The aim of this Research Note is to elucidate the behaviour of a dynamical system in the neighbourhood of a fixed point or invariant torus (in the case of a diffeomorphism) or an equilibrium (in the case of an ordinary differential equation). It is shown that the space has both a horizontal and vertical foliation. Each leaf in the vertical foliation consists of the initial points of those orbits which are asymptotic to a particular orbit on the centre-stable manifold. This structure leads to a linearization of the system.
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The aim of this Research Note is to elucidate the behaviour of a dynamical system in the neighbourhood of a fixed point or invariant torus (in the case of a diffeomorphism) or an equilibrium (in the case of an ordinary differential equation). It is shown that the space has both a horizontal and vertical foliation. Each leaf in the vertical foliation consists of the initial points of those orbits which are asymptotic to a particular orbit on the centre-stable manifold. This structure leads to a linearization of the system.
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