Lower bounds for the local cyclicity of centers using high order developments and parallelization

Orbitally symmetric systems with applications to planar centers

“…Degree three orbitally ϕ-symmetric vector fields corresponding to them are detailed in [40,41]. The cyclicity of the cubic vector field 6.1 labeled by CR (12) 17 in [41] is studied in [21]. Here we prove that the equilibrium point is of center type because it satisfies 1.3.…”

“…The previous best lower bound was 44, see [20]. For other low degree vector fields the best lower bounds for M (n) are given in [21]. Up to our knowledge, these are the first studies of this problem for the presented orbitally ϕ-symmetric systems.…”

“…This section is devoted to study the bifurcation of limit cycles of small amplitude in some of the polynomial families presented in this work. The mechanism is the classical degenerate Hopf-bifurcation, we have closely follow the notation and results in [21]. In fact, the main bifurcation theorems were given by Christopher in [10], where he details in Theorems 2.1 and 3.1 how to use first and higher-order developments, respectively, of the Lyapunov constants to provide the complete unfolding of limit cycles bifurcating from the center equilibrium point.…”

“…Recently, in [18], this conjecture should be updated in one, M (n) = n 2 + 3n − 6, because it is false, at least for n = 3. In the recent work [21] the local cyclicity of some Darboux cubic centers in [41], the ones with the highest codimension, is studied. The embryo of this idea appears in two relatively old works of Chicone & Jacobs [8,9] but better developed in [10].…”

“…The difficulties are related with the fact that we are almost using the total number of essential free parameters. For small degrees n, the best lower bound values for M (n) can be found in [18,21].…”