A globally adaptive explicit numerical method for exploding systems of ordinary differential equations

2013 ◽  
Vol 67 ◽  
pp. 204-219 ◽  
Author(s):  
Nabil R. Nassif ◽  
Noha Makhoul-Karam ◽  
Jocelyne Erhel
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Explicit Numerical Method

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

1993 ◽  
Vol 03 (02) ◽  
pp. 333-361 ◽  
Author(s):  
RENÉ LOZI ◽  
SHIGEHIRO USHIKI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Phase Portrait ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Bifurcation Diagrams ◽  
Accurate Analysis ◽  
Chua's Circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

scholarly journals Reconstruction of the Lorenz system using the method of perspective coefficients

2018 ◽  
Author(s):  
V. G. Gorodetskiy ◽  
N. P. Osadchuk
Keyword(s):  
Time Series ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Lorenz System ◽  
Ode System ◽  
Right Hand ◽  
Observable Variable ◽  
The Lorenz System

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

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