Real Perturbation of Complex Analytic Families: Points to Regions

This study provides some connections between bifurcations of onecomplex-parameter complex analytic families of maps of the complex plane C and bifurcations of more general two-real-parameter families of real analytic (or C or C∞) maps of the real plane R. We perform a numerical study of local bifurcations in the families of maps of the plane given by z 7→ F(C,α)(z, z) = z 2 + C + αz where z is a complex dynamic (phase) variable, z its complex conjugate, C is a complex parameter, and α is a real parameter. For α = 0, the resulting family is the familiar complex quadratic family. For α 6= 0, the map fails to be complex analytic, but is still analytic (quadratic) when viewed as a map of R. We treat α in this family as a perturbing parameter and ask how the two-parameter bifurcation diagrams in the C parameter plane change as the perturbing parameter α is varied. The most striking phenomenon that appears as α is varied is that bifurcation points in the C plane for the quadratic family (α = 0) evolve into fascinating bifurcation regions in the C plane for nonzero α. Such points are the cusp of the main cardioid of the Mandelbrot set and contact points between “bulbs” of the Mandelbrot set. Arnold resonance tongues are part of the evolved scenario. We also provide sufficient conditions for more general perturbations of complex analytic maps of the plane of the form: z 7→ F(C,α)(z, z) = fC(z) + αgα(z, z) to have bifurcation points for α = 0 which evolve into nontrivial bifurcation regions as α grows from zero.