Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

Artur Karimov,Denis Butusov,Erivelton G Nepomuceno,Aleksandra Tutueva

doi:10.3390/math8020300

Artur Karimov, Denis Butusov + Show 2 more

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https://doi.org/10.3390/math8020300

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Abstract

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series.

Highlights

The problem in nonlinear dynamical system reconstruction from data series can be described as follows: For a given data series, a system must be found that produces data series that are close in a certain sense to the initial series
This paper is organized as follows: First, we describe how polynomial dynamical systems (PDSs) can be reconstructed from complete data using the extended Kera-Hasegawa method; second, we explain how these results can be applied to data series lacking some state variables; third, we carry out computational experiments showing the validity of the approach, to be more precise, we reconstruct the Lorenz system from 1-dimensional data series and a 4-dimensional memristive circuit proposed by B
We describe an algebraic approach to the system reconstruction method based on subsequent application of the approximate Buchberger–Möller method, the least-squares method and deleting minor terms method, close to the method proposed by Kera and Hasegawa, but we extend this approach to the Laurent polynomials which can have negative powers of monomials in comparison with ordinary polynomials with nonnegative powers only

Summary

Introduction

The problem in nonlinear dynamical system reconstruction from data series can be described as follows: For a given data series, a system must be found that produces data series that are close in a certain sense to the initial series. O the problem statement as follows: For a given data series {xi } and its time derivative xi a system of ODEs must be reconstructed as x = f(x), (1). Some information about f is known apriori, for instance, the possible elementary operations constituting it. These operations usually construct a group or, a ring. One such ring is a polynomial ring which includes a set of polynomials in several variables with coefficients belonging to a certain field such as a field of real or complex numbers [5,6]

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