Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation

We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(\lambda e^{i \varphi}, \lambda e^{-i \varphi}, \gamma)$, where $0< \lambda < 1 <|\gamma|$ and $|\lambda^2 \gamma|=1$. We show that in a three-parameter family, $g_{\varepsilon}$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\varepsilon = 0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional Hénon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional Hénon maps occupy in the class of three-dimensional quadratic maps with constant Jacobian.