On limit cycles near two centres and a double homoclinic loop in Liénard differential system

This paper studies the number of limit cycles in an ( m + 1 ) th order Lienard differential system, which are bifurcated from the families of periodic orbits near two centres and a double homoclinic loop of a Hamiltonian system with an elliptic Hamiltonian function H = 1 2 y 2 − 1 2 x 2 + 1 4 x 4 . It is proved that this kind of systems can have at least 7 [ m − 2 5 ] + 2 r − [ r 3 ] ⋅ mod ( r + 1 , 3 ) number of limit cycles, where r = mod ( m − 2 , 5 ) and 3 ≤ m ≤ 18 except m = 8 . When the Lienard differential system is of centrally symmetric, the lower bound on the number of limit cycles could be 7 [ m 6 ] + 2 r + [ r 2 ] with r = mod ( m 2 , 3 ) and m ≤ 124 even. In any of the cases the bounds can be reached with the help of Maple programs. Here mod ( a , b ) represents the remainder of a divided by b.