Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms

A class of cubic Hamiltonion system with the higher-order perturbed term of degree n = 5, 7, 9, 11, 13 is investigated. We find that there exist at least 13 limit cycles with the distribution C-9(1) superset of 2[C-3(2) superset of 2C(2)(2)] (let C-m(k) denote a nest of limit cycles which encloses nl singular points, and the symbol 'subset of' is used to show the enclosing relations between limit cycles, while the sign '+' is used to divide limit cycles enclosing different critical points. Denote simply C-m(k) + C-m(k) = 2C(m)(k), etc.) in the Hamiltonian system under the perturbed term of degree 7, and give the complete bifurcation diagrams and classification of the phase portraits by using bifurcation theory and qualitative method and numerical simulations. These results in this paper are useful for the study of the weaken Hilbert 16th problem. (C) 2000 Published by Elsevier Science Ltd.