- Research Article
- Open Access

- Wirot Tikjha
^{1, 2}, - Yongwimon Lenbury
^{1, 2}Email author and - EvelinaGiusti Lapierre
^{3}

**2010**:573281

https://doi.org/10.1155/2010/573281

© Wirot Tikjha et al. 2010

**Received:**23 June 2010**Accepted:**2 December 2010**Published:**13 December 2010

## 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.

## Keywords

- Differential Equation
- Direct Consequence
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis

## 1. Introduction

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 .

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)

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.

and so the proof is complete.

Lemma 2.3.

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

Proof.

and so the proof is complete.

Lemma 2.4.

Proof.

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.

Proof.

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.

Proof.

and so statement (1) is true.

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

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

Lemma 2.7.

Proof.

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.

Thus there exists a unique integer such that .

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

Without loss of generality, we may assume .

Claim 1.

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 .

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.

- (2)
- (3)

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

Thus there exists a unique integer such that .

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.

Case 1.

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

Case 2.

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.

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.

- (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.

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

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.

Proof.

Case 1.

and so statement (1) is true.

Case 2.

Suppose . Then, in particular, .

Subcase 1.

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

Subcase 2.

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.

Proof.

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.

- (1)
- (2)
- (3)
- (4)
- (5)

Proof.

- (1)
- (2)
- (3)
- (4)
- (5)

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.

## 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.

## Declarations

### Acknowledgments

The authors would like to express their gratitude to the Strategic Scholarships Fellowships Frontier Research Networks, the Office of the Commission on Higher Education, and National Center for Genetic Engineering and Biotechnology.

## Authors’ Affiliations

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## Copyright

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