An algorithm for computing a new normal form for dynamical systems
Autonomous dynamical systems \(\dot{x}=F(x)\) are considered, with \(x\in \mathbb{R}^n\), \(F(x)\) is a vector whose components are formal power series and \(F(0)=0\). A new formal normal form for system (1) is proposed which improves the classical normal forms in the sense that it is a further reduction of the classical normal forms. An algorithm for an effective computation of these normal forms is given. All computations are done in a field \(K\), if the coefficients of the system are in \(K\). As a particular case, if the matrix of the linear part is a companion matrix then the dynamical system can be reduced to a single differential equation. The method is applicable for both the nilpotent and the nonnilpotent cases. The algorithm is implemented in MAPLE V and many examples of the further reduced normal forms up to some finite order are obtained.
scientific article; zbMATH DE number 1445834
scientific article