The asymptotics of the return map of a singular point with fixed Newton diagram
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We exhibit a large class of nondegenerate singular points in which necessary and sufficient conditions are given for monodromy. We compute the generalized first Lyapunov value, which is expressed in terms of the Newton diagram of the singular point. The computational algorithm proposed is based on writing the return map as the composition of transition mappings constructed using the diagram. The nonvanishing of the generalized first Lyapunov value is a sufficient condition for the existence of a focus. Bibliography: 8 titles.
KeywordsSingular Point Large Class Transition Mapping Computational Algorithm Newton Diagram
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- 1.A. D. Bryuno,A Local Method of Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979).Google Scholar
- 2.F. S. Berezovskaya, “A complex stationary point of a system in the plane: the structure of a neighborhood and the index,” Preprint, Pushchino, ONTI NTsBI (1978).Google Scholar
- 3.V. I. Arnol'd and Yu. S. Il'yashenko, “Ordinary Differential Equations. I,” in:Sovr. Probl. Matem. Fund. Napr. [in Russian], Vol. 1 ofItogi Nauki i Tekhniki, VINITI AN SSSR (1985), pp. 7–149.Google Scholar
- 4.A. Dyulak (Henri Dulac),On Limit Cycles [in Russian], Nauka, Moscow (1980).Google Scholar
- 5.V. I. Arnol'd,Ordinary Differential Equations, MIT Press, Cambridge (1973).Google Scholar
- 6.A. A. Andronov et al.,The Qualitative Theory of Second-order Dynamical Systems [in Russian], Nauka, Moscow (1956).Google Scholar
- 7.V. S. Samovol, “On the linearization of a system of differential equations in the neighborhood of a singular point,”Dokl. Akad. Nauk,206, No. 3, 545–548 (1972).Google Scholar
- 8.N. B. Medvedeva, “The first focus quantity of a compound monodromic singular point,”Tr. Semin. Petrov., No. 13, 106–122 (1988).Google Scholar