Bifurcations of integrable mechanical systems with magnetic field on surfaces of revolution

E. A. Kudryavtseva ; A. A. Oshemkov

Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 244-265.

Résumé

On a surface homeomorphic to $2$-sphere, we study a natural mechanical system with a magnetic field that is invariant under the $S^1$-action. For singular points of rank $0$ of the momentum mapping, a criterion for non-degeneracy is obtained, the type of non-degenerate singular points (center-center and focus-focus) is determined, bifurcations of typical degenerate singular points are described (integrable Hamiltonian Hopf bifurcation of two types). For families of singular circles of rank $1$ of the momentum mapping (consisting of relative equilibriums of the system) their parametric representation is obtained, nondegeneracy criterion is proved, the type of nondegenerate (elliptic and hyperbolic) and typical degenerate (parabolic) singular circles is determined. The parametric representation of the bifurcation diagram of the momentum mapping is obtained. Geometric properties of the bifurcation diagram and the bifurcation complex are described in the case when the functions defining the system are in general position. The topology of nonsingular isoenergy $3$-dimensional manifolds is determined, the topology of the Liouville foliation on them is described up to the rough Liouville equivalence (in terms of Fomenko's atoms and molecules). The “splitting” hyperbolic singularities of rank $1$ are described, which are topologically unstable bifurcations of the Liouville foliation.

Détail

Keywords: integrable system, Liouville foliation, bifurcation diagram, surface of revolution, magnetic field.

Bibliographie

[1] Besse A. L., Manifolds all of whose Geodesics are Closed, Springer, Berlin, 1988 | MR

[2] Kantonistova E. O., “Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution”, Sb. Math., 207:3 (2016), 358–399 | DOI | MR | Zbl

[3] Kozlov I., Oshemkov A., “Integrable systems with linear periodic integral for the Lie algebra $e(3)$”, Lobachevskii J. Math., 38:6 (2017), 1014–1026 | DOI | MR | Zbl

[4] Van der Meer J.-C., The Hamiltonian Hopf bifurcation, Lecture Notes in Mathematics, 1160, Springer-Verlag, Berlin, 1985 | MR | Zbl

[5] Efstathiou K., Giacobbe A., “The topology associated with cusp singular points”, Nonlinearity, 25 (2012), 3409–3422 | DOI | MR | Zbl

[6] Fomenko A.T., “Topological invariants of Liouville integrable Hamiltonian systems”, Funct. Anal. Appl., 22:4 (1988), 286–296 | DOI | MR | Zbl

[7] Bolsinov A., Guglielmi L., Kudryavtseva E., “Symplectic invariants for parabolic orbits and cusp singularities of integrable systems”, Philos. Trans. A Math. Phys. Eng. Sci., 376:2131 (2018), 20170424 | DOI | MR | Zbl

[8] Kudryavtseva E. A., Lakshtanov E. L., “Classification of singularities and bifurcations of critical points of even functions”, Topological Methods in the Theory of Integrable Systems, eds. Bolsinov A. V., Fomenko A. T., Oshemkov A. A., Cambridge Scientific Publishers, Cambridge, 2006, 173–214 | MR

[9] Bolsinov A. V., Matveev S. V., Fomenko A. T., “Topological classification of integrable Hamiltonian systems with two degrees of freedom. List of systems of small complexity”, Russian Math. Surveys, 45:2 (1990), 59–94 | DOI | MR | Zbl

[10] Bolsinov A. V., Fomenko A. T., Integrable Hamiltonian systems: geometry, topology, classification, Chapman Hall / CRC, Boca Raton–London–N.Y.–Washington, 2004 | MR | Zbl