Perturbations of parallel flows on the sphere in

We consider perturbations of the parallel flows on each invariant sphere x 2 + y 2 + z 2 = h ( > 0 ) by homogeneous polynomials of degree m ( ⩾ 2 ) , and obtain the least upper bound of the number of limit cycles appearing on the sphere by using the Poincaré map and the averaging method.