Journal of East China Normal University(Natural Science) >

2017 >Issue 3: 94 - 98

DOI: https://doi.org/10.3969/j.issn.1000-5641.2017.03.010

Physics and Electronics

Four kinds of gradient representations of autonomous Birkhoffian systems

  • CUI Jin-chao ,
  • LIAO Cui-cui ,
  • MEI Feng-xiang
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  • 1. School of Science, Jiangnan University, Wuxi Jiangsu 214122, China;
    2. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Received date: 2016-04-19

  Online published: 2017-05-18

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Abstract

In order to study the integration and the stability of autonomous Birkhoffian systems, we propose four kinds of gradient systems to represent the autonomous Birkhoffian systems. By analysing the relationship between the gradient systems and the Birkhoffian systems, we obtain the conditions that the Birkhoffian systems can be transformed into a kind of four gradient systems. Then, we use the properties of gradient system to investigate the problems of integration and stability of the Birkhoffian systems. Finally, we give some examples to illustrate the application of the theory.

Key words： Birkhoffian systems; gradient system; integration; stability

Cite this article

CUI Jin-chao , LIAO Cui-cui , MEI Feng-xiang . Four kinds of gradient representations of autonomous Birkhoffian systems[J]. Journal of East China Normal University(Natural Science), 2017 , (3) : 94 -98 . DOI: 10.3969/j.issn.1000-5641.2017.03.010

References

[1] HIRSCH M W, SMALE S, DEVANEY R L. Differential Equations, Dynamical Systems, and an Introduction to Chaos[M]. 3rd ed. Waltham: Academic Press, 2012.
[2] MCLACHLAN R I, QUISPEL G R W, ROBIDOUX N. Geometric integration using discrete gradients[J]. Philo-sophical Transactions of the Royal Society of London: Series A, 1999, 357: 1021-1045.
[3] LOU Z M, MEI F X. A second order gradient representation of mechanics system[J]. Acta Physica Sinica, 2012, 61: 337-340.
[4] MEI F X. On the gradient system[J]. Mechanics in Engineering, 2012, 34: 89-90.
[5] MEI F X, WU H B. A gradient representation for generalized Birkhoffian system[J]. Dynamics and Control, 2012, 10: 289-292.
[6] MEI F X, WU H B. Generalized Hamilton system and gradient system[J]. Science China Physics, Mechanics and Astronomy, 2013, 43: 538-540.
[7] MEI F X, CUI J C, WU H B. A gradient representation and a fractional gradient representation of Birkhoff system[J]. Transactions of Beijing Institute of Technology, 2012, 32: 1298-1300.
[8] MEI F X. Analytical Mechanics/[M]. Beijing: Beijing Institute of Technology Press, 2013.
[9] GUO Y X, LIU C, LIU S X. Generalized Birkhoffian realization of nonholonomic systems [J]. Communications in Mathematics, 2010, 18: 21-35.
[10] SANTILLI R M. Foundations of Theoretical Mechanics/[M]. New York: Springer, 1983.
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