# 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)