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: email@example.com
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.
1 Cremers J., Hubler A. Construction of differential equations from experimental data. Naturforsch, 1987, vol. 42a, pp. 797–802.
2 Breeden J.L., Hubler A. Reconstructing equations of motion from experimental data with unobserved variables. Phys. Rev. A, 1990, vol. 42, pp. 5817–5826. https://doi.org/10.1103/PhysRevA.42.5817
3 Gouesbet G. Reconstruction of standard and inverse vector fields equivalent to the Rossler system. Phys. Rev. A, 1991, vol. 44, pp. 6264–6280. https://doi.org/10.1103/PhysRevA.44.6264
4 Takens F. Detecting strange attractors in turbulence. In: D.A. Rand, L.S. Young (Eds.), Dynamical System and Turbulence, Lecture Notes in Mathematics. Springer, New York. 1981, no. 898, pp. 366–381. https://doi.org/10.1007/BFb0091924
5 Gouesbet G. Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series. Phys. Rev. A, 1991, vol. 43, pp. 5321–5331. https://doi.org/10.1103/PhysRevA.43.5321
6 Gouesbet G., Letellier C. Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets. Phys. Rev. E, 1994, vol. 49, no. 6, pp. 4955–4972. https://doi.org/10.1103/PhysRevE.49.4955
7 Maquet J., Letellier C., Aguirre L. A. Scalar modelling and analysis of a 3D biochemical reaction model. Journal of Theoretical Biology, 2004, vol. 228, pp. 421–430. https://doi.org/10.1016/j.jtbi.2004.02.004
8 Gorodetskyi V., Osadchuk M. Reconstruction of chaotic systems of a certain class. International Journal of Dynamics and Control. Available at: DOI 10.1007/s40435-014-0100-y.
9 Le Sceller L., Letellier C., Gouesbet G. Structure selection for Global vector field reconstruction by using the identification of fixed points. Phys. Rev. E, 1999, vol. 60, no. 2, pp. 1600–1606. https://doi.org/10.1103/PhysRevE.60.1600
10 Strang G. Linear Algebra and Its Applications. Thomson Brooks/Cole, 2006, 487 p.
11 Rossler O.E. An equation for continuous chaos. Phys. Lett. A, 1976, vol. 57, pp. 397–398. https://doi.org/10.1016/0375-9601(76)90101-8
12 Forsythe G. E., Moler C.B. Computer solution of linear algebraic systems. N. J.: Prentice-Hall, Inc., Englewood Cliffs, 1967, 148 p.