On the invariant sphere theorem with emphasis in the round sphere S n $\mathbb S^n$ double struck upper S Superscript n

Abstract This article discusses vector fields in R n $\mathbb R^n$ double struck upper R Superscript n , n ≥ 3 $n \geq 3$ n greater than or equals 3 , that can be written as the sum of a radial repelling vector field and an attracting nonlinear homogeneous vector field of odd degree. The nontrivial dynamics for such vector fields is restricted to an invariant topological sphere. A necessary and sufficient condition for this invariant topological sphere to be the unit sphere centered at the origin of coordinates is given. Examples exhibiting distinct dynamics on these invariant spheres are presented.