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This article is cited in 9 scientific papers (total in 9 papers)
Integrable perturbations of saddle singularities of rank 0 of integrable Hamiltonian systems
A. A. Oshemkov, M. A. Tuzhilin Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Abstract:
We study the stability property of singularities of integrable Hamiltonian systems under integrable perturbations. It is known that among singularities of corank $1$, only singularities of complexity $1$ are stable. As it turns out, in the case of two degrees of freedom, there are both stable and unstable singularities of rank $0$ and complexity $2$. A complete list of singularities of saddle-saddle type of complexity $2$ is known and it consists of 39 pairwise non-equivalent singularities. In this paper we prove a criterion for the stability of multi-dimensional saddle singularities of rank $0$ under component-wise perturbations. Using this criterion, in the case of two degrees of freedom, for each of the 39 singularities of complexity $2$ we obtain an answer to the question of whether this singularity is component-wise stable. For a singularity of saddle-saddle type we analyse the connection between the stability property and the characteristics of its loop molecule.
Bibliography: 27 titles.
Keywords:
integrable Hamiltonian systems, momentum map, nondegenerate singularities, stability.
Received: 20.11.2017 and 18.12.2017
Citation:
A. A. Oshemkov, M. A. Tuzhilin, “Integrable perturbations of saddle singularities of rank 0 of integrable Hamiltonian systems”, Mat. Sb., 209:9 (2018), 102–127; Sb. Math., 209:9 (2018), 1351–1375
Linking options:
https://www.mathnet.ru/eng/sm9040https://doi.org/10.1070/SM9040 https://www.mathnet.ru/eng/sm/v209/i9/p102
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Abstract page: | 420 | Russian version PDF: | 57 | English version PDF: | 15 | References: | 32 | First page: | 21 |
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