Integrable perturbations of saddle singularities of rank 0 of integrable Hamiltonian systems

We study the stability property of singularities of integrable Hamiltonian systems under integrable perturbations. It is known that among singularities of corank , only singularities of complexity are stable. As it turns out, in the case of two degrees of freedom, there are both stable and unstable singularities of rank and complexity . A complete list of singularities of saddle-saddle type of complexity 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 under component-wise perturbations. Using this criterion, in the case of two degrees of freedom, for each of the 39 singularities of complexity 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.

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