On bifurcations of two-dimensional diffeomorphisms with quadratic homoclinic tangency to a nonhyperbolic fixed point

Title	On bifurcations of two-dimensional diffeomorphisms with quadratic homoclinic tangency to a nonhyperbolic fixed point
Authors	O. V. Gordeeva¹, V. E. Gordeev¹ ¹Lobachevsky State University of Nizhny Novgorod
Annotation	We consider a two-parameter family $f_mu$ of two-dimensional diffeomorphisms, where $mu=(mu_1,mu_2)$, such that for $mu=0$ the diffeomorphism $f_0$ has a nonhyperbolic fixed point of arbitrary finite order $ngeq 1$ of degeneracy, the invariant stable and unstable manifolds of this point has quadratically tangent and for $mu_1>0$ the fixed point becomes a hyperbolic saddle. We study some bifurcations of the single-round periodic orbits. It is shown that the first return map reduces to the well-known parabola map.
Keywords	saddle-node, nonhyperbolic saddle, homoclinic orbit, parabola map
Citation	Gordeeva O. V., Gordeev V. E. ''On bifurcations of two-dimensional diffeomorphisms with quadratic homoclinic tangency to a nonhyperbolic fixed point'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 29-31, 2025). Saransk: SVMO Publ, 2025. - pp. 79-81. Available at: https://conf.svmo.ru/files/2025/papers/paper15.pdf. - Date of access: 30.08.2025.