Modeling of Nonlinear Dynamic Systems on the Basis of the System Sensitivity Model to Its Initial Conditions

Abstract

A typical approach for building and analyzing an object model is presented. It is determined that the tasks of analysis of nonlinear systems consist of: calculation of transients and established processes; determination of static and dynamic stability of the found processes; calculation of the sensitivity of the initial characteristics of the system to changes in its internal and external parameters. It is established that the efficiency of the analysis as a whole is determined not only by the efficiency of the algorithms of each of the stages of calculation, but also by the consistency of the mathematical apparatus that underlies them. It is determined that the calculation of transients is reduced to a problem with initial conditions in which the values of dependent variables are set for the same value of the independent variable, namely time. It is determined that nonlinear dynamic systems whose models are built on the qualitative theory of general differential equations are the main tool for solving many practical problems. It is established that this is explained by the following factors: the presence of a well-developed analytical apparatus and numerous methods of solving general differential equations; transparency and naturalness of general differential equations as a mathematical model to describe the process of transition of real objects from one state to another for external and internal causes; The availability of public qualitative methods of studying decisions of general differential equations, in particular methods of evaluation of stability, analysis of behavior within special points and their asymptotic behavior. The circumstances that lead to the fact that the systems described by conventional differential equations are a methodically very convenient material to create general algorithms for the study of dynamic systems. A mathematical model of sensitivity to the initial conditions is constructed on the basis of heterogeneous differential equations of the first variation, which opens up opportunities for solving the basic problems of analysis, which are: calculation of transitional processes and processes that have been established; Determination of static stability and calculation of parametric sensitivity, on the basis of a single algorithm for solving a two-point T-periodic marginal problem for conventional nonlinear differential equations.