On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

Abstract

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x = y, y = Q 1 (x) + ε y Q 2 (x) with Q 1 and Q 2 polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε.And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3, 0) of limit cycles can appear for some values of parameters, where (3, 0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.

The monotonicity of the ratio of two Abelian integrals

“…Their criterion is useful and can work in many independently arising particular cases: elliptic case, hyperelliptic case, and even the non-algebraic case. There have been some results obtained by using this criterion; here we only list [3,12,24] for some hyperelliptic Hamiltonian functions of degrees five and six. However, it is very difficult to use this criteria for most Hamiltonian functions with parameters.…”

Nine Limit Cycles in a 5-Degree Polynomials Liénard System

“…Dumortier and Li studied four Liénard systems with different portraits of type (3,2) in a series of papers [5][6][7][8] and gave the corresponding sharp bound of number of limit cycles by Poincaré bifurcation. e exact bounds of H(4, 3) and H (5,4) for some special Liénard systems were reported in [9][10][11][12][13][14] and references therein. For results on H(7, 6) associated with symmetric system (5), the relatively new works are referred [15][16][17].…”

Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

“…There are abundant results for weak Hilberts 16th problem restricted to Liénard systems of type ( , ), especially for = − 1, such as types (3, 2) [16], (4, 3) [17], (5,4) [18,19], and (7,6) [20]. More details of the relative researches can be seen in [21][22][23][24][25][26][27][28][29][30][31][32].…”