The article discusses emergence of chaotic attractors in the system of three ordinary differential equations arising in the theory of reaction–diffusion models. We studied the dynamics of the corresponding one- and two-dimensional maps and Lyapunov exponents of such attractors. We have shown that chaos is emerging in an unconventional pattern with chaotic regimes emerging and disappearing repeatedly. We had already studied this unconventional pattern for one-dimensional maps with a sharp apex and a quadratic minimum. We applied numerical analysis to study characteristic properties of the system, such as bistability and hyperbolicity zones, crisis of chaotic attractors.
Malinetsky G., Faller D. Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 111-125
The study's aim is to predict main trends and create scenarios for the economic development of BRICS countries (Brazil, India, China, Russia, South Africa) and the USA. Regression autonomous macro models were built, as well as a model of trade between them. The autonomous submodels use Population, Fixed Capital, Gross Domestic Product and Gross Capital Formation as key indicators. Autoregressive equations describe the dynamics of these variables. The resulting system of equations allowed us to describe the historical dynamics of demographic and macroeconomic indicators from 1990 to 2015 and to do a forecast until 2030. In the trade submodel bilateral trade flows link with gross domestic products of the economies. The relationship is described by the power dependence of the export flow on the gross domestic product of both trading partners. Unlike gravity-type models, the regression equation parameters are assumed to be constant for each pair of trading partners over the entire predicted time interval. The calculations showed that the models satisfactorily describe the dynamics of monotonically changing indicators and therefore can be used as a simple tool for forecasting the national and regional economy.
Малинецкий Г.Г., Махов С.А. Динамика макроэкономических показателей и взаимной торговли стран БРИКС и США. Математическое моделирование и численные методы, 2023, No 1, с. 112–123.