# American Institute of Mathematical Sciences

August  2013, 33(8): 3543-3554. doi: 10.3934/dcds.2013.33.3543

## Discretization of dynamical systems with first integrals

 1 Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, 842 48 Bratislava 2 Centre for Research and Utilization of Renewable Energy, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic

Received  May 2012 Revised  September 2012 Published  January 2013

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.
Citation: Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543
##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", $2^{nd}$ edition, 60 (1989).   Google Scholar [2] M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces,", Graduate Texts in Mathematics, 115 (1988).   Google Scholar [3] M. Farkas, "Periodic Motions,", Applied Mathematical Sciences, 104 (1994).   Google Scholar [4] M. Fečkan, Criteria on the nonexistence of invariant Lipschitz submanifolds for dynamical systems,, J. Differential Equations, 174 (2001), 392.  doi: 10.1006/jdeq.2000.3943.  Google Scholar [5] B. M. Garay, The discretized flow on domains of attraction: A structural stability result,, IMA J. Numer. Anal., 18 (1998), 77.  doi: 10.1093/imanum/18.1.77.  Google Scholar [6] B. M. Garay and P. E. Kloeden, Discretization near compact invariant sets,, Random & Comput. Dynamics, 5 (1997), 93.   Google Scholar [7] R. Goldman, Curvature formulas for implicit curves and surfaces,, Computer Aided Geometric Design, 22 (2005), 632.  doi: 10.1016/j.cagd.2005.06.005.  Google Scholar [8] E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,", $2^{nd}$ (2006) edition, 31 (2006).   Google Scholar [9] P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (1964).   Google Scholar [10] M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar [11] Y. Li and J. S. Muldowney, On Bendixon's criterion,, J. Differential Equations, 106 (1993), 27.  doi: 10.1006/jdeq.1993.1097.  Google Scholar [12] C. Pötzsche and M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds,, Numerische Mathematik, 112 (2009), 449.  doi: 10.1007/s00211-009-0215-9.  Google Scholar [13] A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,", Cambridge Monographs on Applied and Computational Mathematics, 2 (1996).   Google Scholar

show all references

##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", $2^{nd}$ edition, 60 (1989).   Google Scholar [2] M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces,", Graduate Texts in Mathematics, 115 (1988).   Google Scholar [3] M. Farkas, "Periodic Motions,", Applied Mathematical Sciences, 104 (1994).   Google Scholar [4] M. Fečkan, Criteria on the nonexistence of invariant Lipschitz submanifolds for dynamical systems,, J. Differential Equations, 174 (2001), 392.  doi: 10.1006/jdeq.2000.3943.  Google Scholar [5] B. M. Garay, The discretized flow on domains of attraction: A structural stability result,, IMA J. Numer. Anal., 18 (1998), 77.  doi: 10.1093/imanum/18.1.77.  Google Scholar [6] B. M. Garay and P. E. Kloeden, Discretization near compact invariant sets,, Random & Comput. Dynamics, 5 (1997), 93.   Google Scholar [7] R. Goldman, Curvature formulas for implicit curves and surfaces,, Computer Aided Geometric Design, 22 (2005), 632.  doi: 10.1016/j.cagd.2005.06.005.  Google Scholar [8] E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,", $2^{nd}$ (2006) edition, 31 (2006).   Google Scholar [9] P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (1964).   Google Scholar [10] M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar [11] Y. Li and J. S. Muldowney, On Bendixon's criterion,, J. Differential Equations, 106 (1993), 27.  doi: 10.1006/jdeq.1993.1097.  Google Scholar [12] C. Pötzsche and M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds,, Numerische Mathematik, 112 (2009), 449.  doi: 10.1007/s00211-009-0215-9.  Google Scholar [13] A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,", Cambridge Monographs on Applied and Computational Mathematics, 2 (1996).   Google Scholar
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