Hyperbolic Chaos of Turing Patterns

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions.Comment: 4 pages, 4 figure

Modern understanding of "geoeconomic position" and the Saint Petersburg agglomeration

This article presents a modern interpretation of the concept of "geoeconomic situation" as applied to one of the most important centers of the Baltic region - the St. Petersburg agglomeration. The coastal location of the agglomeration and close connections with the Leningrad region make it possible to consider the Saint Petersburg coastal region (Baltic Area) as a whole. The article sets out not only to verify, confirm, and explain the features of the geoeconomic position of the coastal region, but also to describe the contiguous geoeconomic space. The position of the St. Petersburg coastal region is of crucial importance for ensuring a steady growth of regional economy, the propagation of industrialization impulses, and modernization in the heart of Russian Northwestern macroregion. At the same time, the specific features of the region's geoeconomic position magnify the "inherited" ad acquired effects of focal industrialization and space polarization, which creates additional prerequisites for the inversion of the Russian economic space - "Russia of the physical space" and "the economic space of Russia". The study uses traditional methodology of economic geography (the territorial, cluster, and spatial approaches) and the geoeconomic approach developed by the authors. The article also addresses recent findings in regional economy and spatial studies. It is aimed at the development of the geoeconomic paradigm in the framework of social geography and that of spatial science. An analysis of the geoeconomic position and the developing spatial relations can be of interest for researchers of geographic clusters, agglomerations, and such cross-border forms of cooperation, as growth triangles, for example

Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

Dynamical equations are formulated and a numerical study is provided for self-oscillatory model systems based on the triple linkage hinge mechanism of Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.Comment: 30 pages, 18 figure

Microconstituents of the Modified Surface Layer of Austenitic Steel With Nanofibres of Aluminium Oxyhydroxide

In the paper the authors provide the results of experimental study of the effect caused by introduction of nanostructured fibres of aluminium oxyhydroxide into the surface layer of austenitic steel upon its microconstituents. The authors show that, due to introduction of given fibres dendrite size is reduced and equilibrium structure is formed

Assignment of Appropriate Conditions for Synthesizing Tungsten Nanopowder by Electric Explosion of Conductors

The paper provides the results of experimental research into properties of tungsten nanopowders synthesized by electric explosion of a conductor in argon at various energies, put into the conductor when exploding. The authors have studied how the conditions of synthesizing the tungsten nanopowder influence on the average size of particles

Microconstituents of the Modified Surface Layer of Austenitic Steel With Nanofibres of Aluminium Oxyhydroxide

In the paper the authors provide the results of experimental study of the effect caused by introduction of nanostructured fibres of aluminium oxyhydroxide into the surface layer of austenitic steel upon its microconstituents. The authors show that, due to introduction of given fibres dendrite size is reduced and equilibrium structure is formed

On application of solution continuation method with respect to the best exponential argument in solving stiff boundary value problems

The problematic of solving stiff boundary value problems permeates numerous scientific and engineering disciplines, demanding novel approaches to surpass the limitations of traditional numerical techniques. This research delves into the implementation of the solution continuation method with respect to the best exponential argument, to address these stiff problems characterized by rapidly evolving integral curves. The investigation was conducted by comparing the efficiency and stability of this novel method against the conventional shooting method, which has been a cornerstone in addressing such problems but struggles with the erratic growth of integral curves. The results indicate a marked elevation in computational efficiency when the problem is transformed using the exponential best argument. This method is particularly pronounced in scenarios where integral curves exhibit exponential growth speed. The main takeaway from this study is the instrumental role of the regularization parameter. Its judicious selection based on the unique attributes of the problem can dictate the efficiency of the solution. In summary, this research not only offers an innovative method to solve stiff boundary value problems but also underscores the nuances involved in method selection, potentially paving the way for further refinements and applications in diverse domains

Plykin-like attractor in non-autonomous coupled oscillators

A system of two coupled non-autonomous oscillators is considered. Dynamics of complex amplitudes is governed by differential equations with periodic piecewise continuous dependence of the coefficients on time. The Poincar\'{e} map is derived explicitly. With exclusion of the overall phase, on which the evolution of other variables does not depend, the Poincar\'{e} map is reduced to 3D mapping. It possesses an attractor of Plykin type located on an invariant sphere. Computer verification of the cone criterion confirms the hyperbolic nature of the attractor in the 3D map. Some results of numerical studies of the dynamics for the coupled oscillators are presented, including the attractor portraits, Lyapunov exponents, and the power spectral density.Comment: 11 pages, 9 figure

Thermal to Nonthermal Energy Partition at the Early Rise Phase of Solar Flares

In some flares the thermal component appears much earlier than the nonthermal component in X-ray range. Using sensitive microwave observations we revisit this finding made by Battaglia et al. (2009) based on RHESSI data analysis. We have found that nonthermal microwave emission produced by accelerated electrons with energy of at least several hundred keV, appears as early as the thermal soft X-ray emission indicative that the electron acceleration takes place at the very early flare phase. The non-detection of the hard X-rays at that early stage of the flares is, thus, an artifact of a limited RHESSI sensitivity. In all considered events, the microwave emission intensity increases at the early flare phase. We found that either thermal or nonthermal gyrosynchrotron emission can dominate the low-frequency part of the microwave spectrum below the spectral peak occurring at 3-10 GHz. In contrast, the high-frequency optically thin part of the spectrum is always formed by the nonthermal, accelerated electron component, whose power-law energy spectrum can extend up to a few MeV at this early flare stage. This means that even though the total number of accelerated electrons is small at this stage, their nonthermal spectrum is fully developed. This implies that an acceleration process of available seed particles is fully operational. While, creation of this seed population (the process commonly called `injection' of the particles from the thermal pool into acceleration) has a rather low efficiency at this stage, although, the plasma heating efficiency is high. This imbalance between the heating and acceleration (in favor of the heating) is difficult to reconcile within most of available flare energization models. Being reminiscent of the tradeoff between the Joule heating and runaway electron acceleration, it puts additional constraints on the electron injection into the acceleration process.Comment: 11 pages, 12 figures, accepted for Ap