# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A perturbation method for computing the simplest normal forms of dynamical systems. (English) Zbl 1237.34074
Summary: A previously developed perturbation method is generalized for computing the simplest normal form (at each level of computation, the minimum number of terms are retained) of general $n$-dimensional differential equations. This “direct” approach, combining the normal form theory with center manifold theory in one unified procedure, can be used to systematically compute the simplest (or unique) normal form. Two particular singularities of the Jacobian of the system are considered in this paper: the first one is associated with one pair of purely imaginary eigenvalues (Hopf-type singularity), and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues (Hopf-zero-type singularity). The approach can be easily formulated and implemented using a computer algebra system. Maple programs have been developed in this paper which can be “automatically” executed by a user without the knowledge of computer algebra. A physical oscillator model is studied in detail to show the computational efficiency of the “direct” method, and the advantage of using the simplest normal form, which greatly simplifies the analysis on dynamical systems, in particular, for bifurcations and stability.
##### MSC:
 34C20 Transformation and reduction of ODE and systems, normal forms 37G05 Normal forms
Maple
Maple