Simultaneous bifurcation of limit cycles from a linear center with extra singular points

Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli

“…As in [24], the integrals (2), in the usual polar coordinates (x, y) = (r cos θ, r sin θ), can be written, for j ∈ {i, e}, as…”

“…Although the most common technique to study simultaneity is the Z n symmetry, see for example [20,21], when it is not considered, see for example [3], more periodic orbits appear. This is the case done in [24] where nonsymmetric perturbations are considered. Following the same procedure we consider now a piecewise polynomial perturbation of the two nested period annuli.…”

“…We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli, first separately and second simultaneously. This problem is a generalization of [24] to the piecewise systems class. When the polynomial perturbation has degree n, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros.…”

Simultaneity of centres in ℤq-equivariant systems

“…For instance, the existence of two simultaneous centres was studied in [39,40] for quadratic systems, and in [41,42] for some particular cubic systems. The simultaneity of centres in planar differential systems is important because perturbations of such systems give a great number of bifurcations of limit cycles; see [30,43,44].…”