Homoclinics : Poincaré-Melnikov type results via a variational approach
Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere.
Let X be a homogeneous polynomial vector field of degree 2 on S2 having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S2, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.
Homotopical dynamics II : Hopf invariants, smoothings and the Morse complex
Hyperbolic sets with the strong limit shadowing property.
Hyperbolicity, stability, and the criterion of homoclinic points.