Genericity of hyperbolic homogeneous vector fields in \(\mathbb{R}^ 3\) (Q1188241)

Consider a homogeneous polynomial vector field \(Q\) of degree \(k \geq 3\) in \(\mathbb{R}^ 3\). The standard blow up of \(0 \in \mathbb{R}^ 3\) produces the vector field \(B(Q)\) (called the blow up of \(Q\)). The field \(B(Q)\) is defined near the sphere \(S^ 2 \times \{0\}\) which is invariant for it. The author shows that any \(Q\) can be approximated (in \(C^ r\) topology) by a vector field \(Q_ 0\) such that \(B(Q_ 0)\) is Morse-Smale on \(S^ 2 \times \{0\}\).