forward computation, which will be omitted, that . Thus and statement (2) is true.

We next give the proof of Theorem 2.1 when is in .

Lemma 2.14.

Suppose there exists an integer such that . Then the following statements are true.

• (1) If , then .

• (2) If , then .

• (3) If , and , then .

• (4) If , , and , then .

• (5) If , , and , then .

Proof.

Now and .

• (1) If , then

  

  (2.32)

  Thus .

• (2) If , then . It follows by a straight forward computation, which will be omitted, that

  

  (2.33)

  Hence .

• (3) If , , and , then . It follows by a straight forward computation, which will be omitted, that

  

  (2.34)

  Thus .

• (4) If , , , and , then . It follows by a straight forward computation, which will be omitted, that

  

  (2.35)

  Thus .

• (5) Finally, suppose that , , , and . Then . It follows by a straight forward computation, which will be omitted, that

  

  (2.36)

  Note that

  

  (2.37)

  and so by the first statement of this Lemma, .

Thus we see that if there exists an integer such that , then the proof of Theorem 2.1 is complete. Finally, we consider the case where the initial condition .

Lemma 2.15.

Suppose there exists an integer such that . Then there exists a positive integer such that .

Proof.

Without loss of generality, it suffices to consider the case where

(2.38)

Now , and hence and .

Thus

(2.39)

We have , and thus

(2.40)

We also have , and hence

(2.41)

Finally, we have , and so

(2.42)

In particular, and hence .

3. Conclusion

We have presented the complete results concerning the global character of the solutions to System (1.1). We divided the real plane into 8 sections and utilized mathematical induction, proof by iteration, and direct computations to show that every solution of System (1.1) is eventually either the prime period-5 solution , the prime period-5 solution , or else the unique equilibrium point . The proofs involve careful consideration of the various cases and subcases.