The model system manifesting phenomena peculiar to complex analytic maps is offered. The system is a non-autonomous ring cavity with nonlinear elements and filters,
A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytic maps (such as the Mandelbrot set and Julia sets). The system is composed of two coupled alternately excited oscillators (or self-sustained oscillators). T
he idea is based on a turn-by-turn transfer of the excitation from one subsystem to another (S.P.~Kuznetsov, Phys.~Rev.~Lett. bf 95 rm, 2005, 144101) accompanied with appropriate nonlinear transformation of the complex amplitude of the oscillations in the course of the process. Analytic and numerical studies are performed. Special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations. The main effect is the rotation of the Mandelbrot-like set in the complex parameter plane; one more effect is the destruction of subtle small-scale fractal structure of the set due to the presence of non-analytic terms in the complex amplitude equations.
It is shown that critical phenomena associated with Siegel disk, intrinsic to 1D complex analytical maps, survives in 2D complex invertible dissipative H{e}non map. Special numerical method of estimation of the Siegel disk scaling center position (fo
r 1D maps it corresponds to extremum) for multi-dimensional invertible maps are developed.
