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Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 6, Pages 64–81 (Mi izv826)  

On quasiperiodic solutions of the matrix Riccati equation

V. S. Pronkin


Abstract: 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.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 43:3, 455–470

Bibliographic databases:

UDC: 517.925.52+517.923
MSC: 34A34, 34C20, 34C28
Received: 17.04.1992

Citation: V. S. Pronkin, “On quasiperiodic solutions of the matrix Riccati equation”, Izv. RAN. Ser. Mat., 57:6 (1993), 64–81; Russian Acad. Sci. Izv. Math., 43:3 (1994), 455–470

Citation in format AMSBIB
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\by V.~S.~Pronkin
\paper On quasiperiodic solutions of the matrix Riccati equation
\jour Izv. RAN. Ser. Mat.
\yr 1993
\vol 57
\issue 6
\pages 64--81
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994IzMat..43..455P}
\transl
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 43
\issue 3
\pages 455--470
\crossref{https://doi.org/10.1070/IM1994v043n03ABEH001575}
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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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