The limit set of trajectory in quasi-homogeneous system in ℝ³

The projective vector field of a kind of three-dimensional quasi-homogeneous system on S2

“…In the paper [8] Llibre and Pessoa also found the upper bound for the number of invariant circles and invariant great circles of Q T with ⟨y, Q (y)⟩ ≡ 0 and δ = n. For more works concerning homogeneous systems, see [9,1,10], etc. For when Q is a general quasi-homogeneous vector field, the authors of the paper [3] generalized the results of Coleman and Sharipov. And in the paper [2] Llibre and Zhang studied the polynomial first integrals for n-dimensional quasihomogeneous systems.…”

“…Furthermore, let dτ = (r δ−1 /⟨y, y⟩)dt. System (2) is topologically equivalent to dr dτ = r · R(y) = r⟨y, Q (y)⟩, (3) dy dτ = Q T (y) = ⟨y, y⟩Q (y) − ⟨y, Q (y)⟩y. (4) System (4) is an independent system on S 2 , which is important to us for analyzing the topology of (2) (see [3]).…”

“…System (2) is topologically equivalent to dr dτ = r · R(y) = r⟨y, Q (y)⟩, (3) dy dτ = Q T (y) = ⟨y, y⟩Q (y) − ⟨y, Q (y)⟩y. (4) System (4) is an independent system on S 2 , which is important to us for analyzing the topology of (2) (see [3]). As definitions in [4], we write Γ , g, and C for trajectories, singularities, and closed orbits of (4), respectively, and W for trajectories of (2).…”

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

“…When Q is a generic quasi-homogeneous polynomial vector field, Huang and Zhao [12] generalized the results of Coleman and Sharipov. For the case that Q has weight (1, 1, α 3 ) with α 3 ≥ 2 and degree δ = 2, they investigated the geometry of Q T and classified Q T into two types [13].…”

“…For the case that Ω Γ = g and R(r, g) ≡ 0, Huang and Zhao give a homogeneous example in paper [12] such that W Γ is an oscillating curve.…”