Parameter Identification and Prediction of the Rőssler System with Complete and Incomplete Information: Two Known and One Unknown State Variables

Abstract

Parameters identification algorithms are formulated for the system of equations of Rőssler's type. The problem is solved for the cases of complete and incomplete information about functions of the system. If there is complete information about the state variables, it is possible to apply the algorithms discussed to any system of linear or nonlinear ordinary differential equations of arbitrary order in the Cauchy form that linearly depends on the unknown parameters (or groups of unknown parameters). The problems of parameter identification in the case of incomplete information about the state variables must be solved individually, depending on the possibility (or impossibility) of eliminating unknown steady states from the system of equations. Most real-world problems in fields such as chemical kinetics, mathematical ecology, predator-prey dynamics in game reserves, and the spread of infectious diseases belong to this class of problems, in which the algorithms discussed demonstrate their applicability. In the present paper the case of one unknown function is considered. Lemmas about possibilities on complete and incomplete parameter identification are formulated and proven. The algorithms of the parameter identification are formulated in the process of the constructive proofs of the lemmas. Numerical examples and graphs of solutions are considered which demonstrate efficiency and accuracy of the developed algorithms. In the proposed paper, the integration approach is used instead of the differential approach because it allows for the smoothing of discrete data, thereby reducing the estimation errors of the unknown parameters.