  We consider a dynamical system which has the hyperbolic structure along an
attracting invariant manifold $M$. The problem is whether every motion starting
in a neighborhood of $M$ possesses an asymptotic phase, i.e. eventually
approaches a particular motion on $M$. Earlier, positive solutions to the
problem were obtained under the condition that the decay rate of solutions
toward the manifold exceeds the decay rate of the solutions within the
manifold. We show that in our case the above condition is not necessary. To
prove that a neighborhood of $M$ is filled with motions for each of which there
exists an asymptotic phase we apply the Brouwer fixed point theorem. An
invariant foliation structure which appears in the neighborhood of $M$ is
discussed.
