Research on the properties of some planar polynomial differential equations (Q426631)

Consider the differential equation NEWLINE\[NEWLINEx' = X(t,x), NEWLINE\]NEWLINE where \(X\) is a differentiable function. Let \(\phi(t;t_0,x_0)\) denote the general solution. The reflecting function is defined by \(F(t,x)=\phi(-t;t,x)\). If the equation is \(2\omega\)-periodic in \(t\) then the Poincaré mapping is \(T(x)=F(-\omega,x)\). Hence, a solution \(\phi(t;-\omega,x_0)\) is \(2\omega\)-periodic if and only if \(x_0\) is a fixed point of \(T\). The author uses the method of reflecting functions to study the behavior of solutions and gives sufficient conditions for the equations to have linear or fractional reflecting functions. The results are used to derive sufficient conditions for a center of polynomial two-dimensional systems.