ABSTRACT
The validity of the linearization procedure in the vicinity of a fixed point goes beyond Liapunov’s linear stability theorem but, necessarily, cannot be used in all situations. Since the structure of orbits near the fixed point can be regarded as a homeomorphic image of the orbits in the linearized problem, the stable, centre and unstable subspaces of the linear problem will be present in the nonlinear problem as local stable, centre and unstable manifolds. The extension of stability property from the linear case to the nonlinear situation is straightforward. It suffices to consider the same quadratic form for nonlinear system, provided the readers restrict ourselves to a sufficiently small neighbourhood of the fixed point so as to be able to disregard higher-order corrections to the time-derivative. The local Poincaré maps obtained choosing different sections are actually conjugated by the flow. This fact is direct consequence of the already mentioned fact that the image of a Poincaré section is Poincaré section.
