Existence, persistence and structure of integral manifolds in the neighbourhood of a periodic solution of autonomous differential systems. (English) Zbl 0611.34045

The authors consider the existence and structure of integral manifolds near a periodic solution of a system of autonomous ordinary differential equations (*) \(dx/dt=f(x,\lambda)\) and their persistence when \(\lambda\) changes in a metric space. To this end special local coordinates are constructed in a neighbourhood of the family of periodic solutions under consideration. So they obtain a nonautonomous system of differential equations whose qualitative behaviour near the stationary solution at the origin uniquely defines the topological structure of the trajectories of the system (*). The authors investigate existence, persistance and structure of integral manifolds of the derived nonautonomous system and prove that these integral manifolds are homeomorphic either to \(R^ k\times S^ 1\) where \(S^ 1\) denotes the unit circle in \(R^ 2\), or to \(R^{k-1}\times {\mathcal M}^ 2\) where \({\mathcal M}^ 2\) is the Möbius strip.


34C45 Invariant manifolds for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37C10 Dynamics induced by flows and semiflows
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