M. Reyes

Computing center conditions for vector fields with constant angular speed

We investigate the planar analytic systems which have a center-focus equilibrium at the origin and whose angular speed is constant. The conditions for the origin to be a center (in fact, an isochronous center) are obtained. Concretely, we find conditions for the existence of a Cw-commutator of the field. We cite several subfamilies of centers and obtain the centers of the cuartic polynomial systems and of the families (-y + x(H1 + Hm), x + y(H1 + Hm)t and (-y + x(H2 + H2n), x + y(H2 + H2n))t, with Hi homogeneous polynomial in x,y of degree i. In these cases, the maximum number of limit cycles which can bifurcate from a fine focus is determined.

Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations

We study the analytic system of differential equations in the plane which can be written, in a suitable coordinates system, as([emailxa0protected]?,[emailxa0protected]?)^[emailxa0protected]?i=0~Fq-p+2is,where p,[emailxa0protected]?N,p=0,[emailxa0protected]?N and Fi=(Pi,Qi)^T are quasi-homogeneous vector fields of type t=(p,q) and degree i, with Fq-p=(y,0)^T and Qq-p+2s(1,0)<0. The origin of this system is a nilpotent and monodromic isolated singular point. We show the Taylor expansion of the return map near the origin for this system, which allow us to generate small amplitude limit cycles bifurcating from the critical point. Also, as an application of the theoretical procedure, we characterize the centers and we generate limit cycles of small amplitude from the origin of several families. Finally, we give a new family integrable analytically which includes the centers of the systems studied.

Centers with degenerate infinity and their commutators

Abstract We study polynomial systems with degeneracy at infinity and a center-focus equilibrium at the origin. We give some general properties related to the existence of polynomial commutators and use these properties in order to characterize uniformly isochronous polynomial centers with polynomial commutator and, also, we show that the commutator of the centers of the analytic systems whose angular speed is constant can be chosen of radial form. Finally, we characterize the systems (−y+Ps+∑j=kn−1xHj,x+Qs+∑j=kn−1yHj)t with polynomial commutator, with Pj,Qj,Hj and Kj homogeneous polynomials.

A new algorithm for determining the monodromy of a planar differential system

Abstract We give a new algorithmic criterium that determines wether an isolated degenerate singular point of a system of differential equations on the plane is monodromic. This criterium involves the conservative and dissipative parts associated to the edges and vertices of the Newton diagram of the vector field.

Non-formally integrable centers admitting an algebraic inverse integrating factor

We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system begin{document}

Algebraic Inverse Integrating Factors for a Class of Generalized Nilpotent Systems

-y^3partial_x+x^3partial_y

Monodromy of a class of analytic generalized nilpotent systems through their Newton diagram

end{document} having an algebraic inverse integrating factor.

The center problem for a family of systems of differential equations having a nilpotent singular point

Usually, the study of differential systems with linear part null is done using quasi-homogeneous expansions of vector fields. Here, we use this technique for analyzing the existence of an inverse integrating factor for generalized nilpotent systems, in general non-integrable, whose lowest-degree quasi-homogeneous term is the Hamiltonian system (y^{2}partial _{x} + x^{3}partial y).

Integrability of two dimensional quasi-homogeneous polynomial differential systems

Newton diagram of a planar vector field allows to determine whether a singular point of an analytic system is a monodromic singular point. We solve the monodromy problem for the nilpotent systems and we apply our method to a wide family of systems with a degenerate singular point, so-called generalized nilpotent cubic systems.