BIFURCATIONS OF LIMIT CYCLES FORMING COMPOUND EYES IN THE CUBIC SYSTEM

Let H(n)be the maximal number of limit cycle of planar real polynomial differentialsystem with the degree n and C_m~k denote the nest of k limit cycles enclosing m singular points.By computing detection functions,tne authors study bifurcation and phase diagrams in theclass of a planar cubic disturbed Hamiltonian system.In particular,the following conclusionis reached:The planar cubic system(E_ε)has 11 limit cycles,which form the pattern ofcompound eyes of C_9~1(?)2C'~ε(?)(2C_1~2)and have the symmetrical structure;so the Hilbertnumber H(3)≥11.

Bifurcations of Limit Cycles Forming Compound Eyes in the Cubic System

Let H(n) be the maximal number of limit cycle of planar real polynomial differential system with the degree n and C_m^k denote the nest of k limit cycles enclosing m singular points.By computing detection functions, tne authors study bifurcation and phase diagrams in the class of a planar cubic disturbed Hamiltonian system.In particular, the following conclusion is reached: The planar cubic system(E_3) has 11 limit cycles, which form the pattern of compound eyes of C^1_9\supseteqq 2C''_3\supseteqq (2C^2_1)] and have the symmetrical structure; so the Hilbert number H(3)\geq 11.