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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]

$2^{nd}$ edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.  Google Scholar

[2]

Graduate Texts in Mathematics, 115, Springer-Verlag, New York, 1988.  Google Scholar

[3]

Applied Mathematical Sciences, 104, Springer-Verlag, New York, 1994.  Google Scholar

[4]

J. Differential Equations, 174 (2001), 392-419. doi: 10.1006/jdeq.2000.3943.  Google Scholar

[5]

IMA J. Numer. Anal., 18 (1998), 77-90. doi: 10.1093/imanum/18.1.77.  Google Scholar

[6]

Random & Comput. Dynamics, 5 (1997), 93-123.  Google Scholar

[7]

Computer Aided Geometric Design, 22 (2005), 632-658. doi: 10.1016/j.cagd.2005.06.005.  Google Scholar

[8]

$2^{nd}$ (2006) edition, Springer Series in Computational Mathematics, 31, Springer, Heidelberg, 2010.  Google Scholar

[9]

John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[10]

Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[11]

J. Differential Equations, 106 (1993), 27-39. doi: 10.1006/jdeq.1993.1097.  Google Scholar

[12]

Numerische Mathematik, 112 (2009), 449-483. doi: 10.1007/s00211-009-0215-9.  Google Scholar

[13]

Cambridge Monographs on Applied and Computational Mathematics, 2, Cambridge Univ. Press, Cambridge, 1996.  Google Scholar

show all references

References:
[1]

$2^{nd}$ edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.  Google Scholar

[2]

Graduate Texts in Mathematics, 115, Springer-Verlag, New York, 1988.  Google Scholar

[3]

Applied Mathematical Sciences, 104, Springer-Verlag, New York, 1994.  Google Scholar

[4]

J. Differential Equations, 174 (2001), 392-419. doi: 10.1006/jdeq.2000.3943.  Google Scholar

[5]

IMA J. Numer. Anal., 18 (1998), 77-90. doi: 10.1093/imanum/18.1.77.  Google Scholar

[6]

Random & Comput. Dynamics, 5 (1997), 93-123.  Google Scholar

[7]

Computer Aided Geometric Design, 22 (2005), 632-658. doi: 10.1016/j.cagd.2005.06.005.  Google Scholar

[8]

$2^{nd}$ (2006) edition, Springer Series in Computational Mathematics, 31, Springer, Heidelberg, 2010.  Google Scholar

[9]

John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[10]

Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[11]

J. Differential Equations, 106 (1993), 27-39. doi: 10.1006/jdeq.1993.1097.  Google Scholar

[12]

Numerische Mathematik, 112 (2009), 449-483. doi: 10.1007/s00211-009-0215-9.  Google Scholar

[13]

Cambridge Monographs on Applied and Computational Mathematics, 2, Cambridge Univ. Press, Cambridge, 1996.  Google Scholar

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