On the application of normal forms near attracting fixed points of dynamical systems

It is shown on an integrable example in the plane, that normal form solutions need not converge over the full basin of attraction of fixed points of dissipative dynamical systems. Their convergence breaks down at a singularity in the complex time plane of the exact solutions of the problem. However, as is demonstrated on a nonintegrable example with 3-dimensional phase space, the region of convergence of normal forms can be large enough to extend almost to a nearby hyperbolic fixed point, whose invariant manifolds “embrace” the attracting fixed point forming a complicated basin boundary. Thus, in such problems, normal forms are shown to be useful in practice, as a tool for finding large regions of initial conditions for which the solutions are necessarily attracted to the fixed point at t → ∞.