Bifurcation Set and Limit Cycles Forming Compound Eyes in a Perturbed Hamiltonian System

BIFURCATION SET AND LIMIT CYCLES FORMING COMPOUND EYES IN A PERTURBED HAMILTONIAN SYSTEM JIBIN LI AND ZHENRONG LIU In this paper we consider a class of perturbation of a Hamiltonian cubic system with 9 finite critical points . Using detection functions, we present explicit formulas for the global and local bifurcations of the flow . We exhibit various patterns of compound eyes of limit cycles . These results are concerned with the weakened Hilbert's 16th problem posed by V.I . Arnold in 1977. 1 . Introduction The weakened Hilbert 16th problem, posed by V.I . Arnold in 1977 [1], is to determine the number of limit cycles that can be generated from a polynomial Hamiltonian system of degree n 1 with perturbed terms of a polynomial of degree m + 1 . The separatrixes and relative positions of the limit cycles for the Hamiltonian system with perturbations play an important role [2] . For a polynomial differential system of degree n, the results of [3] imply that, in order to get more limit cycles and various patterns of their distribution,, one efficient method is to perturb a Hamiltonian system with symmetry which has the maximal number of centers . Thus, to study the weakened Hilbert 16th problem, we should first investigate the property of unperturbed Hamiltonian systems, Le ., determine the global property of the family of planar algebraic curves . Then, by using proper perturbation techniques, we can obtain the global information of the perturbed non-integrable system . Only two particular examples were given in the paper [3] . In this paper we discuss the following system : dx dt =Y(, + x2 ay2) + ex(mx2 + ny 2 d = -x(1 cx2 + y2) +ey(mx2 + ny2 where a > c > 0, ac > 1, 0 < e « 1, m, n, A are parameters . Our object is to reveal the bifurcation set in the 5-parameter space . Since the vector field defined