Focal points and basin fractalization in two classes of rational maps. (English) Zbl 0960.37024

The main goal of this paper is to describe some new kinds of basin bifurcation occuring in noninvertible rational mappings having a vanishing denominator. As an example of such class of maps, the authors consider \((\widetilde{x}, \widetilde{y})= R(x,y)= (F(x,y), G(x,y))\), defined by \[ R: \begin{cases} \widetilde{x}'= \frac{\rho xy+ f(x)} {1+\rho y},\\ \widetilde{y}'= 1+ \rho y, \end{cases} \] where \(\rho\in (0,1)\) and \(f(x)\) is a real function of class \(C^1\). Such new kinds of bifurcations are due to the combined effect of contacts of basin boundaries with criticla curves and with recently defined singularities, specific to maps with vanishing denominator, called sets of focal values.


37G99 Local and nonlocal bifurcation theory for dynamical systems
58K05 Critical points of functions and mappings on manifolds