Osadchuk M. Determining parameters of mathematical models of the dynamical systems from scalar time series

The thesis considers parameters estimation of the ordinary differential equations systems with polynomial right-hand sides by use of scalar time series. The polygonal lines method is proposed for determining embedding dimension. The method is based on phase trajectory approximation by polygonal line and further search of the polygonal line self-intersections. Method was used for embedding dimension determining of the low-dimensional and high-dimensional theoretical systems with noise and without noise, and for real time series. In all cases polygonal lines method results were not less than real value of the embedding dimension. It was shown, that polygonal lines method results depend on time delay, but the least obtained value coincide with real embedding dimension. Using of the analytical relations between coefficients of the original and standard systems for determining a complete family of candidate systems is considered. Relations for third-order differential equations systems class with three nonlinear terms in the right-hand sides of equations were obtained. Example of the analytical search of all possible original system structures, that can reproduce time series under investigation, is demonstrated. The method for system choosing from the candidate family, based on combination of analytical investigations and possibility to interact with system under investigation, is proposed. The perspective coefficients method for the search of the unknown system structure with minimal number of nonzero coefficients in the right-hand sides of equations, in case of the absence of additional information and possibility to interact with system, is proposed. Possibility to determine exactly the values of the part of the model coefficients from the scalar time series is demonstrated, and relations that connect the remaining coefficients may be obtained.