Tzénoff方程解的稳定性
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摘要
提出两类广义梯度系统,即广义斜梯度系统和具有对称负定矩阵的广义梯度系统,并研究了这两类广义梯度系统的性质.在满足一定的条件下,把Tzénoff方程化成这两类广义梯度系统方程.进一步利用这两类广义梯度系统的性质来研究Tzénoff方程解的稳定性,分别给出解是稳定、渐进稳定和不稳定的条件,并举例说明结果的应用.
Two kinds of generalized gradient systems,namely generalized skew gradient system and generalized gradient system with symmetric negative definite matrix,are proposed,and the properties of these two kinds of generalized gradient systems are studied.The conditions under which the Tzénoff equations can be considered as one of the two generalized gradient systems are obtained.The characteristics of the generalized gradient systems can be used to study the stability of solution of the Tzénoff equations,and the conditions for the solution to be stable,asymptotically stable and unstable are given. Some examples are used to illustrate the application of the results.
Two kinds of generalized gradient systems,namely generalized skew gradient system and generalized gradient system with symmetric negative definite matrix,are proposed,and the properties of these two kinds of generalized gradient systems are studied.The conditions under which the Tzénoff equations can be considered as one of the two generalized gradient systems are obtained.The characteristics of the generalized gradient systems can be used to study the stability of solution of the Tzénoff equations,and the conditions for the solution to be stable,asymptotically stable and unstable are given. Some examples are used to illustrate the application of the results.
引文
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[12] LUO S K,HE J M,XU Y L.Fractional Birkhoffian method for equilibrium stability of dynamical systems[J].International Journal of Non-Linear Mechanics,2016,78(1):105-111.
[13]李彦敏.变质量非完整力学系统的共形不变性[J].云南大学学报:自然科学版,2010,32(1):52-57.LI Y M.Conformal invariance for nonholonomic mechanical systems with variable mass[J]. Journal of Yunnan University:Natural Sciences Edition,2010,32(1):52-57.
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[15]李彦敏,陈向炜,吴惠彬,等.广义Birkhoff系统的两类广义梯度表示[J].物理学报,2016,65(8):080201.LI Y M,CHEN X W,WU H B,et al.Two kinds of generalized gradient representations for generalized Birkhoff system[J].Acta Physica Sinica,2016,65(8):080201.
[2]薛问西.Appell方程和Tzénoff方程在一般非完整系统上的推广[J].力学学报,1987,19(2):156-164.XUE W X. Generalization of appell equation and Tzénoff equation of non-holonomic systems[J]. Acta Mechanica Sinica,1987,19(2):156-164.
[3]程丁龙.ЦЕНОВ方程对变质量非完整系统的推广[J].北京工业学院学报,1987,7(3):76-85.CHENG D L.Extension of Tzénoff equations to variable mass non-holonomic mechanical systems[J].Journal of Beijing Institute of Technology,1987,7(3):76-85.
[4]郑世旺,贾利群.非完整系统Tzénoff方程的Mei对称性和守恒量[J].物理学报,2007,56(2):661-665.ZHENG S W,JIA L Q.Mei symmetry and conserved quantity of Tzénoff equations for nonholonomic systems[J].Acta Physica Sinica,2007,56(2):661-665.
[5] HIRSCH M W,SMALE S.Differential equations,dynamical systems and linear algebra[M].New York:Academic Press,1974.
[6] MCLACHLAN R I,QUISPEL G R W,ROBIDOUX N.Geometric integration using discrete gradients[J].Philosophical Transactions of the Royal Society A,1999,357(1754):1 021-1 045.
[7]梅凤翔,崔金超,吴惠彬.Birkhoff系统的梯度表示和分数维梯度表示[J].北京理工大学学报,2012,32(12):1 298-1 300.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(12):1 298-1 300.
[8]楼智美,梅凤翔.力学系统的二阶梯度表示[J].物理学报,2012,61(2):024502.LOU Z M,MEI F X. A second order gradient representation of mechanics system[J]. Acta Physica Sinica,2012,61(2):024502.
[9]梅凤翔,吴惠彬.广义Birkhoff系统的梯度表示[J].动力学与控制学报,2012,10(4):289-292.MEI F X,WU H B.A gradient representation for generalized Birkhoff system[J]. Journal of Dynamics and Control,2012,10(4):289-292.
[10] CHEN X W,ZHAO G L,MEI F X.A fractional gradient representation of the Poincaréequations[J].Nonlinear Dynamics,2013,73(1):579-582.
[11] CHEN X W,MEI F X.Constrained mechanical systems and gradient systems with strong Lyapunov functions[J].Mechanics Research Communications,2016,76:91-95.
[12] LUO S K,HE J M,XU Y L.Fractional Birkhoffian method for equilibrium stability of dynamical systems[J].International Journal of Non-Linear Mechanics,2016,78(1):105-111.
[13]李彦敏.变质量非完整力学系统的共形不变性[J].云南大学学报:自然科学版,2010,32(1):52-57.LI Y M.Conformal invariance for nonholonomic mechanical systems with variable mass[J]. Journal of Yunnan University:Natural Sciences Edition,2010,32(1):52-57.
[14] MARIN A M,ORTIZ R D,RODRIGUEZ J A.A generalization of a gradient system[J].International Mathematical Forum,2013,8:803-806.
[15]李彦敏,陈向炜,吴惠彬,等.广义Birkhoff系统的两类广义梯度表示[J].物理学报,2016,65(8):080201.LI Y M,CHEN X W,WU H B,et al.Two kinds of generalized gradient representations for generalized Birkhoff system[J].Acta Physica Sinica,2016,65(8):080201.