The features of the analytical numerical method of analysis and parametrical synthesis of the nonlinear nonautonomous determined system with lumped parameters of the dynamic system models are described below. The consistent procedure of model generation, specialized calculation algorithms together with the existing software, the method makes up a unique problem-adapted modeling complex.

The progress in the information technologies field as well as in the theory of nonlinear phenomena, system analysis and applied mathematics has increased the role of mathematical modeling in engineering development and scientific investigations, thus turning this universal methodology into the basis of mathematization of the scientific-and-engineering progress. Judging by the means used and the results received mathematical modeling proved to be a factor which unites traditional research methods and new methodology; thus enabling you to receive the necessary information about the character and specific features of the processes in the dynamic systems quickly and easily, to do the right prognosis and to make up recommendations which will be able to guarantee reaching your purpose.

One of the main components of the complex dynamic system modeling is the generated model calculation method, its possibilities and unique features influencing to a great extent the validity level of the received results as well as the complicity and the volume of the calculations made. The essential nonlinearity and nonstationarity of the complex dynamic systems models, their objective stiffness and poor conditionality together with ambiguity and instability of the coordinates’ behavior cause a number of special and obligatory requirements to the analysis and synthesis method used for such models. Taking into consideration the last achievements in the development of the qualitative nonlinear phenomena theory and the basic directions of improving the calculus mathematics methods, the single-step variable order analysis and parametrical synthesis method of nonlinear nonautonomus dynamic system models, named as analytical numerical method, has been developed. The calculus algorithm of the method imposes only two restrictions on the modeling system properties description, demanding determinated and lumped parameters of the generated model, including accountable external influences. The computational model of the analytical part of the method is based on the generalized functions set, generalized Laplace transformation and functional power series. The numerical part of the method is realized according to the principle of analytic continuation of the regular component of the required solution of the model dynamics equation. The calculation step choice procedure provides the solution of all problems concerning the existence and uniqueness of the required solution analysis, the opportunity to majorize the absolute local and full calculation inaccuracy of the approximate solution, the coordination of the step length and the change rate of the regular component of the required solution, and also the observance of the numerical stability conditions. The method order is the function of the interaction result of the model dynamic properties of their own and the specified level of the limited absolute local inaccuracy of its analytical numerical calculation. If necessary, the method order can be arbitrary high, constantly providing the coordination and convergence of the method.

The main difference and the advantage of the developed method is the possibility to receive not only the approximate solutions of the analyzed model dynamics equation in the time optimum grid, as well as the solutions with the specified level of the limited absolute local and full inaccuracy of areas calculation, containing unknown exact values of the desired solutions of this equation. Taking into account the conversion of cause and effect relationship the method can be effectively used for the solution of the parametrical synthesis problem within the framework of the generated system model.

Having in mind to extend the possibilities of the method on the basis of its analytical model the problem-oriented research algorithms of poorly conditioned models, models with non-distinguished linear part and with discontinuous differentiated coordinates; rigid model calculation and the models with complex functional nonlinearity; analysis of limited coordinates states of the selected class of nonautonomous non-stationary models and spectral structure of the distinguished linear part of the model dynamics equations have been generated.

To sum it all up, it is possible to say that the analytical numerical method which makes up the basis of the complex modeling approach of complex dynamic system modeling has the necessary analytical models and calculus algorithms, so that its functional features enable you to carry out the all-round and full research analysis of the properties and peculiarities of the modeling object of its own.

All the procedures and transformations accompanying the dynamic systems modeling process including the generation of their models of a various complexity degree, directed enumeration, the addition and the analytical numerical calculation of these models under the condition of optimization of the calculations volume on the interval unit of the Taylor series convergence for the regular components of the desired solutions are well formalized and correctly coordinated, thus representing together with the developed software the uniform problem-adapted modeling complex.

  
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