Waleed A. Mathematical modeling and computational methods for the analysis of sustainable development processes in nonlinear dynamic systems with competitive interaction

The object of the study is the sustainable development processes in the multiply non-linear dynamic systems with competitive interactions. The purpose of the study is the extraction and the study of a new class of mathematical models of interconnected nonlinear dynamical systems, which includes both a well-known model and multiply dynamic models with different types of perturbations of the right part, the characteristic features of which are non-linear competition or solidarity interaction of subsystems. The methods of systems analysis are the research methods for the classes selection and the construction of mathematical models; the methods of qualitative theory of differential equations were used for the study of the systems models dynamics; to obtain the phase portraits of analytical and numerical methods for solving linear and nonlinear systems of ordinary differential equations and the method of analysis of their sustainability were modified, in order to achieve the sustainable growth of the systems studied and to prevent their chaotic dynamics. Personal computer was used as the equipment. Theoretical and practical results of the research are the derived formulas of systems models dynamics, consisting of both competing and solidarity subsystems, allowing analytically investigate the explicit dependence of the phase portraits from the bifurcation parameters; the scope of application of the phase space method was expanded to the practically important class of non-autonomous systems with perturbation of the change rate of actors. Scientific novelty is in the fact that for the first time for the selected class of nonlinear dynamic models with the competitive interaction, the conditions of the existence of stationary sets of toroidal type and chaotic attractors were obtained and investigated, certain factors that are responsible for the emergence of quasi-chaotic regimes were determined; for the first time a special class of models of interaction dynamics with the parallel connection of pump units of pump station that differs from the well-known by their non-linear links was proposed; mathematical models of competing systems of several actors the differ from the well-known by the fact that they include two sets of (solidarity and antagonistic) actors received further development; the method for numerical analysis of the chaotic models dynamics with a close to a resonant periodic external influence was improved, which allowed to investigate their emergence conditions and to get a visual image in the extended phase space or projections. The results of the thesis are implemented in the educational process in such courses as "Simulation modelling", "Catastrophe theory" and "Synergetic methods in economics". Research results are used in the performance of the state budget themes.