644 research outputs found
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
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