Determining Liouvillian first integrals for dynamical systems in the plane

Abstract

Here we present/implement a semi-algorithm to find Liouvillian first integrals of dynamical systems in the plane. The algorithm is based on a Darboux-type procedure to find the integrating factor for the system. Since the particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE), the Lsolver package presents a set of software routines in Maple for dealing with rational first order ODEs. The package present commands permitting research investigations of some algebraic properties of the system that is being studied. Program summary Program title:Lsolver Catalogue number:ADZF_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZF_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:1513 No. of bytes in distributed program, including test data, etc.:17 367 Distribution format:tar.gz Programming language:Maple Computer:Any computer running Maple Operating system:Windows ME, Windows XP Classification:4.3 Nature of problem: Solution of rational first order ordinary differential equations. Solution method:The method of solution is based on a Darboux/PS type approach. Restrictions:If the integrating factor for the ODE under consideration presents Darboux Polynomials of high degree (>3) in the dependent and independent variables, the package may spend an impractical amount of time to obtain the solution. Unusual features:Our implementation not only searches for Liouvillian conserved quantities, but can also be used as a research tool that allows the user to follow all the steps of the procedure (for example, we can calculate the algebraic invariants curves and associated co-factors, the integrating factor etc). In addition, since our package is based on recent theoretical developments [J. Avellar, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Integrating first-order differential equations with Liouvillian solutions via quadratures: a semi-algorithmic method, J. Comput. Appl. Math. 182 (2005) 327–332], it can successfully solve a class of rational first order ODEs that were not solved by some of the best-known ODE solvers available. Running time:This depends strongly on the ODE, but usually under 4 seconds.