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 conductedthrough the critical manifolds concepts, generalization of criticalpoints and critical lines introducedby Gumowski and Mira [1,2]. The phase plane shared within two open regions: the ﬁrst (denoted Z0)eachpoint having norealpreimage, and the second (denoted Z2) eachpoint having two real preimages. The regions Z0, Z2 are separated by the critical manifolds, locus of points having two coincident preimages. This requires the visualization of critical manifolds in the three dimensional phasesspace.And this work also describes thepassageof invariant or attractor curves towards weakly chaotic attractors then towards hyper-chaotic attractors via the contact bifurcation through the critical manifolds, which disappear after thecontact bifurcation with the its attractionbasinboundary.