41 research outputs found
The period function for quadratic integrable systems with cubic orbits
AbstractThis paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves. It is proved that each such system has a period function with at most one critical point
Perturbations of quadratic centers of genus one
We propose a program for finding the cyclicity of period annuli of quadratic
systems with centers of genus one. As a first step, we classify all such
systems and determine the essential one-parameter quadratic perturbations which
produce the maximal number of limit cycles. We compute the associated
Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit
cycles. To illustrate our approach, we determine the cyclicity of the annuli of
two particular reversible systems.Comment: 28 page
The bifurcation of limit cycles of two classes of cubic isochronous systems
In this paper, we study the bifurcation of limit cycles of the periodic annulus of two classes of cubic isochronous systems. By using complete elliptic integrals of the first, second kinds and the Chebyshev criterion, we show that the upper bound for the number of limit cycles which appear from the periodic annuli of the two systems are at least three under cubic perturbations. Moreover, there exists a perturbation that give rise to exactly i limit cycles bifurcating from the period annulus for each i = 0, 1, 2, 3
Bifurcation of critical periods from Pleshkan's isochrones
Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities â„“3. In this paper we prove that if we perturb any of these isochrones inside â„“3, then at most two critical periods bifurcate from its period annulus. Moreover, we show that, for each k=0, 1, 2, there are perturbations giving rise to exactly k critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers â„“2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in â„“2. We prove that if we perturb three of them inside â„“2, then at most one critical period bifurcates from its period annulus. In addition, for each k=0, 1, we show that there are perturbations giving rise to exactly k critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper
Algebraic and analytical tools for the study of the period function
In this paper we consider analytic planar differential systems having a first integral of the form H(x, y) = A(x) + B(x)y + C(x)y2 and an integrating factor Îş(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results
The bifurcation of limit cycles of two classes of cubic systems with homogeneous nonlinearities
In this paper, we study the bifurcation of limit cycles of the periodic annulus of two classes of cubic isochronous systems. By using complete elliptic integrals of the first, second kinds and the Chebyshev criterion, we show that the upper bound for the number of limit cycles which appear from the periodic annuli of the two systems are at least three under cubic perturbations. Moreover, there exists a perturbation that give rise to exactly limit cycles bifurcating from the period annulus for each