CHAOTIC DYNAMICS OF THREE-DIMENSIONAL H ENON MAPS THAT ORIGINATE FROM A HOMOCLINIC BIFURCATION

AbstractWe study bifurcations of a three-dimensional diffeomorphism, g 0 , that has a quadratichomoclinic tangency to a saddle-focus fixed point with multipliers (λe iϕ ,λe − ,γ), where0 <λ<1 <|γ| and |λ 2 γ| = 1. We show that in a three-parameter family, g e , of dif-feomorphisms close to g 0 , there exist infinitely many open regions near e= 0 where thecorresponding normal form of the first return map to a neighborhood of a homoclinic pointis a three-dimensional H´enon-like map. This map possesses, in some parameter regions, a“wild-hyperbolic” Lorenz-type strange attractor. Thus, we show that this homoclinic bifur-cation leads to a strange attractor. We also discuss the place that these three-dimensionalH´enon maps occupy in the class of quadratic volume-preserving diffeomorphisms. 1 Introduction In thispaperwe areconcerned with thestudyofthethree-dimensional map (¯x,y,¯ z¯) = f(x,y,z) ∈R 3 defined byx¯ = y, ¯y = z, z¯ = M 1 +Bx+M 2 y−z 2 , (1)with three parameters, M 1 , M 2 , and B. This map is a natural generalization of the famous two-dimensional H´enon map [H´76]; indeed, like the latter, the map (1) is quadratic, has co nstantJacobian det(Df) ≡ Band moreover, when B= 0, reduces to the two-dimensional H´enon map.