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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 |
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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