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## Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena

A problem of synchronization of quasiperiodic oscillations is discussed in application to an example of coupled systems with autonomous quasiperiodic dynamics. Charts of Lyapunov exponents are presented that reveal characteristic domains on the parameter plane such as oscillator death, complete synchronization, phase synchronization of quasiperiodic oscillations, broadband synchronization, broadband quasiperiodicity. Features of each kind of dynamical behavior are discussed. Analysis of corresponding bifurcations is presented, including quasiperiodic Hopf bifurcations, saddle–node bifurcations of invariant tori of different dimensions, and bifurcations of torus doublings. Both the case of dominance of quasiperiodic oscillations in one of the generators and the case of pronounced periodic resonances embedded in the region of quasiperiodicity are considered.

In our earlier studies, we found the effect of non-conventional synchronization, which is a specific type of nonlinear stable beating in the system of two weakly coupled autogenerators with hard excitation given by generalized van der Pol-Duffing characteristics. The corresponding synchronized dynamics are due to a new type of attractor in a reduced phase space of the system. In the present work, we show that, as the strength of nonlinear stiffness and dissipation are changing, the phase portrait undergoes a complicated evolution leading to a quite unexpected appearance of difficult to detect “repellers” separating a stable limit cycle and equilibrium points in the phase plane. In terms of the original coordinates, the limit cycle associates with nonlinear beatings while the stationary points correspond to the stationary synchronous dynamics similar to the so-called nonlinear local modes.

**Results:*** The first M-class flare occurred on September 4, 2017 at 05:36-06:05 UT. We found that 100-minute oscillations were observed on September 4 during a few hours before M1.2 flare. At the same time, there were no noticeable oscillations on September 3. The observed effect is similar to the previously detected effect for 3-minute and 10-minute oscillations, namely, before radio burst there was increase of the power of oscillations. The effect can be interpreted as a relationship between MHD waves propagating along the magnetic flux tube of sunspot and beginning of the flares.*

**Methods:*** We used the Nobeyama Radioheliograph (NoRH) daily observations. The radio maps of the whole solar disk were synthesized in non-standard mode with a cadence of ten seconds and ten seconds averaging. We computed the time series of maximum brightness temperature and total flux over selected field-of-view (FOV) and used spectral wavelet analysis of the time series.*

**Context***: We continue research the oscillation parameters in solar active regions (ARs) in connection with their flare activity.*

**Aims:*** The aim of this paper is to study oscillations of microwave emission of AR NOAA 12673 before first M-class flare in September 2017.*

Using video on the Internet has become a common practice, but the television-like ‘passive viewer’ approach misses the benefits of the interactive nature of the Internet. The technological limitations of television can be overridden by the Internet. Having multiple sources of input does not mean they should be merged into one editor-controlled flat output. Treating streams as objects, it is possible to make viewers editors for their screens whenever they want, or let them watch a pre-edited version. Active streams are distributed to viewers to gain control over the scene layout. Recorded scenes can be remastered whenever needed and represented in different views simultaneously. For lectures and conference recordings, inline slide browsing is also possible. This approach was successfully tested in the Viditory.net project for the broadcasting and recording of conferences with multi-camera shots and remote speakers. Despite the Adobe Flash platform becoming obsolete, it is possible to implement similar capabilities on modern platforms and by using modern technologies.

OBJECTIVE:

We present a novel method for the extraction of neuronal components showing cross-frequency phase synchronization.

METHODS:

In general the method can be applied for the detection of phase interactions between components with frequencies f1 and f2, where f2 ≈ rf1 and r is some integer. We refer to the method as cross-frequency decomposition (CFD), which consists of the following steps: (a) extraction of f1-oscillations with the spatio-spectral decomposition algorithm (SSD); (b) frequency modification of the f1-oscillations obtained with SSD; and (c) finding f2-oscillations synchronous with f1-oscillations using least-squares estimation.

RESULTS:

Our simulations showed that CFD was capable of recovering interacting components even when the signal-to-noise ratio was as low as 0.01. An application of CFD to the real EEG data demonstrated that cross-frequency phase synchronization between alpha and beta oscillations can originate from the same or remote neuronal populations.

CONCLUSIONS:

CFD allows a compact representation of the sets of interacting components. The application of CFD to EEG data allows differentiating cross-frequency synchronization arising due to genuine neurophysiological interactions from interactions occurring due to quasi-sinusoidal waveform of neuronal oscillations.

SIGNIFICANCE:

CFD is a method capable of extracting cross-frequency coupled neuronal oscillations even in the presence of strong noise.

Copyright © 2011 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.

We consider a minimal action of a finitely generated semigroup by homeomorphisms of the circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov’s theorem and its corollaries.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.