Computing center conditions for vector fields with constant angular speed
The authors study the planar analytic system \[ dx/dt = -y + xH(x,y), \qquad dy/dt= x+yH(x,y), \tag{1} \] where \(H\) is analytic and \(H(0,0)= 0\). Conditions are obtained to characterize the center at the origin of (1). The equivalence between those conditions and the vanishing of the coefficients of the radial component of the normal form is proved. A recursive algorithm of coefficient conditions is found for the \((0,0)\) to be a center. A few subfamilies having a center are given. The centers of quartic polynomial systems and of two other families are derivated to determine the maximum number of limit cycles which can bifurcate from a fine focus.
scientific article; zbMATH DE number 1915270