Modeling general laws of spatial–temporal evolution of society: hyperbolic growth and historical cycles

The article investigates a distributed model of global evolution of humanity. The model utilizes a onedimensional quasilinear heat equation with a volume source and a nonlinear thermal conductivity coefficient. The model is applied to describe cyclic processes that unfolded over the entire span of human evolution against the backdrop of general population growth with blowup. The model parameters are chosen so that they satisfy the following requirements: the space integral (total population size) increases hyperbolically; the evolution of the system goes through 11 stages corresponding to the main historical epochs in accepted classification and matches actual quantitative indicators.     

Numerical Analysis of the Synchronization of Coupled Chemical Oscillators

A reaction–diffusion model describing a system of coupled oscillators is constructed and investigated. The oscillators in this study are chemical oscillators that represent an oscillatory heterogeneous catalytic reaction in a granular catalyst layer. The oscillators are arranged serially in the reagent stream and are coupled through the gaseous phase. The dynamic behavior of the system is investigated as a function of the main external parameter — the partial pressure of one of the reagents in the gaseous phase. Existence regions of regular and chaotic oscillations are identified. Synchronization conditions are established for the oscillations in such a chain of coupled chemical oscillators.     

Blowup solutions in a problem for the nonlinear heat equation on a small interval

The article considers a one-dimensional quasi-linear heat equation with a volume heat source and a nonlinear thermal conductivity. The analysis is conducted for parameter values where selfsimilar solutions of the equations evolve in an LS-regime with blowup. Heat localization is observed in this case, and the combustion process in the developed stage is in the form of simple or complex structures of contracting half-width. We study the evolution dynamics of various initial distributions and their achievement of the self-similar regime, and also the dependence of the size of the localization region on the shape of the initial compactly supported distribution. The possibility of cyclic evolution of solutions against the background of overall growth with blowup is demonstrated. We particularly focus on the case when the size of the spatial region is much less than the characteristic size of the localization region, and heat flow is obstructed by the physical boundaries. In this case all initial perturbations achieve the self-similar regime, but the corresponding scenario has certain specific features. We present an example of formation of a complex spatial structure that evolves with blowup on a small interval. Translated from Prikladnaya Matematika i Informatika, No. 29, 2008, pp. 88–112.