| Summary: | 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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