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Brikhoff system is a kind of basic dynamical system. The theory and method of Brikhoff system dynamics have been applied to the hadron physics, quantum physics, relativity and rotational relativistic system. The properties of gradient system not only play an important role in revealing the internal structure of dynamical system, but also help to explore the dynamical behavior of the system. In this paper, two kinds of generalized gradient representations for generalized Birkhoff system are studied. First, two kinds of generalized gradient systems, i. e., the generalized skew gradient system and the generalized gradient system with symmetric negative definite matrix, are proposed and the characteristics of the systems are studied. Second, the relations of stability between these two kinds of gradient system and the dynamical system are discussed. Third, the condition under which a generalized Birkhoff system can be considered as one of the two generalized gradient systems is obtained. Fourth, the gradient discrimination method of stability of the generalized Brikhoff system is given, and the characteristics of the generalized gradient systems can be used to study the stability of the generalized Birkhoff system. Finally, some examples are given to illustrate the application of the result. Therefore, once the mechanical system is expressed as the generalized gradient system, the stability and the asymptotic stability can be conveniently studied by using the properties of generalized gradient system. The difficulty in constructing Lyapunov functions is avoided, and a convenient method of analyzing the stability of mechanical system is provided.
[1] Santilli R M 1978 Foundations of Theoretical Mechanics I (New York: Springer) pp182-191
[2] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer) pp253-267
[3] Hirsch M W, Smale S 1974 Differential Equations, Dynamical Systems, and Linear Algebra (New York: Academic Press) pp199-203
[4] Mc Lachlan R I, Quispel G R W, Robidoux N 1999 Phil. Trans. R. Soc. Lond. A 357 1021
[5] Mei F X, Wu H B 2012 J. Dynam. Control 10 289 (in Chinese) [梅凤翔, 吴惠彬 2012 动力学与控制学报 10 289]
[6] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 024502 (in Chinese) [楼智美, 梅凤翔 2012 物理学报 61 024502]
[7] Hirsch M W, Smale S, Devaney R L 2008 Differential Equations, Dynamical Systems, and an Introduction to Chaos (Singapore: Elsevier) pp203-206
[8] Mei F X, Cui J C, Wu H B 2012 Trans. Beijing Inst. Tech. 32 1298 (in Chinese) [梅凤翔, 崔金超, 吴惠彬 2012 北京理工大学学报 32 1298]
[9] Tom B, Ralph C, Eva F 2012 Monatsh Math. 166 57
[10] Mei F X, Wu H B 2013 Sci. Sin.: Phys. Mech. Astron. 43 538 (in Chinese) [梅凤翔, 吴惠彬 2013 中国科学: 物理学 力学 天文学 43 538]
[11] Chen X W, Zhao G L, Mei F X 2013 Nonlinear Dyn. 73 579
[12] Mei F X 2013 Analytical Mechanics II (Beijing: Beijing Inst. Tech. Press) pp564-581 (in Chinese) [梅凤翔 2013 分析力学II(北京: 北京理工大学出版社) 第 564-581 页]
[13] Marin A M, Ortiz R D, Rodriguez J A 2013 International Mathematical Forum 8 803
