Real continuation from the complex quadratic family: Fixed-point bifurcation sets

This paper is primarily a numerical study of the fixed-point bifurcation lo ci - saddle-node, period-doubling and Hopf bifurcations - present in the fa mily: z --> f((C,A)) (z, (z) over bar) = z + z(2) + C + A<(zover bar> where z is a complex dynamic (phase) variable, (z) over bar its complex con jugate, and C and A are complex parameters. We treat the parameter C as a p rimary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varie d. For A = 0, the resulting two-real-parameter family is a familiar one-com plex-parameter quadratic family, and the local fixed-point bifurcation locu s is the main cardioid of the Mandlebrot set. For A not equal 0, the result ing two-real-parameter families are not complex analytic, but are still ana lytic (quadratic) when viewed as a map of R-2. Saddle-node and period-doubl ing loci evolve from points on the main cardioid for A = 0 into closed curv es for A not equal 0. As A is varied further from 0 in the complex plane, t he three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some param eters designated as primary and the rest as auxiliary. The auxiliary parame ter space is then divided into equivalence classes with respect to a specif ied set of bifurcation loci. This equivalence is defined by the existence o f a diffeomorphism of corresponding primary parameter spaces which preserve s the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurca tion loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-on e bifurcations one should expect in general studies of families of two-para meter families of maps of the plane. Comments on numerical continuation tec hniques are provided as well.