Parametric Qualitative Analysis of Ordinary Differential Equations: Computer Algebra Methods for Excluding Oscillations (Extended Abstract) (Invited Talk)

2010 ◽  
pp. 267-279 ◽  
Author(s):  
Andreas Weber ◽  
Thomas Sturm ◽  
Werner M. Seiler ◽  
Essam O. Abdel-Rahman
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Computer Algebra

scholarly journals Symmetries and Conservation Laws for Some Compacton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. L. Gandarias ◽  
M. S. Bruzón ◽  
M. Rosa
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Dynamical Behaviour ◽  
Point Of View ◽  
Nonlinear Dispersion ◽  
Multipliers Method

We consider some equations with compacton solutions and nonlinear dispersion from the point of view of Lie classical reductions. The reduced ordinary differential equations are suitable for qualitative analysis and their dynamical behaviour is described. We derive by using the multipliers method some nontrivial conservation laws for these equations.

A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters

1988 ◽  
Vol 416 (1851) ◽  
pp. 361-389 ◽  
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Parameter Space ◽  
Bifurcation Theory ◽  
Parameter Plane ◽  
Phase Portraits ◽  
Relevant Parameter ◽  
Bifurcation Diagrams ◽  
Independent Parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODEs) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear non-linearly) can be used to parametrize the relevant loci.

scholarly journals The symbolic problems associated with Runge-Kutta methods and their solving in Sage

2019 ◽  
Vol 27 (1) ◽  
pp. 33-41
Author(s):  
Yu Ying
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computer Algebra ◽  
Numerical Experiments ◽  
Affine Space ◽  
Computer Algebra System ◽  
Algebraic Sets ◽  
Algebraic Properties ◽  
Runge Kutta ◽  
Symplectic Schemes

Runge-Kutta schemes play a very important role in solving ordinary differential equations numerically. At first we want to present the Sage routine for calculation of Butcher matrix, we call it an rk package. We tested our Sage routine in several numerical experiments with standard and symplectic schemes and verified our result by corporation with results of the calculations made by hand.Second, in Sage there are the excellent tools for investigation of algebraic sets, based on Gröbner basis technique. As we all known, the choice of parameters in Runge- Kutta scheme is free. By the help of these tools we study the algebraic properties of the manifolds in affine space, coordinates of whose are Butcher coefficients in Runge-Kutta scheme. Results are given both for explicit Runge-Kutta scheme and implicit Runge-Kutta scheme by using our rk package. Examples are carried out to justify our results. All calculation are executed in the computer algebra system Sage.

Sign in / Sign up

Export Citation Format

Share Document