Robustness, convergence, and homoclinic classes in the dynamics of the Hénon map
Year: 2026
Type: article
Abstract: 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.
Source: Chaos Solitons & Fractals
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,970
OpenAlex ID: W7215898173