Computing center conditions for vector fields with constant angular speed

AbstractWe 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

Cooperative Games on Antimatroids

AMS classification: 90D12;game theory;cooperative games;antimatroids

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

AbstractWe study the analytic system of differential equations in the plane(x˙,y˙)t=∑i=0∞Fq−p+2is, where p,q∈N, p⩽q, s=(n+1)p−q>0, n∈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 prove for this system the existence of a Lyapunov function and we solve theoretically the center problem for such system. Finally, as an application of the theoretical procedure, we characterize the centers of several subfamilies

The Myerson Value for Union Stable Systems

AMS classification: 90D12