Robustness, convergence, and homoclinic classes in the dynamics of the Hénon map

We analyze the dynamics of the Hénon map 𝑇 𝑎 , 𝑏 ⁡ ( 𝑥 , 𝑦 ) = ( 𝑦 + 1 − 𝑎 ⁢ 𝑥 2 , 𝑏 ⁢ 𝑥 ) from three complementary perspectives. First, we prove the persistence of a trapping region for parameters ( 𝑎 ′ , 𝑏 ′ ) near ( 1 . 4 , 0 . 3 ) , showing that the associated attractor 𝛬 𝑎 ′ , 𝑏 ′ is well defined and varies upper semicontinuously with respect to the Hausdorff metric; under standard assumptions on the classical Hénon attractor, we deduce topological transitivity and sensitivity to initial conditions for certain parameters ( 𝑎 ′ , 𝑏 ′ ) . Second, extending results of Devaney–Nitecki, we construct a Smale horseshoe for 𝑏 = 0 . 3 and sufficiently large values of 𝑎 , and establish its hyperbolicity via invariant cone fields. Third, we prove that this horseshoe is a homoclinic class, relating its geometric structure to fundamental invariants of hyperbolic dynamics.