
TIDES: a Taylor Integrator for Differential Equations
TIDES is a free software to integrate numerically Ordinary Differential Equations by using
a Taylor Series method. This software, developed by
A. Abad,
R Barrio,
F. Blesa,
M. Rodríguez,
(GME), consists on a C (Fortran) library, libTIDES,
and a Mathematica package, MathTIDES. (MathTIDES requires Mathematica version >= 7.0)
The main features of TIDES are the following:
 TIDES permits to integrate numerically ODE problems with multiple precision,
that means that you can solve ODE problems up to any precision level in a reasonable computer time.
 TIDES may solve directly sensitivity equations with respect to initial conditions
or parameters up to any any order.
 TIDES integrates by using the Taylor Series method with an optimized variablestepsize and variableorder formulation, and
extended formulas for variational equations.
 The software has been done to be extremely easy to use: with MathTIDES we write,
in a natural way, the ODE and their parameters, together with the parameters of the integration.
Then, MathTIDES writes the C (Fortran) code, that, compiled and linked with libTIDES, integrates the ODE.
 The derivatives and partial derivatives are obtained by using Automatic Differentiation (AD) techniques.
 There are four different versions of the code generated by MathTIDES:
 Two minimal versions (in C or Fortran language). These are faster than standard
versions and easy to use as subprograms, but they do not have most of the possibilities of the standard versions.
 A standard version (only in C language) for double precision computation.
 A standard version (only in C language) for multiple precision computations.
This version uses MPFR,
and GMP libraries to integrate ODEs with any arbitrary precision.
 MathTIDES writes automatically the code to compute partial derivatives of the solution of the ODE
with respect to any variable or parameter (using AD and avoiding the use of any variational equation or
sensitivity with respect to the parameters).
 TIDES may detect events of ODEs, i. e. points where a function of the solution of the ODE
satisfies an event function, like it becomes zero or reaches an extremum.
References
 A. Abad, R. Barrio, F. Blesa, M. Rodríguez, 2012.
Algorithm 924: TIDES, a Taylor series Integrator for Differential EquationS,
ACM Transactions on Mathematical Software. 39, no. 1, article 5. (Use this reference for TIDES citation.)
 A. Abad, R. Barrio, F. Blesa, M. Rodríguez, 2011. TIDES tutorial: Integrating ODEs by using the Taylor Series Method., Monografías de la Academia de Ciencias
de la Universidad de Zaragoza. 36 pp. 1  116.
 R. Barrio, 2005. Performance of the Taylor series method for ODEs/DAEs,
Applied Mathematics and Computation 163, no. 2, 525545.
 R. Barrio, F. Blesa, M. Lara, 2005.
VSVO formulation of the Taylor method for the numerical solution of ODEs, Computers and Mathematics and Applications 50 (12) 93111.
 R. Barrio, 2006.
Sensitivity analysis of ODE's/DAE's using the Taylor series method, SIAM J. on Scientific Computing 27 (6) 19291947.
Related references
The following papers use TIDES. If you use TIDES, please send a reference of your paper to
tides.taylor@gmail.com
and it will be added to this list
 A. Abad, R. Barrio and A. Dena, 2011. Computing periodic orbits with arbitrary precision,
Physical Review E. 84  016701.
In this paper we compute four periodic orbits with 1000 digits of precision.
The periodic orbits and the programs to check that they are really periodic orbits may be found here
 R. Barrio, F. Blesa, M. Lara, M. 2003.
Highprecision numerical solution of ODE with highorder Taylor methods in parallel.
Analytic and numerical techniques in orbital dynamics (in Spanish),
Monografías de la Real Academia de Ciencias de Zaragoza 22 pp 6774.
 R. Barrio,F. Blesa, S. Serrano, 2010.
Bifurcations and chaos in Hamiltonian systems.
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
20, no. 5, pp 12931319.
 R. Barrio, M. Rodríguez, A. Abad, S. Serrano, 2011. Uncertainty propagation or box propagation,
Mathematical and Computer Modeling. 54 (1112) 2602  2615.
 R. Barrio, M. Rodríguez, A. Abad, F. Blesa, 2011.
Breaking the limits: the Taylor series method, Applied Mathematics and Computation 217 (20) 7940  7954.
 R. Barrio, A. Shilnikov, 2011.
Parametersweeping techniques for temporal dynamics of neuronal systems: case study of HindmarshRose model.
The Journal of Mathematical Neuroscience. 1, Art. 6, 20 pp.
 R. Barrio, A. Shilnikov, L. Shilnikov, 2012.
Kneadings, symbolic dynamics and painting Lorenz chaos.
International Journal of Bifurcations and Chaos 22, no. 4, 1230016.
 A. Dena, M. Rodríguez, S. Serrano, R. Barrio. 2012
Highprecision continuation of periodic orbits.
Abstract and Applied Analysis 2012, Art. ID 716024, 12pp.
 E. Gerlach, Ch. Skokos, 2011.
Comparing the efficiency of numerical techniques for the integration of variational equations.
in "Dynamical Systems, Differential Equations and Applications",
Discr.Cont. Dyn. Sys.Supp. 2011 (dedicated to the 8th AIMS Conference),
eds. Feng W., Feng Z., Grasselli M., Ibragimov A., Lu X, Siegmund S. & Voirt J., AIMS, 475484.
 E. Gerlach, S. Eggl and Ch. Skokos, 2012.
Efficient integration of the variational equations of multidimensional Hamiltonian systems: Application to the FermiPastaUlam lattice.
International Journal of Bifurcations and Chaos 22, no. 9, 1250216.
 M. Rodríguez, R. Barrio, 2012.
Reducing rounding errors and achieving Brouwer's law with Taylor Series Method.
Applied Numerical Mathematics, 62, no. 8, pp 10141024.
How to obtain the software
Download it from this sourceforge web page
