In the paper, the problems of identification of focus or center for a class of nonlinear systems are studied. As results, the identifying rule for the type of singular point of the system and the calculating formula f...In the paper, the problems of identification of focus or center for a class of nonlinear systems are studied. As results, the identifying rule for the type of singular point of the system and the calculating formula for the focus value are obtained by the classical theory of Poincare and Lyapunov. At the end, the numerical simulation for a polynomial system of seven-degree are showed to corroborate the theoretical results of the method.展开更多
Fix a collection of polynomial vector fields on R3with a singularity at the origin,for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue. Some such systems admit a loc...Fix a collection of polynomial vector fields on R3with a singularity at the origin,for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue. Some such systems admit a local analytic first integral,which then defines a local center manifold of the system. Conditions for existence of a first integral are given by the vanishing certain polynomial or rational functions in the coefficients of the system called focus quantities. In this paper we prove that the focus quantities have a structure analogous to that in the two-dimensional case and use it to formulate an efficient algorithm for computing them.展开更多
文摘In the paper, the problems of identification of focus or center for a class of nonlinear systems are studied. As results, the identifying rule for the type of singular point of the system and the calculating formula for the focus value are obtained by the classical theory of Poincare and Lyapunov. At the end, the numerical simulation for a polynomial system of seven-degree are showed to corroborate the theoretical results of the method.
基金VR acknowledges the support of this work by the Slovenian Research Agency and by a Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme,FP7-PEOPLE-2012-IRSES-316338
文摘Fix a collection of polynomial vector fields on R3with a singularity at the origin,for every one of which the linear part at the origin has two pure imaginary and one non-zero eigenvalue. Some such systems admit a local analytic first integral,which then defines a local center manifold of the system. Conditions for existence of a first integral are given by the vanishing certain polynomial or rational functions in the coefficients of the system called focus quantities. In this paper we prove that the focus quantities have a structure analogous to that in the two-dimensional case and use it to formulate an efficient algorithm for computing them.
