Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation
oleh: Tarek F. Ibrahim, Zehra Nurkanović
| Format: | Article |
|---|---|
| Diterbitkan: | MDPI AG 2019-08-01 |
Deskripsi
By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>α</mi> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>+</mo> <mi>β</mi> <msubsup> <mi>t</mi> <mrow> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>−</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="1.em"></mspace> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where are <inline-formula> <math display="inline"> <semantics> <msub> <mi>t</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.