View source for Dichotomy

The property that for a linear system of ordinary differential equations

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with bounded continuous coefficients, there are positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316004.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316005.png" /> such that there exists a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316006.png" /> for which

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(exponential dichotomy; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160011.png" />, one has ordinary dichotomy). The presence of exponential dichotomy is equivalent to saying that the inhomogeneous system

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has, for any bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160014.png" />, at least one bounded solution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160015.png" /> [[#References|[1]]]. The theory of dichotomy [[#References|[2]]], transferred to equations in Banach spaces, is also employed in the study of flows and cascades on smooth manifolds [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Perron,   "Stability of differential equations"  ''Math. Z.'' , '''32''' :  5  (1930)  pp. 703–728</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Scheffer,   "Linear differential equations and function spaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Daletskii,   M.G. Krein,   "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.V. Anosov,   "Geodesic flows on closed Riemann manifolds with negative curvature"  ''Proc. Steklov Inst. Math.'' , '''90'''  (1969)  ''Trudy Mat. Inst. Steklov.'' , '''90'''  (1967)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Oseledec,   "A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1969)  pp. 197–232  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 179–210</TD></TR></table>