Complex periodic orbits, renormalization, and scaling for quasiperiodic golden-mean transition to chaos

At the critical point of the golden-mean quasiperiodic transition to chaos we show the presence of an infinite sequence of unstable orbits in complex domain with periods given by the Fibonacci numbers. The Floquet eigenvalues (multipliers) are found to converge fast to a universal complex constant. We explain this result on the basis of the renormalization group approach and suggest using it for accurate estimates of the location of the golden-mean critical points in parameter space for a class of nonlinear dissipative systems defined analytically. As an example, we obtain data for the golden-mean critical point in the two-dimensional dissipative invertible map of Zaslavsky. We give a set of graphical illustrations for the scaling properties and emphasize that demonstration of self-similarity on two-dimensional diagrams of Arnold tongues requires the use of a properly chosen curvilinear coordinate system. We discuss a procedure of construction of the appropriate local coordinate system in the parameter plane and present the corresponding data for the circle map and Zaslavsky map.