Abstract
In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularity.
Highlights
The theory of normal forms is a basic tool for the study of several problems in differential equations: bifurcations, analysis of stability, the center problem, the reversibility problem, the integrability problem, etc
An important fact is that the homological operator depends on the linearization matrix DFp0q, and the structure of the normal form is determined by this matrix
The main goal of this paper is to show that the simplest normal form using equivalence can be characterized by means of linear procedures, with reduced time-reparametrizations and near-identity transformations
Summary
The theory of normal forms is a basic tool for the study of several problems in differential equations: bifurcations, analysis of stability, the center problem, the reversibility problem, the integrability problem, etc. The use of quasi-homogeneous expansions gives rise to a theory, similar to the classical one, but the homological equation depends on the lowest-degree quasi-homogeneous term (called the principal part) of Fpxq, which plays the role of the linear part Another possibility of obtaining further simplifications in the classical normal form is based on the use of transformations in the state variables and in the time (i.e., one can use equivalence instead of conjugation). This procedure is essentially nonlinear, we show that the simplest normal form is characterized by means of a suitable linear homological operator (see Theorem 3).
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