## ISSUE 179, article 5

Kibern. vyčisl. teh., 2015, Issue 179, pp 56-69.

Gorodetskyi Victor G., PhD (Physics & Mathematics), Associate Professor of the National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Ave., 37, Kiev, 03056, Ukraine , e-mail: v.gorodetskyi@ukr.net

Osadchuk Mykola P., Assistant of the National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Ave., 37, Kiev, 03056, Ukraine, email: 13717421@ukr.net

ALGORITHM FOR RECONSTRUCTING THE DYNAMICAL SYSTEMS USING ONE OBSERVABLE VARIABLE

Introduction. We consider the problem of reconstructing the system of ordinary differential equations by using one observable variable. The data under investigation is a scalar time series of some process data. It is assumed that the dynamics of the process can be described by an original system of ordinary differential equations with polynomial right-hand sides. We replace the original system by standard system of known type in which the unknown variables are replaced by derivatives of the observable variable, and one of the variables of the standard system is the same as observable variable. We use standard systems which have the ratio of polynomials with unknown coefficients in the right-hand sides.

The purpose of this work is to simplify and improve the accuracy of G. Gouesbet algorithm for determining the coefficients of the standard system.

Methods. As well as in the prototype algorithm, the time series is differentiated to find all the variables of the standard system. Then we form the system of linear algebraic equations which are solved with respect to the unknown coefficients of the standard system. The algorithm uses such novelties: ability to assign the known values for any coefficient of the standard system, solving the over determined algebraic system by using least square method, possibility to use different methods of differentiation.

Results. Algorithm was utilized to reconstruct standard system by use of one variable of Rossler system and other systems with chaotic evolution. All the results confirmed the effectiveness of the algorithm improvements.

Conclusion. The proposed novelties allow to improve the accuracy of computing the coefficients of the standard system and simplify the algorithm.

Keywords: original system, standard system, reconstructing, least square method, observable variable.

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