Necessary And Sufficient Conditions For The Time Reversibility Of Cubic Systems With Respect To Quadratic Involution

In this thesis,we consider the quadratic involution time reversibility and the central problem for the system of polynomial central focus type.On the basis of the condition that the quadratic involution time is invertible in the plane quadratic system,by changing the order of variables,by using the classification discussion of regular chains and semi-algebraic sets and so on,the necessary and sufficient conditions for the complete cubic system to be reversible with respect to the quadratic involution time are obtained.At the same time,the necessary and sufficient conditions for the equilibrium point to be the center of the corresponding system are obtained under the reversible condition of the quadratic involution time of the cubic system.