Orbital Hypernormal Forms

In this paper, we analyze the problem of determining orbital hypernormal forms—that is, simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in state variables. We show computation forms carried out degree by using quasi-homogeneous expansions vector field means reduced near-identity transformations, achieving important reduction computational effort. Moreover, although form procedure is essentially nonlinear nature, our results are characterized linear operators. Some applications considered: case planar fields, with emphasis on Takens–Bogdanov singularity.