Cr-Lohner algorithm

Daniel Wilczak,

Piotr Zgliczyński


We present a Lohner type algorithm for the computation of rigorous bounds for the solutions of ordinary differential equations and its derivatives with respect to the initial conditions up to an arbitrary order.

Słowa kluczowe: rigorous integration of ODEs, variational equations.

Alefeld G.; Inclusion methods for systems of nonlinear equations – the interval Newton method and modifications, in: Herzberger J. (ed.), Topics in Validated Computations, Elsevier Science B.V., 1994, pp. 7–26.

Broer H.W., Huitema G.B., Sevryuk M.B.; Quasi-periodicity in families of dynamical systems: order amidst chaos, Lecture Notes in Mathematics, 1645, Springer Verlag, 1996.

Berz M., Makino K.; New Methods for High-Dimensional Verified Quadrature, Reliable Computing, 5, 1999, pp. 13–22.

CAPD – Computer Assisted Proofs in Dynamics group; a C++ package for rigorous numerics. Available via

Griewank A.; Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Frontiers in Applied Mathematics, 19, SIAM, 2000.

Galias Z., Zgliczy´nski P.; Computer assisted proof of chaos in the Lorenz system, Physica D, 115(3–4), 1998, pp. 165–188.

Jorba `A., Zou M.; A software package for the numerical integration of ODE by means of high-order Taylor methods, Experimental Mathematics, 14, 2005, pp. 99–117.

Hardy M.; Combinatorics of Partial Derivatives, Electronic Journal of Combinatorics, 13, 2006.

Hairer E., Nørsett S.P., Wanner G.; Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag, Berlin–Heidelberg, 1987.

Hassard B., Zhang J., Hastings S., Troy W.; A computer proof that the Lorenz equations have ”chaotic” solutions, Applied Mathematics Letters, 7, 1994, pp. 79–83.

Kapela T., Zgliczy´nski P.; The existence of simple choreographies for N-body problem – a computer assisted proof, Nonlinearity, 16, 2003, pp. 1899–1918.

Kokubu H., Wilczak D., Zgliczy´nski P.; Rigorous verification of cocoon bifurcations in the Michelson system, Nonlinearity, 20, 2007, pp. 2147–2174.

Lohner R.J.; Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems, in: Cash J.R., Gladwell I. (ed.), Computational Ordinary Differential Equations, Clarendon Press, Oxford 1992.

Michelson D.; Steady solutions of the Kuramoto–Sivashinsky equation, Physica D, 19, 1986, pp. 89–111.

Moore R.E.; Interval Analysis, Prentice Hall, 1966.

Mischaikow K., Mrozek M.; Chaos in the Lorenz equations: A computer assisted proof, Mathematics of Computation, 67, 1998, pp. 1023–1046.

Mrozek M., Zgliczy´nski P.; Set arithmetic and the enclosing problem in dynamics, Annales Polonici Mathematici, 2000, pp. 237–259.

Nedialkov N.S., Jackson K.R.; An Interval Hermite – Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation, in: Csendes T. (ed.), Developments in Reliable Computing, Kluwer, Dordrecht, Netherlands, 1999, pp. 289–310.

Neumeier A.; Interval methods for systems of equations, Cambridge University Press, 1990.

Rall L.B.; Automatic Differentiation: Techniques and Applications, Lecture Notes in Computer Science, 120, 1981.

Rage T., Neumaier A., Schlier C.; Rigorous verification of chaos in a molecular model, Phys. Rev. E, 50, 1994, pp. 2682–2688.

R¨ossler O.E.; An Equation for Continuous Chaos, Physics Letters A, 57(5), 1976, pp. 397–398.

Tucker W.; A Rigorous ODE solver and Smale’s 14th Problem, Foundations of Com- putational Mathematics, 2(1), 2002, pp. 53–117.

Walter W.; Differential and integral inequalities, Springer-Verlag, New York 1970.

Wilczak D.; Rigorous normal forms and the existence of KAM invariant curves for Poincar´e maps, in review.

Wilczak D.; Symmetric heteroclinic connections in the Michelson system – a computer assisted proof, SIAM Journal on Applied Dynamical Systems, 4(3), 2005, pp. 489–514.

Wilczak D., Zgliczy´nski P.; Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem – A Computer Assisted Proof, Commu- nications in Mathematical Physics, 234, 2003, pp. 37–75.

Wilczak D., Zgliczy´nski P.; Computer assisted proof of the existence of homoclinic tangency for the Henon map and for the forced-damped pendulum, SIAM Journal on Applied Dynamical Systems, 8(4), 2009, pp. 1632–1663.

Wilczak D., Zgliczy´nski. P.; Period doubling in the R¨ossler system – a computer assisted proof, Foundations of Computational Mathematics, 9, 2009, pp. 611–649.

Zgliczy´nski P.; Computer assisted proof of chaos in the H´enon map and in the R¨ossler equations, Nonlinearity, 10(1), 1997, 243–252.

Zgliczy´nski P.; C1-Lohner algorithm, Foundations of Computational Mathematics, 2, 2002, pp. 429–465.

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.

Wersja papierowa czasopisma dostępna na