Parameter Identification of Lorenz System with Incomplete Information: Case of One Unknown Function

Abstract

Inverse problem of the Lorenz system parametric identification is considered in the case of incomplete information about solutions of the system. In the present paper, it is assumed that only two solutions of the system from three are known in different combinations. The problem of the parameter identification of the system is solved by means of elimination of unknown functions from the original system. The obtained system of equations has the same order as the original one, but contains the unknown original parameters in new combinations. Sometimes, the number of new unknown parameters is higher than number of the original unknowns. In this case, the method of the constrained least squares minimization is used in the special formulation, developed by the authors. This novel formulation exploits linearity of the system with respect to the new unknown parameters, by means of which the number of nonlinear equations becomes equal to the number of the constraints between the new parameters. Two methods of the constraint minimization are considered: the classical method of Lagrange’s multipliers and a novel method of the auxiliary parameters. Numerical simulations demonstrate effectiveness of the algorithms.