Asymptotic phase for flows with exponentially stable partially hyperbolic invariant manifolds

We consider an autonomous system admitting an invariant manifold $\mathcal{M}$. The following questions are discussed: (i) what are the conditions ensuring exponential stability of the invariant manifold? (ii) does every motion attracting by $\mathcal{M}$ tend to some motion on $\mathcal{M}$ (i.e. have an asymptotic phase)? (iii) what is the geometrical structure of the set formed by orbits approaching a given orbit? We get an answer to (i) in terms of Lyapunov functions omitting the assumption that the normal bundle of $\mathcal{M}$ is trivial. An affirmative answer to (ii) is obtained for invariant manifold $\mathcal{M}$ with partially hyperbolic structure of tangent bundle. In this case, the existence of asymptotic phase is obtained under new conditions involving contraction rates of the linearized flow in normal and tangential to $\mathcal{M}$ directions. To answer the question (iii), we show that a neighborhood of $\mathcal{M}$ has a structure of invariant foliation each leaf of which corresponds to motions with common asymptotic phase. In contrast to theory of cascades, our technique exploits the classical Lyapunov–Perron method of integral equations.