Vol 31, No 1 (2023)

Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants

Purpose of this study is to investigate the problem of how typical (or, conversely, unique) is the phenomenon of explosive synchronization in networks of nonlinear oscillators with topologies of links such as “ring” and “small world”, and, in turn, how the partial frequencies of the interacting oscillators must correlate with each other for the phenomenon of explosive synchronization in these networks can be possible. Methods. In this paper, we use an analytical description of the synchronous behavior of networks of nonlinear elements with “ring” and “small world” link topologies. To confirm the obtained results the numerical simulation is used. Results. It is shown that in networks of nonlinear oscillators with topologies of links such as “ring” and “small world”, the phenomenon of explosive synchronization can be observed for the different distributions of partial frequencies of network oscillators. Conclusion. The paper considers an analytical description of the behavior of network oscillators with “ring” and “small world” topologies of links and shows that the phenomenon of explosive synchronization in such networks is atypical, but not unique.

The study of the problem of damping vibrations of a solid body mounted on viscoelastic supports is an urgent task. The paper considers the problem of reducing the level of vibrations on the paws of electric machines using dynamic vibration dampers. For this purpose, the paw of electric machines is represented in the form of a subamortized solid body with six degrees of freedom mounted on viscoelastic supports. The aim of the work is to develop calculation methods and algorithms for studying the oscillations of the resonant amplitudes of a solid body mounted on viscoelastic supports. Dynamic oscillation (vibration) damping method consists in attaching a system to the protected object, the reactions of which reduce the scope of vibration of the object at the points of attachment of this system. Applying the D’Alembert principle, the equations of small vibrations of a solid with dampers are derived. For practical calculations, a simplified system of equations was obtained that takes into account only three degrees of freedom. Numerical calculations were carried out on a computer to determine the amplitude-frequency characteristics of the main body. Numerical experiments were carried out using the Matlab mathematical package. Considering that a solid body is characterized by vibration, as a rule, in a continuous and wide frequency range, therefore, dynamic vibration dampers are used to protect a solid body mounted on viscoelastic supports. It was found that when the damper is set at a frequency of 50 Hz, the vibration level at the left end of the frequency interval of rotary motion of the rotor-converter, decreases to 37.5 dB, and at the right end — to 42.5 dB. At a frequency of 50 Hz, the paws do not oscillate. When setting the dampers to a frequency of 51.5 Hz, the maximum vibration level does not exceed 40 dB. The optimal setting of the dampers is within the frequency of 50.60...50.70 Hz, and a two-mass extinguisher is 10–15% more efficient than a single-mass one. Thus, the paper sets the tasks of dynamic damping of vibrations of a solid body mounted on viscoelastic supports, develops solution methods and an algorithm for determining the dynamic state of a solid body with passive vibration of the object in question. 

Purpose of this work is to study a computational model of working memory formation based on spiking neural network with plastic connections and to study the capacity of working memory depending on the time scales of synaptic facilitation and depression and the background excitation of the network. Methods. The model imitates working memory formation within synaptic theory: memorized items are stored in form of short-term potentiated connections in selective population but not in form of persistent activity. Integrate-And-Fire neuron model in excitable mode are used as network elements. Connections between excitatory neurons demonstrates the effect of short-term plasticity. Results. It is shown that the working memory capacity increases as calcium recovery time parameter grow up or the capacity increases with neurotransmitter recovery time parameter becomes lower. Working memory capacity is found to decrease to zero with decrease of the background excitation as a result of lower values of both the mean and the variance of the external noise. Conclusion. Working memory capacity was studied as a function of time scales of synaptic facilitation and depression and background excitation of the network. Estimated working memory capacity is shown to be possibly larger than classical experimental estimations of four items. But capacity strongly depends on intrinsic parameters of neural networks.