Reconstruction of the Lorenz system using the method of perspective coefficients

Reconstruction of the Lorenz ordinary differential equations system is performed by using perspective coefficients method. Four systems that have structures different from Lorenz system and can reproduce time series of one variable of Lorenz system were found. In many areas of science, the problem of identifying a system of ordinary differential equations (ODE) from a time series of one observable variable is relevant. If the right-hand sides of an ODE system are polynomials, then solving such a problem only by numerical methods allows to obtain a model containing, in most cases, redundant terms and not reflecting the physics of the process. The preliminary choice of the structure of the system allows to improve the precision of the reconstruction. Since this study considers only the single time series of the observable variable, and there are no additional requirements for candidate systems, we will look only for systems of ODE's that have the least number of terms in the equations. We will look for candidate systems among particular cases of the system with quadratic polynomial right-hand sides. To solve this problem, we will use a combination of analytical and numerical methods proposed in [12, 11]. We call the original system (OS) the ODE system, which precisely describes the dynamics of the process under study. We also use another type of ODE system-standard system (SS), which has the polynomial or rational function only in one equation. The number of OS variables is equal to the number of SS variables. The observable variable of the SS coincides with the observable variable of the OS. The SS must correspond to the OS. Namely, all the SS coefficients can be analytically expressed in terms of the OS coefficients. In addition, there is a numerical method [12], which allows to determine the SS coefficients from a time series. To find only the simplest OS, one can use the perspective coefficients method [10], which means the following. Initially, the SS is reconstructed from a time series using a numerical method. Then, using analytical relations and the structure of the SS, we determine which OS coefficients are strictly zero and strictly non-zero and form the initial system (IS), which includes only strictly non-zero coefficients. After that, the IS is supplemented with OS coefficients until the corresponding SS coincides with the SS obtained by a numerical method. The result will be one or more OS’s. Using this approach, we have found 4 OS structures with 7 coefficients that differ from the Lorenz system [17], but are able to reproduce exactly the time series of X variable of the Lorenz system. Numerical values of the part of the coefficients and relations connecting the rest of the coefficients were found for each OS