Abstract
In this paper, we study the number of limit cycles in the perturbed Hamiltonian system \(dH=\varepsilon F_1+\varepsilon ^2 F_2+\varepsilon ^3 F_3\) with \(F_i\), the vector valued homogeneous polynomials of degree i and \(4-i\) for \(i=1,2,3\), and small positive parameter \(\varepsilon \). The Hamiltonian function has the form \(H=y^2/2+U(x)\), where U is a univariate polynomial of degree four without symmetry. We compute higher order Melnikov functions until we obtain reversible perturbations. Then we find the upper bounds for the number of limit cycles that can bifurcate from the periodic orbits of \(dH=0\).
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Acknowledgements
The authors would like to thank the reviewers for their valuable comments and helpful suggestions, which improve the presentation of the paper. This work is supported by Isfahan University of Technology (IUT).
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Asheghi, R., Nabavi, A. Higher Order Melnikov Functions for Studying Limit Cycles of Some Perturbed Elliptic Hamiltonian Vector Fields. Qual. Theory Dyn. Syst. 18, 289–313 (2019). https://doi.org/10.1007/s12346-018-0284-1
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DOI: https://doi.org/10.1007/s12346-018-0284-1