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

This paper is primarily a numerical study of the fixed-point bifurcation loci - saddle-node, period-doubling and Hopf bifurcations -present in the family: z → f(C,A)(z, z̄) ≡ z + z2 + C + Az̄ where z is a complex dynamic (phase) variable, z̄ its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varied. For A = 0, the resulting two-real-parameter family is a familiar one-complex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A ≠ 0, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of R2. Saddle-node and period-doubling loci evolve from points on the main cardioid for A = 0 into closed curves for A ≠ 0. As A is varied further from O in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.