Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

Abstract

We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincare) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map \begin{document}$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$\end{document} which, as known [ 14 ], exhibits discrete Lorenz attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of systems possessing discrete Lorenz attractors near the original diffeomorphism.