Calculating the Poincaré map for two-dimensional periodic systems and Riccati equations (Q2064223)

For the study of periodic solutions of non-autonomous periodic systems \(dx/dt=X(t,x)\), A.I. Mironov introduced in 1984 the so-called reflecting function which can be used to represent the corresponding Poincaré map. In the present paper, the authors consider two-dimensional linear non-autonomous systems \[ \frac{dx}{dt} = P(t)x. \tag{1} \] They derive two groups of conditions on the entries of the matrix \(P\) such that the reflecting function can be determined explicitely. In case that \(P\) is periodic, the corresponding Poincaré map can also be determined explicitely, which yields analytic conditions for the existence of periodic solutions of (1). Additionally, the authors show that the same approach can be applied to the Riccati equation \[ \frac{dx}{dt} = a(t)+b(t)x + c(t)x^2. \]