A rigorous ODE solver and Smale’s 14th problem: 5. The RODES Program: The RODES (Rigorous ODE Solver) program is a highly adaptive, multiprocessorprogram. As we pointed out earlier, the computations are performed in intervalarithmetic with directed rounding when necessary. This functionality is providedby the PROFIL/BIAS package (see [8]) which is supported on all architecturesutilized in the proof. The program was executed on 20 machines working in parallel.Data was passed between the processes via a common text file. All floating pointnumbers were passed with 17 digits of precision, which converts exactly according o the IEEE standard. The computers employed for the task were a variety ofSUN Sparc stations, with models ranging from LX, Sparc 4 to Ultra 1. The totalcomputational time in this setting was about 100 hours. Other setups with fewercomputers equipped with stronger processors have been performed with similarresults. In the sections to come, we will give an overview of the program’s globalstructure and the computations carried out.

References in zbMATH (referenced in 127 articles , 1 standard article )

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  1. Beck, Margaret; Jaquette, Jonathan: Validated spectral stability via conjugate points (2022)
  2. Jaquette, Jonathan; Lessard, Jean-Philippe; Takayasu, Akitoshi: Global dynamics in nonconservative nonlinear Schrödinger equations (2022)
  3. Kapela, Tomasz; Wilczak, Daniel; Zgliczyński, Piotr: Recent advances in a rigorous computation of Poincaré maps (2022)
  4. van den Berg, Jan Bouwe; Duchesne, Gabriel William; Lessard, Jean-Philippe: Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: a Taylor-Chebyshev series approach (2022)
  5. van den Berg, Jan Bouwe; Groothedde, Chris; Lessard, Jean-Philippe: A general method for computer-assisted proofs of periodic solutions in delay differential problems (2022)
  6. Evstigneev, N. M.; Ryabkov, O. I.; Shul’min, D. A.: Use of shrink wrapping for interval Taylor models in algorithms of computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (2021)
  7. Façanha, Tiago S.; Barreto, Guilherme A.; Costa Filho, José T.: A novel Kalman filter formulation for improving tracking performance of the extended kernel RLS (2021)
  8. Fang, Wei; Giles, Mike B.: Importance sampling for pathwise sensitivity of stochastic chaotic systems (2021)
  9. Gonchenko, Sergey; Gonchenko, Alexander; Kazakov, Alexey; Samylina, Evgeniya: On discrete Lorenz-like attractors (2021)
  10. Gonchenko, Sergey; Kazakov, Alexey; Turaev, Dmitry: Wild pseudohyperbolic attractor in a four-dimensional Lorenz system (2021)
  11. Graça, Daniel S.; Zhong, Ning: Computability of differential equations (2021)
  12. Hart, Allen G.; Hook, James L.; Dawes, Jonathan H. P.: Echo state networks trained by Tikhonov least squares are (L^2(\mu)) approximators of ergodic dynamical systems (2021)
  13. José Pacifico, Maria; Yang, Fan; Yang, Jiagang: Entropy theory for sectional hyperbolic flows (2021)
  14. Lessard, Jean-Philippe; James, J. D. Mireles: A rigorous implicit (C^1) Chebyshev integrator for delay equations (2021)
  15. Naudot, Vincent; Kepley, Shane; Kalies, William D.: Complexity in a hybrid van der Pol system (2021)
  16. van den Berg, Jan Bouwe; Breden, Maxime; Lessard, Jean-Philippe; van Veen, Lennaert: Spontaneous periodic orbits in the Navier-Stokes flow (2021)
  17. Yampolsky, Michael: Towards understanding the theoretical challenges of numerical modeling of dynamical systems (2021)
  18. Zhang, Xu: On the omega-limit sets of planar nonautonomous differential equations with nonpositive Lyapunov exponents (2021)
  19. Akhmedov, Odiljon S.; Azamov, Abdulla A.; Ibragimov, Gafurjan I.: Four-dimensional Brusselator model with periodical solution (2020)
  20. Bahsoun, Wael; Melbourne, Ian; Ruziboev, Marks: Variance continuity for Lorenz flows (2020)

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