Finding NHIM in 2 and 3 degrees-of-freedom with H\'enon-Heiles type potential

We present the capability of Lagrangian descriptors for revealing the high dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include normally hyperbolic invariant manifolds and their stable and unstable manifolds, and act as codimenision-1 barriers to phase space transport. The method is applied to classical two and three degrees-of-freedom Hamiltonian systems which have implications for myriad applications in physics and chemistry.Comment: 15 pages, 6 figures. This manuscript is better served as dessert to the main course: arXiv:1903.1026

A computer-assisted proof of symbolic dynamics in Hyperion's inner rotation model

The rotation of Hyperion is often modelled by equations of motion of an ellipsoidal satellite. The model is expected to be chaotic for large range of parameters. The paper contains a rigorous computer-assisted proof of the existence of symbolic dynamics in its dynamics by the use of CAPD C++ library.Comment: 14 pages, 11 figure

An improved cosmic crystallography method to detect holonomies in flat spaces

A new, improved version of a cosmic crystallography method for constraining cosmic topology is introduced. Like the circles-in-the-sky method using CMB data, we work in a thin, shell-like region containing plenty of objects. Two pairs of objects (quadruplet) linked by a holonomy show a specific distribution pattern, and three filters of \emph{separation, vectorial condition}, and \emph{lifetime of objects} extract these quadruplets. Each object $P_i$ is assigned an integer $s_i$, which is the number of candidate quadruplets including $P_i$ as their members. Then an additional device of $s_i$-histogram is used to extract topological ghosts, which tend to have high values of $s_i$. In this paper we consider flat spaces with Euclidean geometry, and the filters are designed to constrain their holonomies. As the second filter, we prepared five types that are specialized for constraining specific holonomies: one for translation, one for half-turn corkscrew motion and glide reflection, and three for $n$-th turn corkscrew motion for $n=4, 3,$ and 6. {Every multiconnected space has holonomies that are detected by at least one of these five filters.} Our method is applied to the catalogs of toy quasars in flat $\Lambda$-CDM universes whose typical sizes correspond to $z\sim 5$. With these simulations our method is found to work quite well. {These are the situations in which type-II pair crystallography methods are insensitive because of the tiny number of ghosts. Moreover, in the flat cases, our method should be more sensitive than the type-I pair (or, in general, $n$-tuplet) methods because of its multifilter construction and its independence from $n$.}Comment: 12 pages, 8 figures, accepted for publication in A&A (2011