Influence of parameters inhomogeneity on the existence of chimera states in a ring of nonlocally coupled maps

The aim of the research is to study the influence of inhomogeneity in a control parameter of all partial elements in a ring of nonlocally coupled chaotic maps on the possibility of observing chimera states in the system and to compare the changes in regions of chimera realization using different methods of introducing the inhomogeneity. Methods. In this paper, snapshots of the system dynamics are constructed for various values of the parameters, as well as spatial distributions of cross-correlation coefficient values, which enable us to determine the regime observed in the system for these parameters. To improve the accuracy of the obtained results, the numerical studies are carried out for fifty different realizations of initial conditions of the ring elements. Results. It is shown that a fixed inhomogeneous distribution of the control parameters with increasing noise intensity leads to an increase in the range of the coupling strength where chimera states are observed. With this, the boundary lying in the region of strong coupling changes more significantly as compared with the case of weak coupling strength. The opposite effect is provided when the control parameters are permanently affected by noise. In this case increasing the noise intensity leads to a decrease in the interval of existence of chimera states. Additionally, the nature of the random variable distribution (normal or uniform one) does not strongly influence the observed changes in the ring dynamics. The regions of existence of chimera states are constructed in the plane of «coupling strength – noise intensity» parameters. Conclusion. We have studied how the region of existence of chimeras changes when the coupling strength between the ring elements is varied and when different characteristics of the inhomogeneous distribution of the control parameters are used. It has been shown that in order to increase the region of observing chimera states, the control parameters of the elements must be distributed inhomogeneously over the entire ensemble. To reduce this region, a constant noise effect on the control parameters should be used.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

Study of gyrotron synchronization by an external harmonic signal based on a modified quasi-linear theory

Topic. The paper is devoted to the study of synchronization of a gyrotron by an external harmonic signal. A theoretical study of gyrotron synchronization processes by means of a computational experiment based on certain traditional models of microwave electronics does not provide a complete description of the synchronization pattern. Therefore, the goal of the paper is to develop a modified quasi-linear model based on an approximation of the electron susceptibility by rational functions. Methods. The developed model allows for bifurcation analysis of synchronization processes. On its basis, stationary states are determined and their stability analysis is carried out. The results are in good agreement with numerical simulation based on the non-stationary theory of a gyrotron with a fixed Gaussian high-frequency field structure. Results and discussion. Resonance curves and synchronization bounds are built on the plane of parameters “amplitude – frequency of external signal”. The case where the gyrotron is in the hard excitation mode is considered, since the maximum efficiency is usually achieved in the hard excitation mode. In general, the results are in qualitative agreement with the picture described earlier for a simpler quasi-linear model of a oscillator with hard excitation, in the case of a sufficiently strong phase nonlinearity.

Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another.

Common mechanisms of attractorless oscillatory regimes in radioengineering models of brain thalamocortical network

This work aims to show that long transient processes in mesascale models of thalamocortical brain network can appear in very general case, in particular for different number of elements in the ensemble (different level of detalization) and different initial phase of external driving, with these regimes surviving at small variations of number and structure of couplings. Methods. Thalamocortical brain networks are modelled using electronic circuit realized using computer SPICE eluating software. FitzHugh – Nagumo analog generator is used as a single circuit element. Results. Long quasiregular and nonregular oscillation processes with stationary amplitude were shown to occur in ensembles of 14, 28 and 56 model FitzHug – Nagumo generators. The dependency of transient process length on the external driving initial phase and particular coupling matrix structure was studied. Conclusion. The proposed electronic models of thalamocortical system were proved to reproduce the pathological regimes of brain activity in similar way despite the number of elements in the circuit, connectivity matrix and initial driving phase.

Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

Generation of double-frequency radiation in monotron with three-gap cavity

Purpose of this work is to study modes and conditions that make it possible to excite the highest type of microwave oscillations, the frequency of which is a multiple of the frequency of the main type, in a monotron with a three-band resonator. Method of the investigation is a numerical 3D modeling, used to calculate the dimensions and electrodynamic parameters of the resonator (characteristic impedance, coupling coefficient, relative electronic conductivity); operation modes of the monotron are considered, which are characterized by excitation of oscillations in the highest type oscillations. Result. In the resonator under consideration, it is possible to achieve a multiple (equal to three) ratio between the frequency of the 25th highest type of oscillations and the frequency of the π/2-type. It was shown that in such resonator simultaneous excitation of electromagnetic field on those frequencies can be made. The maximum of an output power achieved at 100.22 GHz is 15.4 W with an accelerating voltage of 7825 V and an electronic beam microperveance 0.36 µA/V3/2 . The maximal efficiency on a third harmonic is 0.83% while the total efficiency (generating electromagnetic waves of the first and the third harmonics) is up to 17%. Conclusion. It was set that the described method of generation of terahertz range radiation is promising for further investigation, as it solves problem that orthodox microwave devices meet in the millimeter wavelength range, such as small linear dimensions of the components and critical current density of the electronic beam.

Increasing the sensitivity of real-time method for diagnostic of autogenerators phase synchronization based on their non-stationary time series

Purpose of this work is to of the research – Increasing the sensitivity of a method for diagnosing phase synchronization of autogenerators based on their non-stationary time series in real time, and also a comparison of the statistical properties of the proposed modification of the method with the well-known method for diagnostics of loop synchronization, which has proven itself in the analysis of experimental data. Methods.The paper compares the probabilities of the appearance of an error of the second kind of the developed modified method for diagnostics of phase synchronization with the probabilities of occurrence of an error of the second kind of the known method at equal values of sensitivity. When comparing the methods, generated test time realizations with a priori known boundaries of the phase synchronization sections are used, which repeat the statistical properties of the experimental data. It also compares the computational complexity of the two methods. Results. A modification of the method for diagnosing phase synchronization of autonomic regulation circuits in real time is proposed. It is shown that the proposed modification provides similar values of sensitivity and probability of appearance of errors of the second kind as the previously proposed approach. The developed method has less computational complexity than the previously proposed method. The values of free parameters corresponding to different values of sensitivity and probability of appearance of errors of the second kind are obtained. Conclusion. The area of application of the developed method with modification is formulated. The low computational complexity of the proposed method, as well as the possibility of switching devices to integer computations in calculations, makes it possible to use it for wearable registrations performing calculations in real time, based on small-sized low-power processors that do not support floating-point arithmetic operations.

Influence of coupling on the dynamics of three delayed oscillators

The purpose of this study is to construct the asymptotics of the relaxation regimes of a system of differential equations with delay, which simulates three diffusion-coupled oscillators with nonlinear compactly supported delayed feedback under the assumption that the factor in front of the feedback function is large enough. Also, the purpose is to study the influence of the coupling between the oscillators on the nonlocal dynamics of the model. Methods. We construct the asymptotics of solutions of the considered model with initial conditions from a special set. From the asymptotics of the solutions, we obtain an operator of the translation along the trajectories that transforms the set of initial functions into a set of the same type. The main part of this operator is described by a finite-dimensional mapping. The study of its dynamics makes it possible to refine the asymptotics of the solutions of the original model and draw conclusions about its dynamics. Results. It follows from the form of the constructed mapping that for positive coupling parameters of the original model, starting from a certain moment of time, all three generators have the same main part of the asymptotics — the generators are “synchronized”. At negative values of the coupling parameter, both inhomogeneous relaxation cycles and irregular regimes are possible. The connection of these modes with the modes of the constructed finite-dimensional mapping is described. Conclusion. From the results of the work it follows that the dynamics of the model under consideration is fundamentally influenced by the value of the coupling parameter between the generators.