Artemenko V.A., Andryukhin A.I.

Complex solutions of matrix equations are constructed.The connection of modern fractal studies and the results of Poincaré on qualitative research of systems is shown. The visualizations of complex solutions of the basic equation of oscillatory dynamics of systems are presented. Visualizations were obtained using the modern package Wolfram Mathematics 11.0. The properties of the mapping eiz are studied. The results obtained in the work are of a constructive nature. Basically they are the fruits of computer experiments. It is shown that the difference between the real parts of the neighboring solutions of the equation eiz = z tends to 2 as z increases. It is also shown that the coefficient value for the imaginary part is less than zero, excluding z0. A transcendental equation is constructed which makes it easy to find fixed points of this map on a computer. Specific values of the solutions of the equation eiz = z with allowance for e 2i = 1 can be determined from K = ... 1.0, + 1, .. from the transcendental equation Ln (cos (a) / (a-2K) - (a-2K) tg (a) = 0. The values of the first solutions of this equation are presented. Also these first solutions are presented in the complex area. It is shown that the map has one stable and infinite number of unstable equilibrium positions, there are an infinite number of repelling 2-periodic cycles. The problem of determining periodic cycles of length n for the map eiz reduces to solving a system of transcendental equations (a0 = an, b0 = bn), a i + 1 = fa (ai, bi), b i + 1 = fb (ai, bi) for i = 0, n-1.Examples of periodic cycles of length 2,3,4,5 are constructed. The Julia sets for various hyperbolic images are visualized. The latter are asymptotic expansions of eiz. Keywords: complex number, limit, matrix, oscillations, Poincaré, fractal.

Send an article