Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

Extended quasi-homogeneous polynomial system in $${\mathbb{R}^{3}}$$

“…In particular, when δ = 2 (i.e. Q is quasi-homogeneous with weight (1, 1, 1) and degree 2), Camacho [7] in 1981 and Llibre and Pessoa [8] in 2006 studied the geometry of Q T . Camacho obtained seven different topological classes of Q T which in all cases turns out to be a Morse-Smale vector field with no limit cycles.…”

“…Later on, Llibre and Pessoa determined the maximum number of invariant circles when y, Q(y) ≡ 0 and Q T has finitely many invariant circles. In the paper [9] Llibre and Pessoa also found an upper bound for the number of invariant circles, invariant great circles of Q T with y, Q(y) ≡ 0 and δ = n. For more works about homogeneous system, see [1,10,11] and [19], etc.…”

“…(ii) If δ = 3 and r(τ ) = r(τ, r 0 , y 0 ) > 0 is the isolated T-periodic solution of (9), then the multiplicity of r(τ ) is no more than 2. Moreover, assume that…”

“…If F (T ) = 0, then A 1 (T ) = 1 and r(τ ) is a hyperbolic periodic solution of (9). Otherwise if F (T ) = 0, then A 1 (T ) = 1 and r(τ ) is non-hyperbolic.…”

“…Proposition 2.1 tells us that ψ α (C) is also a closed orbit of (6b) with expression y τ, ψ α (y 0 ) = ψ α y((−1) δ−1 τ, y 0 ) . Using (9) we have the following two equations for C and ψ α (C):…”