Dichotomy

The property that for a linear system of ordinary differential equations

$$ \dot{x} = A ( t) x ,\ x \in E ^ {n} ,\ t \geq 0 , $$

with bounded continuous coefficients, there are positive constants $ K $, $ L $, $ \alpha $, and $ \beta $ such that there exists a decomposition $ E ^ {n} = E ^ {m} + E ^ {n-} m $ for which

$$ x ( 0) \in E ^ {m} \Rightarrow \| x ( t) \| \leq K \| x ( \tau ) \| \ \mathop{\rm exp} [ - \alpha ( t - \tau ) ] , $$

$$ t \geq \tau \geq 0 ; $$

$$ x ( 0) \in E ^ {n-} m \Rightarrow \| x ( t) \| \leq L \| x ( \tau ) \| \mathop{\rm exp} [ - \beta ( \tau - t ) ] , $$

$$ \tau \geq t \geq 0 $$

(exponential dichotomy; if $ \alpha = \beta = 0 $, one has ordinary dichotomy). The presence of exponential dichotomy is equivalent to saying that the inhomogeneous system

$$ \dot{x} = A ( t) x + f ( t) $$

has, for any bounded continuous function $ f ( t) $, $ t \geq 0 $, at least one bounded solution on $ [ 0 , \infty ) $[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].

References

[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)

References

[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