The nonlinear dynamics in generalized activator-inhibitor systems with space fractional derivatives is studied. As an example, the Brusselator model and the reaction-diffusion model with cubic nonlinearity, in which the classical spatial differential operators are replaced by their fractional analogues, are considered. The fractional operator reflects the nonlocal behavior of superdiffusion. The spatially homogeneous, time independent solution has been found for each system. We have also studied its linear stability and determined instability conditions of both Hopf and Turing. It was established that the anomalous diffusion (superdiffusion) leads to the qualitative change of nonlinear dynamics in mentioned systems.

Досліджено узагальнену модель типу активатор - інгібітор із кубічною нелінійністю, в якій класичний оператор Лапласа замінено дробовим аналогом. Дробовий оператор відображує нелокальну поведінку супердифузії. Знайдено просторово-однорідний стаціонарний розв'язок і вивчено його лінійну стійкість. Проведено слабконелінійний аналіз та отримано систему амплітудних рівнянь. Отримані рівняння надають можливість аналізувати типи структур, які виникають у розглядуваній реакційно-дифузійній системі.

The object of this study is underground gas storage facilities (UGSF) as part of the gas transmission system (GTS), in the process of joint operation of which a significant synergistic effect is achieved. The problem under consideration is to ensure the joint effective operation of the integrated UGSF and GTS complex as a single thermal-hydraulic one. A method of daily calculation of the maximum productivity of underground gas storage facilities has been devised. The optimization potential of UGSF operating modes has been studied. It is shown to range within 11 - 20 %. The problems of planning the operation of UGSF have been stated and solved both under gas pumping modes and gas withdrawal modes. An algorithm for planning gas withdrawal modes at intervals of peak-free UGSF operation been developed. The achieved computational complexity of problem-solving algorithms is in the range of 2 - 10 seconds. The problem of combining simultaneous operation of UGSF under an optimal mode for fuel gas and ensuring the necessary peak operation of UGSF at projected time intervals has also been considered. The joint UGSF performance was calculated at the projected time intervals according to the established criteria. At the same time, thermal-hydraulic coordination of UGSF operating modes with the operation of the GTS main gas pipeline system with which they are integrated was ensured. The problem was solved as a result of the implementation of a universal approach to the construction of functional models of complex systems - a single information support, the representation of the structure of the system in terms of graphs, the statement of proper mathematical problems, the development of methods for guaranteed convergence of systems with different mathematical representations of equations, the development of computational algorithms for combinatorial optimization of minimum complexity processes with discrete and irregular influences on their behavior.