On quasiperiodic solutions of the matrix Riccati equation

V. S. Pronkin

Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 455-470

Résumé

The matrix Riccati equation \begin{equation} \dot X+Xf(t)X+(A_0+A(t))X+\lambda l(t)=0 \tag{1} \end{equation} is considered, where $X$ is an unknown vector, $A_0$ is a constant diagonal matrix whose elements are pairwise distinct imaginary numbers, the coefficients $f(t)$, $A(t)$, and $l(t)$ are matrices whose elements are Arnold'd functions, and $\lambda$ is a small complex parameter. Newton's method is used to prove that (1) has quasiperiodic solutions with the exception of finitely many rays. By using the quasiperiodic solutions obtained it is proved that, with the exception of finitely many rays, the system of differential equations $\dot X=(P_0+\lambda P(t))X$ is reducible, where $P(t)$ is a matrix whose elements are Arnol'd functions, and $\lambda$ is a small complex parameter.