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COMBINED NUMERICAL AND ANALYTICAL METHOD FOR SIMPLIFICATION OF THE MATHEMATICAL MODEL STRUCTURE
03.03.2023 19:21
[1. Information systems and technologies]
Author: Viktor Gorodetskyi, assoc. prof., PhD, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”; Mykola Osadchuk, assistant, PhD, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
ORCID: 0000-0003-4642-3060 Viktor Gorodetskyi
ORCID: 0000-0002-3409-9315 Mykola Osadchuk
In paper [1], a combined numerical and analytical method is proposed, which allows to simplify the model, which is obtained on the basis of a single observed variable of the object under study, and which may have redundancy. A system of ordinary differential equations with polynomial right-hand sides is considered as a model for investigation. A model is considered as redundant if the number of terms in the right-hand sides of its equations is more than the minimum necessary for the model to generate a time series of the observed variable with a given precision.
Suppose that some system of ordinary differential equations (ODE) with polynomial right-hand sides was obtained by identifying some object by a single observed variable x1(t) and has K1 terms in the right-hand sides of the equations, some of which may be redundant. We will call such an ODE system a unoptimized original system (UOS). Suppose that there also exists an ODE system, which is a particular case of UOS and contains K2 coefficients, and K2<K1. We will call such a system a minimized original system (MOS). Instead of direct simplification of the UOS, we will use the auxiliary type of systems proposed in [2]. We will call the ODE system a differential model (DM) for the UOS (or MOS) by the variable x1(t), if it has an observed variable y1(t) º x1(t) and a polynomial function or ratio of polynomials in only one of the equations, and the rest of the equations express derivatives of different orders of y1(t):
y1=y2, y2=y3, ..., yn-1=yn, yn= PN(y1,..., yn/PD(y1,..., yn) (1)
where PN, PD are polynomials. At the same time, each coefficient of DM can be analytically expressed in terms of coefficients of UOS (or MOS) [2, 3]. Also, DM coefficients can be determined not only analytically, but also numerically [2] from the time series. According to [4, 5], the ODE systems, having the same observable variables, also have the same DM. If we allow an approximate equality y1(t) » x1(t) , then it is possible to use a simpler DM than the DM corresponding to the UOS. At the same time, a simpler DM can correspond to a MOS, which will be simpler than a UOS. Therefore, in order to simplify the UOS, first, for the time series of its observed variable x1(t), the identification of the DM is carried out by a numerical method, and then the simplification of the DM and the analytical transition from the simplified DM to the MOS are performed.
The proposed approach was applied to system (39) from [6], which was obtained as a result of identification by one observed variable of the Lorenz system [7] using the "Ansatz library" method. This UOS had three ODEs and contained 21 nonzero coefficients. By means of a numerical method [2], a DM was obtained that reproduced the observed UOS variable with a relative error of 0.29%.
On the basis of the significance value [8], a part of the coefficients was excluded from the DM in several stages and the DM was re-identified. In this way, a simplified DM was obtained which reproduced the observed UOS variable a relative error of 2.45% and contained 12 coefficients. Next, an analytical transition was made from the simplified DM to the MOS, which contained 9 coefficients.
To check the correctness of the simplification, UOS and MOS were integrated by the 4th order Runge-Kutta method at a time interval of 20 s and a comparison of their observed variables was performed. It turned out that after the first 3.7 s, the graphs for UOS and MOS become completely different due to the chaotic nature of the oscillations and the impossibility of analytically determining the corresponding initial conditions for SOS. After adjusting the initial conditions for the unobserved MOS variables by means of a numerical method, the graphs for UOS and MOS began to coincide within 20 seconds with a relative error of 3.32%.
The proposed method allows to choose the most compact structure of the ODE system, containing the smallest number of terms. At the same time, the error of reproduction of the observed variable remains within the specified limits even for systems with deterministic chaos, despite their high sensitivity to initial conditions. In the considered example, it was possible to reduce the number of coefficients in the right-hand sides of the equations from 21 to 9. The method is advisable to use in the case when the researcher has a model that describes some physical process, and which may have redundant terms in the equations.
References
1. В.Г. Городецький, М.П. Осадчук, Розв'язання проблеми надлишковості математичних моделей деяких нелінійних коливальних систем // Системні дослідження та інформаційні технології. - 2021.- N 3. - С. 135-148. - DOI:10.20535/SRIT.2308-8893.2021.3.11
2. Gouesbet G. Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series // Phys. Rev. A. – 1991. – 43. – P. 5321-5331.
3. Gouesbet G. Reconstruction of standard and inverse vector fields equivalent to the Rössler system // Phys. Rev. A. – 1991. – 44. – P. 6264-6280.
4. Lainscsek C. A class of Lorenz-like systems // Chaos. – 2012. – 22. – 013126.
5. Gorodetskyi V., Osadchuk M. Analytic reconstruction of some dynamical systems // Phys. Lett. A. – 2013. – 377. – P. 703–713.
6. Lainscsek C., Letellier C., Schürrer F. Ansatz library for global modeling with a structure selection // Phys. Rev. E. – 2001. – 64. – 016206. – P. 1-15.
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