The Reflective Matrix Of Third-order Linear Differential Systems

Many of the issues in Physics and Engineering Technology can be converted into discussing the behavior problems of the solutions of the periodic linear differential system a number of practical ways of periodic motion described by nonlinear differential equations are also discussed around the system (1). In short, in terms of theory but a practical context, the stability of the solutions to periodic linear differential equations are important for the research.In [4] , using Lyapunov transformation ,the periodic linear system (1) is converted into a constant coefficient system to study the behavior of its solution, and to discuss the value of the characteristic equation and the characteristics for the system stability, which is helpful for studying periodic system. But, it's difficult to manipulate for the methods'limitation. In 1980, The Russian mathematician Mironenko first established the method of reflecting function which provided a new way to study the geometric properties of the solutions of the periodic system. the theory of reflecting function provided a new method for looking for the PoincaréMapping of system. After years of intensive research, experts have made many new findings, which offer the new theoretical basis and evaluation criteria for further explaining the complex motion law of the objects .On the Basis of work which has been done in literature [23] [33] ,this article will further study the form of the reflection matrix of third-order linear differential system.Firstly,it discusses the reflctive matrix F (t)of ( )when to meet .Secondly, further studies F (t) when to meet F ( t)FT (t)=β2(t)E, problem into studying the reflective matric G (t)of the system when to meet G (t )GT (t)=E.And according to the definition and nature of the reflective matrix,we have introduced the specific expression of G (t)and the relationship between G (t) and the coefficient matrix,then discusses the determining method of the stability of periodic solution.Finally, we give some examples to verify the correctness of the conclusion above.