On the bifurcations of plane quadratic maps (Q2758284)

This paper is devoted to the study of a two-dimensional family of real noninvertible maps defined by two polynomials of the second degree, depending on two parameters. According to the region of the phase plane, a point has either no real rank one preimage (region \(Z_0\)), or two such preimages (region \(Z_2\)), or four such preimages (region \(Z_4\)). So the maps considered here belong to a type called \(Z_0-Z_2-Z_4\). The paper title (related to quadratic maps) is not sufficient to characterize the maps studied, because quadratic maps may be of the (\(Z_0-Z_2\)) type. For this problem the regions \(Z_i\), \(i=0, 2,4,\) are separated by arcs belonging to the `critical curve', locus of the points of two rank-one preimages merging. NEWLINENEWLINENEWLINEA first part deals with the determination of the curve of the Neimark (frequently wrongly named Hopf) bifurcation in the parameter plane. This curve is related to a fixed point, the eigenvalues of which are complex with a modulus equal to one. Crossing through this curve changes the stability of the fixed point, and generates an invariant closed curve in the phase plane. The second part considers some qualitative modifications of the phase plane domain giving rise to bounded orbits, when one of the two parameters varies. The paper shows how this domain, when it is simply connected, can become multiply connected, or not connected. Such bifurcations result from a contact between the critical curve and the domain boundary.