Abstract:
It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.
\Bibitem{RadRom08}
\by M.~Radnovi{\'c}, V.~Rom-Kedar
\paper Foliations of isonergy surfaces and singularities of curves
\jour Regul. Chaotic Dyn.
\yr 2008
\vol 13
\issue 6
\pages 645--668
\mathnet{http://mi.mathnet.ru/rcd606}
\crossref{https://doi.org/10.1134/S1560354708060117}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2465729}
\zmath{https://zbmath.org/?q=an:1229.37074}
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https://www.mathnet.ru/eng/rcd/v13/i6/p645
This publication is cited in the following 13 articles:
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E. A. Kudryavtseva, L. M. Lerman, “Bifurcations in Integrable Systems with Three Degrees of Freedom. I”, Proc. Steklov Inst. Math., 327 (2024), 130–207
Vladimir Dragović, Sean Gasiorek, Milena Radnović, “Billiard Ordered Games and Books”, Regul. Chaotic Dyn., 27:2 (2022), 132–150
Pnueli M. Rom-Kedar V., “On the Structure of Hamiltonian Impact Systems”, Nonlinearity, 34:4 (2021), 2611–2658
M. P. Kharlamov, P. E. Ryabov, I. I. Kharlamova, “Topological Atlas of the Kovalevskaya–Yehia Gyrostat”, J. Math. Sci. (N. Y.), 227:3 (2017), 241–386
Souhail Wahid, Hedi Khammari, Mohammed Faouzi Mimouni, 2015 16th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), 2015, 480
M.P. Kharlamov, “Phase topology of one system with separated variables and singularities of the symplectic structure”, Journal of Geometry and Physics, 87 (2015), 248
A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318
V. Dragović, M. Radnović, “Integrable billiards and quadrics”, Russian Math. Surveys, 65:2 (2010), 319–379
V Rom-Kedar, D Turaev, “The symmetric parabolic resonance”, Nonlinearity, 23:6 (2010), 1325
K Efstathiou, D Sugny, “Integrable Hamiltonian systems with swallowtails”, J. Phys. A: Math. Theor., 43:8 (2010), 085216
Eli Shlizerman, Vered Rom-Kedar, “Classification of solutions of the forced periodic nonlinear Schrödinger equation”, Nonlinearity, 23:9 (2010), 2183
V. Dragović, M. Radnović, “Bifurcations of Liouville tori in elliptical billiards”, Regul. Chaotic Dyn., 14:4 (2009), 479–494
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