Chaotic dynamics of a three-dimensional endomorphism

The present work describes the phases plane bifurcations of some attractors given by a noninvertible three-dimensional map. This study is conducted through the critical manifolds concepts, generalization of critical points and critical lines introduced by Gumowski and Mira [1, 2]. The phase plane shared within two open regions: the first (denoted $Z_{0}$) each point having no real preimage, and the second (denoted  $Z_{2}$) each point having two real preimages. The regions $Z_{0}$, $Z_{2}$ are separated by the critical manifolds, locus of points having two coincident preimages. This requires the visualization of critical manifolds in the three dimensional phases space. And this work also describes the passage of invariant or attractor curves towards weakly chaotic attractors then towards hyper-chaotic attractors via the contact bifurcation through the critical manifolds, which disappear after the contact bifurcation with the its attraction basin boundary.