Abstract
Consider an autonomous differential system x ̇ = f(x) of dimension n that admits a k-dimensional invariant manifold Γ in x-space represented by a k-parameter family of periodic solutions. Then k characteristic multipliers of the corresponding variational equation are equal to 1. Under the hypothesis that none of the remaining n − k characteristic multipliers has modulus 1, the behavior of the solutions of x ̇ = f(x) near Γ is investigated. The main results are the description of a neighborhood of Γ in terms of stable and unstable manifolds and the statement that whenever a solution of x ̇ = f(x) approaches the invariant manifold Γ as t → ∞, it tends to Γ with asymptotic amplitude and phase.
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