Bifurcation of limit cycles from quadratic isochrones

Periodic orbits from second order perturbation via rational trigonometric integrals

“…First we provide the expressions of r i (θ; ρ), i = 1, 2, for the solutions of (6) (see Proposition 6). Second we prove Theorem 1, which gives conditions for the existence of hyperbolic periodic solutions bifurcating from the period annulus of equation (6). In this result, additionally, the shape of these solutions is also given.…”

“…This can be seen in all the examples of next sections. Moreover, the functions F 1 and F 2 defined in (7) are the first and second Poincaré-Pontryagin-Melnikov functions for equation (6). As an application, next proposition bounds, up to second order study, the number of periodic solutions of a polynomial perturbation of equation (6) for a class of Abel equations with a(θ) = sin θ.…”

“…In this section we are concerned with both the number and shape of the hyperbolic periodic solutions, bifurcating from the period annulus around r = 0, of the perturbed equation (6). First we provide the expressions of r i (θ; ρ), i = 1, 2, for the solutions of (6) (see Proposition 6).…”

“…These three families are written, respectively, as Figure 1 shows the phase portraits of these families and only the first one presents two continua of periodic orbits. The first order perturbation problem for the above families is considered in [6]. Next result revisits this study and goes on to the second order quadratic perturbation.…”

“…The latter proposition deals with the simultaneous bifurcation of limit cycles from the two centers of S 1 , using first and second order perturbations. There are not to many studies of simultaneous bifurcation from more than one continuum of periodic orbits, some works related on simultaneity on quadratic vector fields are [5,6,11,23].…”