Abstract:
The talk is devoted to bifurcations arising in integrable systems with two degrees of freedom, dependent on parameters. Examples of such systems include many integrable cases of rigid body dynamics. Suppose that an unperturbed Hamiltonian system (with a zero parameter value) has an equilibrium. This is a point at which the linear parts of the Hamilton function $H$ and the additional first integral are zero. Then, the quadratic parts of these functions at this point generate a pair of commuting Hamiltonian linear operators. These operators generate a commutative subalgebra (with respect to the commutator operation) of the Lie algebra $\mathrm{sp}(4,\mathbb R)$. An equilibrium is called non-degenerate if this commutative subalgebra satisfies two conditions: 1) it is two-dimensional, 2) it contains an operator with a simple spectrum. Non-degenerate equilibria are well known in any dimension: by the Eliasson–Wey theorem, they are given by a set of quadratic first integrals of the form
$f_i = (p_i^2+q_i^2)/2$ (elliptic type),
$f_j = p_jq_j$ (hyperbolic type),
$f_{2k-1} = \Re((p_{2k-1}+ip_{2k})(q_{2k-1}-iq_{2k}))$, $f_{2k} = \Im ((p_{2k-1}+ip_{2k})(q_{2k-1}-iq_{2k}))$ (focus-focus type) in some local curvilinear symplectic coordinate system.
The speaker introduced the notion of a semi-toric singularity of an integrable system. Together with L.M. Lerman, a semi-local classification of such singularities (in small neighborhoods of compact orbits of the Hamiltonian $\mathbb R^n$-action) was obtained (2024) for “generic” real-analytic integrable systems with $n=2,3$ degrees of freedom. It turned out that, in addition to non-degenerate singularities, singularities arise at which only one of the non-degeneracy conditions 1) and 2) is violated and the spectrum is resonant. Specifically, the spectrum contains either a pair of coinciding eigenvalues ??(a 1:1 or 1:-1 resonance, so condition 2) for the spectrum to be simple is violated) or a pair of commensurate eigenvalues ??(an m:n resonance, and condition 1) for the subalgebra to be two-dimensional is violated).
We compute resonances corresponding to (degenerate) equilibria of systems in rigid body dynamics. We give formulae for standard momentum maps describing many of these bifurcations. We will also describe semi-local and semi-global topological invariants for many bifurcations.
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