http://publications.mfo.de/handle/mfo/4389 Wed, 08 Apr 2026 15:51:56 GMT 2026-04-08T15:51:56Z http://publications.mfo.de/handle/mfo/4411 Transverse Foliations for Two-Degree-of-Freedom Mechanical Systems de Paulo, Naiara V.; Kim, Seongchan; Salomão, Pedro A. S.; Schneider, Alexsandro We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-2 Lyapunov orbits in the neck region about the rest points and a particular index-3 orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above 1/6. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction. NdP was partially supported by CAPES/MATH-AMSUD 88881.878892/2023-01. SK was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2025-16070003). A part of this work was done during SK’s visit to the Mathematisches Forschungsinstitut Oberwolfach (MFO) as an Oberwolfach Leibniz Fellow in 2020. SK cordially thanks the MFO for its excellent support and stimulating working atmosphere. PS acknowledges the support of the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai and the 2022 National Foreign Experts Program. PS was partially supported by FAPESP (2016/25053-8) and CNPq (306106/2016-7). PS was partially supported by the National Natural Science Foundation of China (grant number W2431007). PS thanks the support of the Shenzhen International Center for Mathematics - SUSTech. AS thanks the Instituto de Matemática Pura e Aplicada (IMPA) for the post-doc position. Part of this work was conducted during visits to the Southern University of Science and Technology (SUSTech) and the Kongju National University (KNU). AS thanks both institutes for their hospitality.; [MSC 2020] Primary 37J55; Secondary 53D35. Sun, 01 Mar 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4411 2026-03-01T00:00:00Z de Paulo, Naiara V. Kim, Seongchan Salomão, Pedro A. S. Schneider, Alexsandro We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-2 Lyapunov orbits in the neck region about the rest points and a particular index-3 orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above 1/6. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction. http://publications.mfo.de/handle/mfo/4390 On Constructing Small Subgraphs in the Budget-Constrained Random Graph Process Antoniuk, Sylwia; Espuny Díaz, Alberto; Petrova, Kalina; Stojaković, Miloš Consider the budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli, where each time an edge is offered through the (standard) random graph process we must irrevocably decide whether to "purchase" this edge or not, with our goal being to construct a graph which satisfies some property within a given time $t$ and while purchasing at most $b$ edges. We consider the problem of constructing graphs containing certain fixed small subgraphs. We provide an optimal strategy for building a graph which contains a copy of $K_4$, showing that budget $b=\omega(\max\{n^8/t^5,n^2/t\})$ suffices and that if $b=o(\max\{n^8/t^5,n^2/t\})$ then no strategy can a.a.s. produce a graph containing a copy of $K_4$. This resolves a problem raised by Iľkovič, León and Shu. More generally, we obtain analogously tight results for containing a wheel of any fixed size, or a graph consisting of a tree plus one additional universal vertex. We also tackle the problem of constructing graphs containing a copy of $K_5$, obtaining both lower and upper bounds on the optimal budget, though a gap remains in this case. This research was supported by the Oberwolfach Research Institute for Mathematics through its Oberwolfach Research Fellows (OWRF) program. S. Antoniuk was supported by Narodowe Centrum Nauki, grant 2024/53/B/ST1/00164. A. Espuny Díaz was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 513704762. K. Petrova was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101034413 . M. Stojaković was partly supported by the Science Fund of the Republic of Serbia, Grant #7462: Graphs in Space and Time: Graph Embeddings for Machine Learning in Complex Dynamical Systems (TIGRA), and partly supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (grants 451-03-33/2026-03/200125 & 451-03-34/2026-03/200125). Sun, 01 Feb 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4390 2026-02-01T00:00:00Z Antoniuk, Sylwia Espuny Díaz, Alberto Petrova, Kalina Stojaković, Miloš Consider the budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli, where each time an edge is offered through the (standard) random graph process we must irrevocably decide whether to "purchase" this edge or not, with our goal being to construct a graph which satisfies some property within a given time $t$ and while purchasing at most $b$ edges. We consider the problem of constructing graphs containing certain fixed small subgraphs. We provide an optimal strategy for building a graph which contains a copy of $K_4$, showing that budget $b=\omega(\max\{n^8/t^5,n^2/t\})$ suffices and that if $b=o(\max\{n^8/t^5,n^2/t\})$ then no strategy can a.a.s. produce a graph containing a copy of $K_4$. This resolves a problem raised by Iľkovič, León and Shu. More generally, we obtain analogously tight results for containing a wheel of any fixed size, or a graph consisting of a tree plus one additional universal vertex. We also tackle the problem of constructing graphs containing a copy of $K_5$, obtaining both lower and upper bounds on the optimal budget, though a gap remains in this case.