Analysis of the second order matrix Riccati equations

摘要

This paper is concerned with periodic solutions of 2x2 autonomous matrix Riccati differential equations. The author had given a necessary and sufficient condition for periodicity of solutions of matrix Riccati differential equations of general type and some examples. However, it is not so simple to verify whether this condition is satisfied or not. So this paper simplifies the verification by restricting to special cases. In particular, we show that there may exist periodic solutions for any case where the coefficient matrix of the linear part of the equation has complex eigenvalues if we choose an initial value suitably. Many examples having a periodic solution are also shown by systematic analysis; such examples are seldom seen in the literature.