Discretization of dynamical systems with first integrals

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

Discretization of dynamical systems with first integrals

Michal Fečkan ^1, and Michal Pospíšil ^2,

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

Abstract
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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.

Keywords: first integral, curvature, Numerical approximation, mechanical system..

Mathematics Subject Classification: Primary: 34C25, 65L20; Secondary: 34C1.

Citation: Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete and Continuous Dynamical Systems, 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, 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.

show all references

References:

[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.

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