# 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 $[ 0 , \infty )$. The theory of dichotomy , transferred to equations in Banach spaces, is also employed in the study of flows and cascades on smooth manifolds .

How to Cite This Entry:
Dichotomy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dichotomy&oldid=46647
This article was adapted from an original article by R.A. Prokhorova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article