Article Contents
Article Contents

# Discretization of dynamical systems with first integrals

• We find conditions under which the first integral of an ordinary differential equation becomes a discrete Lyapunov function for its numerical discretization. This result is applied for precluding periodic and bounded orbits under discretization for several cases.
Mathematics Subject Classification: Primary: 34C25, 65L20; Secondary: 34C14.

 Citation:

•  [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," $2^{nd}$ edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. [2] M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces," Graduate Texts in Mathematics, 115, Springer-Verlag, New York, 1988. [3] M. Farkas, "Periodic Motions," Applied Mathematical Sciences, 104, Springer-Verlag, New York, 1994. [4] M. Fečkan, Criteria on the nonexistence of invariant Lipschitz submanifolds for dynamical systems, J. Differential Equations, 174 (2001), 392-419.doi: 10.1006/jdeq.2000.3943. [5] B. M. Garay, The discretized flow on domains of attraction: A structural stability result, IMA J. Numer. Anal., 18 (1998), 77-90.doi: 10.1093/imanum/18.1.77. [6] B. M. Garay and P. E. Kloeden, Discretization near compact invariant sets, Random & Comput. Dynamics, 5 (1997), 93-123. [7] R. Goldman, Curvature formulas for implicit curves and surfaces, Computer Aided Geometric Design, 22 (2005), 632-658.doi: 10.1016/j.cagd.2005.06.005. [8] E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations," $2^{nd}$ (2006) edition, Springer Series in Computational Mathematics, 31, Springer, Heidelberg, 2010. [9] P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964. [10] M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. [11] Y. Li and J. S. Muldowney, On Bendixon's criterion, J. Differential Equations, 106 (1993), 27-39.doi: 10.1006/jdeq.1993.1097. [12] C. Pötzsche and M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds, Numerische Mathematik, 112 (2009), 449-483.doi: 10.1007/s00211-009-0215-9. [13] A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis," Cambridge Monographs on Applied and Computational Mathematics, 2, Cambridge Univ. Press, Cambridge, 1996.