[14] Mei F X, Wu H B 2015 J. Dynam. Control 13 329 (in Chinese) [梅凤翔, 吴惠彬 2015 动力学与控制学报 13 329]
[15] Yin X W, Li D S 2015 Acta Mathematica Scientia 35A 464 (in Chinese) [尹逊武, 李德生 2015 数学物理学报 35A 464]
[16] Mei F X, Wu H B 2015 Chin. Phys. B 24 104502
[17] Wu H B, Mei F X 2015 Acta Phys. Sin. 64 234501 (in Chinese) [吴惠彬, 梅凤翔 2015 物理学报 64 234501]
[18] Zhang Y 2015 J. Suzhou Univ. Sci. Tech. (Natural Science) 32 1 (in Chinese) [张毅 2015 苏州科技学院学报(自然科学版) 32 1]
[19] Mei F X, Wu H B 2015 Acta Phys. Sin. 64 184501 (in Chinese) [梅凤翔, 吴惠彬 2015 物理学报 64 184501]
[20] Li L, Luo S K 2013 Acta Mechanica 224 1757
[21] Luo S K, He J M, Xu Y L 2016 Inter. J. Non-Linear Mech. 78 105
[22] Mei F X 2013 Dynamics of Generalized Birkhoff Systems (Beijing: Science Press) pp31-36 (in Chinese) [梅凤翔 2013 广义Birkhoff系统动力学 (北京: 科学出版社) 第 31-36 页]
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[1] Santilli R M 1978 Foundations of Theoretical Mechanics I (New York: Springer) pp182-191
[2] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer) pp253-267
[3] Hirsch M W, Smale S 1974 Differential Equations, Dynamical Systems, and Linear Algebra (New York: Academic Press) pp199-203
[4] Mc Lachlan R I, Quispel G R W, Robidoux N 1999 Phil. Trans. R. Soc. Lond. A 357 1021
[5] Mei F X, Wu H B 2012 J. Dynam. Control 10 289 (in Chinese) [梅凤翔, 吴惠彬 2012 动力学与控制学报 10 289]
[6] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 024502 (in Chinese) [楼智美, 梅凤翔 2012 物理学报 61 024502]
[7] Hirsch M W, Smale S, Devaney R L 2008 Differential Equations, Dynamical Systems, and an Introduction to Chaos (Singapore: Elsevier) pp203-206
[8] Mei F X, Cui J C, Wu H B 2012 Trans. Beijing Inst. Tech. 32 1298 (in Chinese) [梅凤翔, 崔金超, 吴惠彬 2012 北京理工大学学报 32 1298]
[9] Tom B, Ralph C, Eva F 2012 Monatsh Math. 166 57
[10] Mei F X, Wu H B 2013 Sci. Sin.: Phys. Mech. Astron. 43 538 (in Chinese) [梅凤翔, 吴惠彬 2013 中国科学: 物理学 力学 天文学 43 538]
[11] Chen X W, Zhao G L, Mei F X 2013 Nonlinear Dyn. 73 579
[12] Mei F X 2013 Analytical Mechanics II (Beijing: Beijing Inst. Tech. Press) pp564-581 (in Chinese) [梅凤翔 2013 分析力学II(北京: 北京理工大学出版社) 第 564-581 页]
[13] Marin A M, Ortiz R D, Rodriguez J A 2013 International Mathematical Forum 8 803
[14] Mei F X, Wu H B 2015 J. Dynam. Control 13 329 (in Chinese) [梅凤翔, 吴惠彬 2015 动力学与控制学报 13 329]
[15] Yin X W, Li D S 2015 Acta Mathematica Scientia 35A 464 (in Chinese) [尹逊武, 李德生 2015 数学物理学报 35A 464]
[16] Mei F X, Wu H B 2015 Chin. Phys. B 24 104502
[17] Wu H B, Mei F X 2015 Acta Phys. Sin. 64 234501 (in Chinese) [吴惠彬, 梅凤翔 2015 物理学报 64 234501]
[18] Zhang Y 2015 J. Suzhou Univ. Sci. Tech. (Natural Science) 32 1 (in Chinese) [张毅 2015 苏州科技学院学报(自然科学版) 32 1]
[19] Mei F X, Wu H B 2015 Acta Phys. Sin. 64 184501 (in Chinese) [梅凤翔, 吴惠彬 2015 物理学报 64 184501]
[20] Li L, Luo S K 2013 Acta Mechanica 224 1757
[21] Luo S K, He J M, Xu Y L 2016 Inter. J. Non-Linear Mech. 78 105
[22] Mei F X 2013 Dynamics of Generalized Birkhoff Systems (Beijing: Science Press) pp31-36 (in Chinese) [梅凤翔 2013 广义Birkhoff系统动力学 (北京: 科学出版社) 第 31-36 页]
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