The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems
This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><...
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MDPI AG
2022-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/9/1483 |
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| author | Lijun Wei Yun Tian Yancong Xu |
| author_facet | Lijun Wei Yun Tian Yancong Xu |
| author_sort | Lijun Wei |
| collection | DOAJ |
| description | This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mi>a</mi><mn>4</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>≠</mo><mn>0</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> under two types of polynomial perturbations of degree <i>m</i>, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mfrac><mrow><mn>3</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mn>4</mn></mfrac><mo>]</mo></mrow></semantics></math></inline-formula> small limit cycles near the center, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≤</mo><mn>101</mn></mrow></semantics></math></inline-formula>, and that the related near-Hamiltonian system with general polynomial perturbations can have at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mo>[</mo><mfrac><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>]</mo><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> small-amplitude limit cycles, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≤</mo><mn>16</mn></mrow></semantics></math></inline-formula>. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo>·</mo><mo>]</mo></mrow></semantics></math></inline-formula> represents the integer function. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:56:30Z |
| publishDate | 2022-04-01 |
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| series | Mathematics |
| spelling | doaj.art-4bc520af0e4f4c1fbfffc30b44242db72023-11-23T08:45:02ZengMDPI AGMathematics2227-73902022-04-01109148310.3390/math10091483The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential SystemsLijun Wei0Yun Tian1Yancong Xu2School of Mathematics, Hangzhou Normal University, Hangzhou 310036, ChinaDepartment of Mathematics, Shanghai Normal University, Shanghai 200234, ChinaSchool of Mathematics, Hangzhou Normal University, Hangzhou 310036, ChinaThis paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mi>a</mi><mn>4</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>≠</mo><mn>0</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> under two types of polynomial perturbations of degree <i>m</i>, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mfrac><mrow><mn>3</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mn>4</mn></mfrac><mo>]</mo></mrow></semantics></math></inline-formula> small limit cycles near the center, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≤</mo><mn>101</mn></mrow></semantics></math></inline-formula>, and that the related near-Hamiltonian system with general polynomial perturbations can have at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mo>[</mo><mfrac><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>]</mo><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> small-amplitude limit cycles, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≤</mo><mn>16</mn></mrow></semantics></math></inline-formula>. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo>·</mo><mo>]</mo></mrow></semantics></math></inline-formula> represents the integer function.https://www.mdpi.com/2227-7390/10/9/1483Liénard systemnear-Hamiltonian systemHopf bifurcationelementary center |
| spellingShingle | Lijun Wei Yun Tian Yancong Xu The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems Mathematics Liénard system near-Hamiltonian system Hopf bifurcation elementary center |
| title | The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems |
| title_full | The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems |
| title_fullStr | The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems |
| title_full_unstemmed | The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems |
| title_short | The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems |
| title_sort | number of limit cycles bifurcating from an elementary centre of hamiltonian differential systems |
| topic | Liénard system near-Hamiltonian system Hopf bifurcation elementary center |
| url | https://www.mdpi.com/2227-7390/10/9/1483 |
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