-lohner Algorithm

We present a modification of the Lohner algorithm for the computation of rigorous bounds for solutions of ordinary differential equations together with partial derivatives with respect to initial conditions. The modified algorithm requires essentially the same computational effort as the original one. We applied the algorithm to show an existence of several periodic orbits for Rössler equations and the 14-dimensional Galerkin projection of the Kuramoto-Sivashinsky PDE. MSC numbers: 65G20, 65L05

[1]  K. Mischaikow,et al.  Chaos in the Lorenz equations: a computer-assisted proof , 1995, math/9501230.

[2]  Rage,et al.  Rigorous verification of chaos in a molecular model. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  G. Sivashinsky.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations , 1977 .

[4]  Warwick Tucker,et al.  Foundations of Computational Mathematics a Rigorous Ode Solver and Smale's 14th Problem , 2022 .

[5]  Y. Kuramoto,et al.  Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium , 1976 .

[6]  Konstantin Mischaikow,et al.  Chaos in the Lorenz equations: A computer assisted proof. Part II: Details , 1998, Math. Comput..

[7]  Piotr Zgliczyński.  MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS AND STABILITY OF SARKOVSKĬ ORDERING , 1999 .

[8]  G. Alefeld.  lnclusion methods for systems of nonlinear equations-the interval Newton method and modifications , 2022 .

[9]  Konstantin Mischaikow,et al.  Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation , 2001, Found. Comput. Math..

[10]  P. Pilarczyk.  COMPUTER ASSISTED METHOD FOR PROVING EXISTENCE OF PERIODIC ORBITS , 1999 .

[11]  D. Wilczak,et al.  Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three-Body Problem – A Computer Assisted Proof , 2002, math/0201278.

[12]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[13]  Nedialko S. Nedialkov,et al.  An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation , 1999, Reliab. Comput..

[14]  R. Lohner.  Einschliessung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen , 1988 .

[15]  Marian Mrozek,et al.  Set arithmetic and the enclosing problem in dynamics , 2000 .

[16]  S. P. Hastings,et al.  A computer proof that the Lorenz equations have “chaotic” solutions , 1994 .