Periodic solutions of arbitrary length in a simple integer iteration
<p>We prove that all solutions to the nonlinear second-order difference equation in integers <mml:math alttext="$y_{n+1}=lceil ay_{n} ceil -y_{n-1}$"> <mml:mrow> <mml:msub> <mml:mi>y</mml:mi> <mml:mrow> <mml:mi>n</mml:mi><mml:mo...
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| Language: | English |
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SpringerOpen
2006-01-01
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| Series: | Advances in Difference Equations |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/ADE/2006/35847 |
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doaj-9d28b4c7704a4167a5df29857f9310d72020-11-24T21:04:31ZengSpringerOpenAdvances in Difference Equations1687-18392006-01-012006Periodic solutions of arbitrary length in a simple integer iteration<p>We prove that all solutions to the nonlinear second-order difference equation in integers <mml:math alttext="$y_{n+1}=lceil ay_{n} ceil -y_{n-1}$"> <mml:mrow> <mml:msub> <mml:mi>y</mml:mi> <mml:mrow> <mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo><mml:mrow><mml:mo>⌈</mml:mo> <mml:mrow> <mml:mi>a</mml:mi><mml:msub> <mml:mi>Y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⌉</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msub> <mml:mi>y</mml:mi> <mml:mrow> <mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math>,<mml:math alttext="${ain mathbb{R} : |a|<2, a eq 0,pm 1}$"> <mml:mrow> <mml:mrow><mml:mo>{</mml:mo> <mml:mrow> <mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi><mml:mo>:</mml:mo><mml:mrow><mml:mo>|</mml:mo> <mml:mi>a</mml:mi> <mml:mo>|</mml:mo></mml:mrow><mml:mo><</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn> </mml:mrow> <mml:mo>}</mml:mo></mml:mrow> </mml:mrow> </mml:math>,<mml:math alttext="$y_{0},y_{1}in mathbb{Z}$"> <mml:mrow> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mo>,</mml:mo><mml:msub> <mml:mi>y</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>∈</mml:mo><mml:mi>ℤ</mml:mi> </mml:mrow> </mml:math>, are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane. </p>http://www.hindawi.com/GetArticle.aspx?doi=10.1155/ADE/2006/35847 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| title |
Periodic solutions of arbitrary length in a simple integer iteration |
| spellingShingle |
Periodic solutions of arbitrary length in a simple integer iteration Advances in Difference Equations |
| title_short |
Periodic solutions of arbitrary length in a simple integer iteration |
| title_full |
Periodic solutions of arbitrary length in a simple integer iteration |
| title_fullStr |
Periodic solutions of arbitrary length in a simple integer iteration |
| title_full_unstemmed |
Periodic solutions of arbitrary length in a simple integer iteration |
| title_sort |
periodic solutions of arbitrary length in a simple integer iteration |
| publisher |
SpringerOpen |
| series |
Advances in Difference Equations |
| issn |
1687-1839 |
| publishDate |
2006-01-01 |
| description |
<p>We prove that all solutions to the nonlinear second-order difference equation in integers <mml:math alttext="$y_{n+1}=lceil ay_{n} ceil -y_{n-1}$"> <mml:mrow> <mml:msub> <mml:mi>y</mml:mi> <mml:mrow> <mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo><mml:mrow><mml:mo>⌈</mml:mo> <mml:mrow> <mml:mi>a</mml:mi><mml:msub> <mml:mi>Y</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⌉</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msub> <mml:mi>y</mml:mi> <mml:mrow> <mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math>,<mml:math alttext="${ain mathbb{R} : |a|<2, a eq 0,pm 1}$"> <mml:mrow> <mml:mrow><mml:mo>{</mml:mo> <mml:mrow> <mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℝ</mml:mi><mml:mo>:</mml:mo><mml:mrow><mml:mo>|</mml:mo> <mml:mi>a</mml:mi> <mml:mo>|</mml:mo></mml:mrow><mml:mo><</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn> </mml:mrow> <mml:mo>}</mml:mo></mml:mrow> </mml:mrow> </mml:math>,<mml:math alttext="$y_{0},y_{1}in mathbb{Z}$"> <mml:mrow> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mo>,</mml:mo><mml:msub> <mml:mi>y</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>∈</mml:mo><mml:mi>ℤ</mml:mi> </mml:mrow> </mml:math>, are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane. </p> |
| url |
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/ADE/2006/35847 |
| _version_ |
1716770801670160384 |