Bounded solutions of linear almost periodic differential equations

The paper deals with bounded (on $\mathbb R_+$ or $\mathbb R$) solutions of the equation $\dot x=\mathcal A(t)x$ with recurrent (almost periodic) coefficients. We show that the zero solution of this equation is uniformly stable (bistable) if and only if all its solutions and the solutions of its limit equations are bounded on $\mathbb R_+$ ($\mathbb R$). These results are generalizations of the well-known theorem of Cameron–Johnson.