On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert-Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.Comment: 16 pages, no figure

On Symbolic Solutions of Algebraic Partial Differential Equations

The final version of this paper appears in Grasegger G., Lastra A., Sendra J.R. and\ud Winkler F. (2014). On symbolic solutions of algebraic partial differential equations, Proc.\ud CASC 2014 SpringerVerlag LNCS 8660 pp. 111-120. DOI 10.1007/978-3-319-10515-4_9\ud and it is available at at Springer via http://DOI 10.1007/978-3-319-10515-4_9In this paper we present a general procedure for solving rst-order autonomous\ud algebraic partial di erential equations in two independent variables.\ud The method uses proper rational parametrizations of algebraic surfaces\ud and generalizes a similar procedure for rst-order autonomous ordinary\ud di erential equations. We will demonstrate in examples that, depending on\ud certain steps in the procedure, rational, radical or even non-algebraic solutions\ud can be found. Solutions computed by the procedure will depend on\ud two arbitrary independent constants

Birational transformations preserving rational solutions of algebraic ordinary differential equations

We characterize the set of all rational transformations with the property of pre- serving the existence of rational solutions of algebraic ordinary di erential equations (AODEs). This set is a group under composition and, by its action, partitions the set of AODEs into equivalence classes for which the existence of rational solutions is an invariant property. Moreover, we describe how the rational solutions, if any, of two different AODEs in the same class are related.Ministerio de Economía y CompetitividadVietnam Institute for Advanced Study in Mathematics (VIASM)Austrian Science Fund (FWF)Research Group ASYNAC

Transforming ODEs and PDEs from radical coefficients to rational coefficients

We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables so that solutions correspond one-to-one. Our method also applies to systems of linear ODEs. It is based on previous work on reparametrization of radical algebraic varieties.Agencia Estatal de InvestigaciónUniversidad de AlcaláJunta de Extremadur