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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

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 & 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.  Google Scholar

[2]

M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces," Graduate Texts in Mathematics, 115, Springer-Verlag, New York, 1988.  Google Scholar

[3]

M. Farkas, "Periodic Motions," Applied Mathematical Sciences, 104, Springer-Verlag, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. M. Garay and P. E. Kloeden, Discretization near compact invariant sets, Random & Comput. Dynamics, 5 (1997), 93-123.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[10]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[11]

Y. Li and J. S. Muldowney, On Bendixon's criterion, J. Differential Equations, 106 (1993), 27-39. 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), 449-483. 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, Cambridge Univ. Press, Cambridge, 1996.  Google Scholar

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.  Google Scholar

[2]

M. Berger and B. Gostiaux, "Differential Geometry: Manifolds, Curves, and Surfaces," Graduate Texts in Mathematics, 115, Springer-Verlag, New York, 1988.  Google Scholar

[3]

M. Farkas, "Periodic Motions," Applied Mathematical Sciences, 104, Springer-Verlag, New York, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. M. Garay and P. E. Kloeden, Discretization near compact invariant sets, Random & Comput. Dynamics, 5 (1997), 93-123.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[10]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[11]

Y. Li and J. S. Muldowney, On Bendixon's criterion, J. Differential Equations, 106 (1993), 27-39. 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), 449-483. 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, Cambridge Univ. Press, Cambridge, 1996.  Google Scholar

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