Time-asymptotic dynamics of Hermitian Riccati Differential Equations

The matrix Riccati differential equation (RDE) raises in a wide variety of applications for science and applied mathematics. We are particularly interested in the Hermitian Riccati Differential Equation (HRDE). Radon’s lemma gives a solution representation to HRDE. Although solutions of HRDE may show the finite escape time phenomenon, we can investigate the time asymptotic dynamical behavior of HRDE by its extended solutions. In this paper, we adapt the Hamiltonian Jordan canonical form to characterize the time asymptotic phenomena of the extended solutions for HRDE in four elementary cases. The extended solutions of HRDE exhibit the dynamics of heteroclinic, homoclinic and periodic orbits in the elementary cases under some conditions.