This work examines the control and stabilization problems of vibrations in a hierarchical chain of oscillators with hysteresis couplings. Hysteresis coupling is formalized within the Bouc-Wen phenomenological model. The mass, stiffness, and damping properties of nonlinear oscillators are set to follow a specific scaling rule and decrease exponentially along the circuit. The model is verified using Kolmogorov's law, similar to what is done for formed turbulent flows. For this purpose, energy spectra are constructed at different amplitudes of the external excitation. The amplitude-frequency characteristics of the system under hysteresis damping are calculated using the frequency scanning method. A comparative analysis is carried out and the regulatory role of hysteresis on the dynamics of the system is noted. It is concluded that as soon as a chain of oscillators enters the chaotic response mode, the energy spectra describing the energy transfer between the elements of the chain seems like (but are not identical to) Kolmogorov’s power law.