Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.Comment: 7 pages, 5 figure

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