Vol 30, No 1 (2022)

The purpose of this work is to numerically study of the generalized Rabinovich–Fabrikant model. This model is obtained using the Lagrange formalism and describing the three-mode interaction in the presence of a general cubic nonlinearity. The model demonstrates very rich dynamics due to the presence of third-order nonlinearity in the equations. Methods. The study is based on the numerical solution of the obtained analytically differential equations, and their numerical bifurcation analysis using the MаtCont program. Results. For the generalized model we present a charts of dynamic regimes in the control parameter plane, Lyapunov exponents depending on parameters, portraits of attractors and their basins. On the plane of control parameters, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. It is shown that the dynamics of the generalized model depends on the signature of the characteristic expressions presented in the equations. A comparison with the dynamics of the Rabinovich– Fabrikant model is carried out. We indicated a region in the parameter plane in which there is a complete or partial coincidence of dynamics. Conclusion. The generalized model is new and describes the interaction of three modes, in the case when the cubic nonlinearity that determines their interaction is given in a general form. In addition, since the considered model is a certain natural extension of the well-known Rabinovich–Fabrikant model, then it is universal. And it can simulate systems of various physical nature (including radio engineering), in which there is a three-mode interaction and there is a general cubic nonlinearity

Nowadays, the dynamic cutting system is represented in the form of two subsystems — tool and workpiece, interacting through a nonlinear relationship formed by the cutting process. Such a representation determines the importance of studying the dynamics of the cutting process as the main factor influencing the efficiency of machines, the trajectories of the executive elements of which are set by CNC and are provided with high accuracy. However, in order to improve the efficiency of cutting, it is necessary to align the trajectories of the executive elements are defined by CNC with the changing dynamics of cutting, which introduces deviations in the program-defined trajectories. Purpose of this article is to consider the dependence of the dynamics of the cutting process on the spatial orientation of the cutting tool elasticity and the regenerative effect, and to find out the effect of the proposed dependence on the efficiency of the cutting process. All the issues discussed in the article are analyzed using the example of external shaft turning. Methods. The study is based on the methods of mathematical modeling and experimental dynamics. In contrast to the known studies, the dependence of the turnover lag time on the oscillatory displacements in the direction of the cutting speed, as well as the influence of the positive feedback formed in this case, is taken into account. In addition, changes in the sign of the internal feedback from the direction of deformations, as well as the influence of the regenerative effect on the generated attracting sets of deformations are taken into account. Results. Dependence of the system evolution on the elements of the stiffness matrix at different spindle speeds is disclosed. The properties of the system evolution depending on the ratio of the spindle rotation frequency and the eigenfrequencies of the tool subsystem, as well as the spatial distribution of the stiffness are studied. Conclusion. The frequency and time characteristics of the system are discussed. Conclusion is made about the possibility of efficiency increasing of the cutting process based on the coordination of the CNC program with the dynamic properties of the system.

The ability to name trivial everyday objects is a key cognitive function that is tested after head injuries or brain surgeries. Although quite a lot of long-standing knowledge on this topic has accumulated over the past few decades and many theoretical models have been created, the underlying neural substrate and brain functioning are still not fully aligned. As far as we know, there have been no studies on this topic using magnetoencephalography (MEG), which allows recording electrophysiological activity with a high temporal resolution. Therefore, to study the underlying spatio-temporal brain activations during the sensory and semantic processing of object naming, we conducted MEG experiments with 15 subjects grouped into three equal-sized groups with different types of language training and skills. Using boundary element methods for modelling cortical surfaces and dynamic statistical parametric mapping to solve the inverse problem, we reconstructed the cortical source activity from the recorded MEG data. The reconstructed cortical maps showed a homogeneous brain response in all three groups at the sensory processing stage, while the responses between the three groups at the semantic processing stage were different. In addition, average time courses were constructed for key brain regions such as the lateral occipital cortex (LO), fusiform gyrus (FG), Broca’s area (BA), and Wernicke’s area (WA). The obtained results assume unimodal forms for LO and WA time series, and bimodal forms for FG and BA. The only LO curve peak and the first FG peak resided in the time interval for the sensory processing stage, whereas, the only WA peak, the second FG peak and the second BA peak resided in the semantic processing stage. The first BA peak was located at the boundary separating the two stages. In addition to segregating regions involved in sensory and semantic processing, this study confirmed the involvement of FG in object naming (for the first time using MEG) that is at risk of resection during mesial temporal lobe epilepsy interventions. However, the results from this work are preliminary due to the limited sample size, and future research with a larger cohort of subjects are needed to verify/strengthen the findings of this study.

Purpose of this work is to determine regularities of formation of spatial structures in an ensemble of chaotic systems with non-local diffusion couplings and to study how these structures depend on the wave response of the digital filter formed by the ensemble couplings structure. Methods. The study was carried out by numerical simulation of an ensemble of logistic maps, calculation of its typical oscillatory regimes and their spectral analysis. The network was considered as a digital filter with a frequency response depending on the coupling parameters. Correlation between the spatial spectra and the amplitude-frequency response of the coupling filter and the mutual coherence of oscillations when the coupling parameters change were considered. Results. The system of couplings between chaotic maps behaves like a wave filter with selective properties, allowing spatial modes with certain wavelengths to exist and suppressing others. The selection of spatial modes is based on the wave characteristic of the coupling filter, the type of which is determined by the radius and the magnitude of couplings. At strong coupling the wave characteristics for ensembles with local and non-local couplings are qualitatively different, therefore they demonstrate essencially different behavior. Discussion. Using spectral methods for dynamics analysis systems with complex network topologies seems to be a promising approach, especially for research of synchronization and multistability in ensembles of chaotic oscillators and maps. The found regularities generalize the results known for ensembles of maps with local couplings. They also can be applied to ensembles of self-sustained oscillators.