| Citation: |
Xia Liu. THE NUMBER OF LIMIT CYCLES FROM ELLIPTIC HAMILTONIAN VECTOR FIELDS BY HIGHER ORDER MELNIKOV FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1239-1254. doi: 10.11948/20220063
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THE NUMBER OF LIMIT CYCLES FROM ELLIPTIC HAMILTONIAN VECTOR FIELDS BY HIGHER ORDER MELNIKOV FUNCTIONS
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Abstract
In this paper, the perturbed Hamiltonian system $ dH=\epsilon F_4+\epsilon^2F_3+\epsilon^3F_2+\epsilon^4F_1 $, with $ F_i $ the vector valued homogeneous polynomials of degree $ i $. The Hamiltonian function is $ H=y^2/2+U(X), $ where $ U $ is a univariate polynomial of degree four without symmetry. By computing higher order Melnikov functions, the upper bounds for the number of limit cycles that bifurcate from $ dH=0 $ are deserved.
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Keywords:
- Melnikov functions /
- bifurcation /
- limit cycles /
- generators
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References
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Xia Liu. THE NUMBER OF LIMIT CYCLES FROM ELLIPTIC HAMILTONIAN VECTOR FIELDS BY HIGHER ORDER MELNIKOV FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1239-1254. doi: 10.11948/20220063
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$ \frac{8}{9}<a<1 $ $ a<0 $