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). Available at Sourceforge.

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 variable-stepsize and variable-order 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

1. A. Abad, R. Barrio, F. Blesa and M. Rodríguez. Algorithm 924: TIDES, a Taylor series integrator for differential equationS, ACM Transactions on Mathematical Software (TOMS) 39 1 (2012) 5.
2. A. Abad, R. Barrio, F. Blesa and M. Rodríguez. TIDES tutorial: Integrating ODEs by using the Taylor Series Method, Monografías de la Academia de Ciencias de la Universidad de Zaragoza 36. RACUZ, 2011.
3. R. Barrio. Performance of the Taylor series method for ODEs/DAEs, Applied Mathematics and Computation 163 2 (2005) 525–545.
4. R. Barrio, F. Blesa and M Lara. VSVO formulation of the Taylor method for the numerical solution of ODEs, Computers & mathematics with Applications 50 1 (2005) 93–111.
5. R. Barrio. Sensitivity analysis of ODEs/DAEs using the Taylor series method, SIAM Journal on Scientific Computing 27 6 (2006) 1929–1947.

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.

1. A. Abad, R. Barrio and A. Dena. Computing periodic orbits with arbitrary precision, Physical review E 84 1 (2011) 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.
2. R. Barrio, F. Blesa, M. Lara, M. 2003. High-precision numerical solution of ODE with high-order Taylor methods in parallel. Analytic and numerical techniques in orbital dynamics (in Spanish), Monografías de la Real Academia de Ciencias de Zaragoza 22 pp 67-74.
3. R. Barrio, F. Blesa and S. Serrano. Bifurcations and chaos in Hamiltonian systems, International Journal of Bifurcation and Chaos 20 5 (2010) 1293–1319.
4. R. Barrio, M. Rodríguez, A. Abad and S. Serrano. Uncertainty propagation or box propagation, Mathematical and Computer Modelling 54 11 (2011) 2602–2615.
5. R. Barrio, M. Rodríguez, A. Abad and F. Blesa. Breaking the limits: the Taylor series method, Applied mathematics and computation 217 20 (2011) 7940–7954.
6. R. Barrio and A. Shilnikov. Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model, The Journal of Mathematical Neuroscience 1 (2011) 6.
7. R. Barrio, A. Shilnikov and L. Shilnikov. Kneadings, symbolic dynamics and painting Lorenz chaos, International Journal of Bifurcation and Chaos 22 4 (2012) 1230016.
8. Á. Dena, M. Rodríguez, S. Serrano and R. Barrio. High-precision continuation of periodic orbits, Abstract and Applied Analysis 2012 (2012) 716024.
9. A. Gerlach and Ch. Skokos. 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, 2011. 475-484
10. E. Gerlach, S. Eggl and Ch. Skokos. Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice, International Journal of Bifurcation and Chaos 22 9 (2012) 1250216.
11. M. Rodríguez and R. Barrio. Reducing rounding errors and achieving Brouwerʼs law with Taylor Series Method, Applied Numerical Mathematics 62 8 (2012) 1014–1024.