Abstract
We study vector fields of the plane preserving the Liouville form. We present their local models up to the natural equivalence relation and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given according to a relation stricter than contact equivalence. In addition, we discuss their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given.
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Sternberg, Sh., Local Cn Transformations of the Real Line, Duke Math. J., 1957, no. 24, pp. 97–102.
Boothby, W.M. and Wang, H.C., On Contact Manifolds, Ann. of Math. (2), 1958, no. 68, pp. 721–734.
Bröcker, Th. and Lander, L., Differentiable Germs and Catastrophes, Cambridge: Cambridge Univ. Press, 1975.
Lychagin, V.V., On Sufficient Orbits of a Group of Contact Diffeomorphisms, Math. USSR-Sb., 1977, vol. 33, no. 2, pp. 223–242; see also: Mat. Sb. (N. S.), 1977, vol. 104(146), no. 2(10), pp. 248–270, 335.
Damon, J., The Unfolding and Determinacy Theorems for Subgroup of A and K, Mem. Amer. Math. Soc., 1984, vol. 50, no. 306, 88 pp.
Chaperon, M., Géométrie différentielle et singularités des systèmes dynamiques, Astérisque, nos. 138–139, Paris: Société mathématique de France, 1986.
Arnold V. I., Gusein-Zade S. M., Varchenko A. N. Singularities of differentiable maps: In 2 vols., Monogr. Math., vols. 82&83, Basel: Birkhäuser, 1985, 1988.
Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.
Banyaga, A., de la Llave, R., and Wayne, C.E., Cohomology Equations Near Hyperbolic Points and Geometric Versions of Sternberg Linearization Theorem, J. Geom. Anal., 1996, vol. 6, no. 4, pp. 613–649.
Haller, G. and Mezić, I., Reduction of Three-Dimensional, Volume-Preserving Flows with Symmetry, Nonlinearity, 1998, vol. 11, no. 2, pp. 319–339.
Chaperon, M., Singularities in Contact Geometry, in Geometry and Topology of Caustics (CAUSTICS’ 02): Proc. of the Banach Center Symposium Dedicated to the Memory of Peter Slodowy (Warsaw, June 17–29, 2002), S. Janeczko, D. Siersma (Eds.), Banach Center Publ., vol. 62, Warsaw: Polish Acad. Sci., 2004, pp. 39–55.
Geiges, H., An Introduction to Contact Topology, Cambridge Stud. Adv. Math., vol. 109, Cambridge: Cambridge Univ. Press, 2008.
Dullin, H.R. and Meiss, J. D., Nilpotent Normal Form for Divergence-Free Vector Fields and Volume- Preserving Maps, Phys. D, 2008, vol. 237, no. 2, pp. 156–166.
Takahashi, M., A Sufficient Condition That Contact Equivalence Implies Right Equivalence for Smooth Function Germs, Houston J. Math., 2009, vol. 35, no. 3, pp. 829–833.
Kourliouros, K., Singularities of Functions on the Martinet Plane, Constrained Hamiltonian Systems and Singular Lagrangians, J. Dyn. Control Syst., 2015, vol. 21, no. 3, pp. 401–422.
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Anastassiou, S. Dynamical systems on the Liouville plane and the related strictly contact systems. Regul. Chaot. Dyn. 21, 862–873 (2016). https://doi.org/10.1134/S1560354716070091
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DOI: https://doi.org/10.1134/S1560354716070091