A Study On The Qualitative Relationship Between Several Dynamical Systems And Time - Varying Systems

When we study the motion laws of objects in the objective world, the changing rules of the species, the changing curves of stock market, and the orbits of the satellite, we are inseparable from the qualitative characteristic researches of differential models. These differential models from practical production problems have autonomous and nonautonomous differential system. For autonomous differential system, a large number of mathematicians from both at home and abroad have spent a lot of enthusiasm to solve many problems and achieved rich results. While for the solution of the time-varying differential model, how to study its qualitative characteristic? It makes the research extremely difficult because of the complexity of the model. But can we turn a time-varying system into autonomous system to research? The answer is yes. Lyapunov transform realizes the time-varying periodic linear systems turning into linear system with constant coefficients to research. So for nonlinear periodic system or time-varying system, can we transform into autonomous system to research?In the 1980s, Mironenko professor have established the theory of reflecting function. By applying the theory, we can establish the qualitative equivalent relation of two differential systems. If two periodic differential systems are equivalent, then their periodic solutions are at the same qualitative behavior.If a non periodic time-varying system is equivalent with a periodic time-varying system, the solutions which satisfy some boundary value problems are one to one correspondence. In this paper, I mainly study the qualitative equivalence between several kinds of nonautonomous nonlinear differential systems and autonomous systems.We will give some sufficient and necessary conditions of their equivalence, and apply the conclusions to study the geometric properties of the periodic solutions.The first part of the paper, we mainly study the differential system with the approximate linear equations we study the necessary and sufficient conditions of (1) and (2) with the same reflecting function, and the structure forms of X(t,x,y),Y(t,x,y).In particular, when X(t,x,y) Y(t,x,y) are quadratic polynomial functions, they have the necessary and sufficient conditions of (1) and (2) with the same reflecting function. We apply the conclusions to study the geometric properties of the periodic solutions.In addition, we give the necessary and sufficient conditions when the n-order polynomial differential system is equivalent to the equations (2), and the geometric properties of their periodic solutions. Where aij(t), bij (t) (z+y=k,k= 1,2,3, …,n) are continuously differentiable functions, and α1, (t), a2 (t)are continuously differentiable odd functions.At the second part, we mainly study the equivalence of a n-order time-varying polynom-ial differential system x=X1(x,y)+X2(x,y)+α1(t)X1(x,y)+a2(t)X2(x,y) (4)With a nonlinear autonomous differential system (5) give the necessary and sufficient conditions when (4) is equivalent to (5) and the structure forms of the system (5), when...