L.G.S. Duarte

Computer algebra solving of second order ODEs using symmetry methods

Abstract A Maple V R.3/4 computer algebra package, ODEtools , for the analytical solving of first order ODEs using Lie group symmetry methods is presented. The set of commands includes a first order ODE solver and mutines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant first order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results.

A method to tackle first-order ordinary differential equations with Liouvillian functions in the solution

The usual Prelle–Singer (PS) approach misses many first-order ordinary differential equations presenting Liouvillian functions in the solution (LFOODEs). We point out why and propose a method extending the PS method to solve a class of these previously unsolved LFOODEs. Although our method does not cover all the LFOODEs, it maintains the semi-decision nature of the usual PS method.

Analysing the structure of the integrating factors for first-order ordinary differential equations with Liouvillian functions in the solution

Here we demonstrate a theorem concerning the general structure of the integrating factor for first-order ordinary differential equations whose solutions contain Liouvillian functions. This result assures the generality of a method presented in a forthcoming paper extending the usual Prelle-Singer approach.

An extension of the Prelle–Singer method and a Maple implementation☆

Abstract The Prelle–Singer method is a semi-decision algorithm which can be used to solve analytically first order ordinary differential equations which have solutions in terms of elementary functions. In this paper we develop an extension to the Prelle–Singer method which deals with first order ordinary differential equations whose solutions lie outside the scope of the standard Prelle–Singer method. We present a software package in Maple V, Release 5 which implements both the Prelle–Singer method in its original form and our extension. Tests with ordinary differential equations taken from standard references show that our package is able to solve equations where Maples standard solution routines fail.

Finding elementary first integrals for rational second order ordinary differential equations

Here we present a semialgorithm to find elementary first integrals of a class of rational second order ordinary differential equations. The method is based on a Darboux-type procedure and it is an attempt to construct an analogous of the method built by Prelle and Singer [“Elementary first integral of differential equations,” Trans. Am. Math. Soc. 279, 215 (1983)] for rational first order ordinary differential equations.

Numerical analysis of dynamical systems and the fractal dimension of boundaries

Abstract A set of MapleV R.4/5 software routines for calculating the numerical evolution of dynamical systems and flexible plotting the results is presented. The package consists of an initial condition generator (on which the user can impose quite general constraints), a numerical solving manager, plotting commands that allow the user to locate and focus in on regions of possible interest and, finally, a set of routines that calculate the fractal dimension of the boundaries of those regions. A special feature of the software routines presented here is an optional interface in C, permitting fast numerical integration using standard Runge—Kutta methods, or variations, for high precision numerical integration.

A Maple package for improved global mapping forecast

Abstract We present a Maple implementation of the well known global approach to time series analysis and some further developments designed to improve the computational efficiency of the forecasting capabilities of the approach. This global approach can be summarized as being a reconstruction of the phase space, based on a time ordered series of data obtained from the system. After that, using the reconstructed vectors, a portion of this space is used to produce a mapping, a polynomial fitting, through a minimization procedure, that represents the system and can be employed to forecast further entries for the series. In the present implementation, we introduce a set of commands, tools, in order to perform all these tasks. For example, the command VecTS deals mainly with the reconstruction of the vector in the phase space. The command GfiTS deals with producing the minimization and the fitting. ForecasTS uses all these and produces the prediction of the next entries. For the non-standard algorithms, we here present two commands: IforecasTS and NiforecasTS that, respectively deal with the one-step and the N -step forecasting. Finally, we introduce two further tools to aid the forecasting. The commands GfiTS and AnalysTS , basically, perform an analysis of the behavior of each portion of a series regarding the settings used on the commands just mentioned above. Program summary Program title: TimeS Catalogue identifier: AERW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AERW_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.: 3001 No. of bytes in distributed program, including test data, etc.: 95018 Distribution format: tar.gz Programming language: Maple 14. Computer: Any capable of running Maple Operating system: Any capable of running Maple. Tested on Windows ME, Windows XP, Windows 7. RAM: 128 MB Classification: 4.3, 4.9, 5 Nature of problem: Time series analysis and improving forecast capability. Solution method: The method of solution is partially based on a result published in [1]. Restrictions: If the time series that is being analyzed presents a great amount of noise or if the dynamical system behind the time series is of high dimensionality (Dim≫3), then the method may not work well. Unusual features: Our implementation can, in the cases where the dynamics behind the time series is given by a system of low dimensionality, greatly improve the forecast. Running time: This depends strongly on the command that is being used. References: [1] Barbosa, L.M.C.R., Duarte, L.G.S., Linhares, C.A. and da Mota, L.A.C.P., Improving the global fitting method on nonlinear time series analysis, Phys. Rev. E 74, 026702 (2006).

3D polynomial dynamical systems with elementary first integrals

Here we present a semi-algorithm to find elementary first integrals of 3D polynomial dynamical systems. It is a Darboux type procedure that extends the method built by Prelle and Singer for 2D systems. Although it cannot deal with the general case, the method presents a direct/simple way to find elementary first integrals.

Determining liouvillian first integrals for dynamical systems in the plane

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.

Alternative predictors in chaotic time series

In the scheme of reconstruction, non-polynomial predictors improve the forecast from chaotic time series. The algebraic manipulation in the Maple environment is the basis for obtaining of accurate predictors. Beyond the different times of prediction, the optional arguments of the computational routines optimize the running and the analysis of global mappings.