Study On Combined Gradient Systems And Nonholonomic Systems And Their Stability

In this paper,the stability of solutions for non-autonomous Birkhoff systems and Chetaev-type nonholonomic systems is studied based on the combined gradient system method.A class of weakly nonlinear coupled nonholonomic systems is simulated by Matlab calculation method.The Poincaré section of the system in phase space is observed.The graph judges its dynamic behavior.Firstly,the differential equations of four basic gradient systems and four generalized gradient systems and their properties are introduced.Secondly,a class of generalized combined gradient systems is constructed.The non-autonomous Birkhoff system and the non-autonomous generalized Birkhoff system are respectively represented as such generalized combined gradient systems under certain conditions.The non-autonomous Birkhoff system is determined by the properties of such generalized combined gradient systems.The stability of the solution.Thirdly,four types of triple-combined gradient systems are constructed.The stationary Chetaev-type non-holonomic systems are transformed into these four-type triple-combined gradient systems under certain conditions,and the stability of the solutions of these mechanical systems is studied by the properties of triple-combined gradient systems.Finally,the Lyapunov first method and the combined gradient method are used to analyze the multi-fixed point stability of a class of weakly nonlinear coupled nonholonomic systems.The Poincaré map is used to qualitatively understand that the motion of the system in phase space is quasi-periodic motion.