Newton Equation Study Of Periodic Solutions

Differential equation has been widely used in practice, most issues related to the rate of the change can use differential equation model to study. To find out the law of the system’s change with respect to time, we need to discuss the periodic solution of differential equation. By solving the equations or qualitative analysis, we study the behaviors of the solutions of differential equations. When it is a2ω-periodic system, we usually use the Poincare mapping to study the periodic solution. This article mainly used Mironenko’s reflective function theory to study periodic solution of the linear Newton equation. Then we apply the results obtained to establish the Poincare mapping of the linear Newton equation and study the qualitative behaviors of the periodic solutions.First of all, We studied the structure of reflection matrix F(t) of differential system x’=A(t)x which is equivalent to the equation xn+a(t)x’+b(t)x=0, and we concluded that the F(t)=H(t)eα(t), det H(t)=1,.h1and h3are odd function. Then we studied the sufficient conditions of the differential system x’=B(t)x has the reflective function shaped like and which satisfies h1(-t)=h1eβ or h2(t)=-h3(t). Here we give the number of the periodic solutions of the system x’=B(t)x and the qualitative behavior when x’=B(t) x is a2ω-periodic system.Further we discussed the sufficient conditions of the Non linear Newton equation x"+a(t)x’+b(t)x=φ(t) and its equivalent system x’=A(t)x+Φ(t) has the reflective function F (t,x,y)=F1(t)x+F0(t). Applying the results we got the qualitative behavior of the periodic solutions of the Non linear system when it is a2ω-periodic system.Finally, one simple example is exploited to show the upper conclusions.