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The property that for a linear system of ordinary differential equations

with bounded continuous coefficients, there are positive constants , , , and such that there exists a decomposition for which

(exponential dichotomy; if , one has ordinary dichotomy). The presence of exponential dichotomy is equivalent to saying that the inhomogeneous system

has, for any bounded continuous function , , at least one bounded solution on [1]. The theory of dichotomy [2], transferred to equations in Banach spaces, is also employed in the study of flows and cascades on smooth manifolds [4].


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



[a1] 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
How to Cite This Entry:
Dichotomy. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by R.A. Prokhorova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article