Related papers: Finding extremal periodic orbits with polynomial o…
We consider the problem of finding the shortest possible period for an exactly periodic solution to some given autonomous ordinary differential equation. We show that, given a pair of Lyapunov-like observable functions defined over the…
Piecewise smooth maps are known to exhibit a wide range of dynamical features including numerous types of periodic orbits. Predicting regions in parameter space where such periodic orbits might occur and determining their stability is…
Periodic orbits (POs) play a central role in the circular restricted three-body problem (CRTBP). This paper introduces a method to search for POs by identifying single- and multiple-revolution fixed points in chosen Poincare maps that…
Non-commutative polynomial optimization (NPO) problems seek to minimize the state average of a polynomial of some operator variables, subject to polynomial constraints, over all states and operators, as well as the Hilbert spaces where…
These notes were written during the 9th and 10th sessions of the subject Dynamical Systems II coursed at DTU (Denmark) during the Winter Semester 2015-2016, and later extended in February 2017. They aim to provide students with a…
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based…
Periodic orbits are fundamental to understand the dynamics of nonlinear systems. In this work, we focus on two aspects of interest regarding periodic orbits, in the context of a dissipative mapping, derived from a prototype model of a…
A dynamical system description of the transition process in shear flows with no linear instability starts with a knowledge of exact coherent solutions, among them travelling waves (TWs) and relative periodic orbits (RPOs). We describe a…
We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed…
Many extrasolar systems possessing planets in mean-motion resonance or resonant chain have been discovered to date. The transit method coupled with transit timing variation analysis provides an insight into the physical and orbital…
Many of exoplanetary systems consist of more than one planet and the study of planetary orbits with respect to their long-term stability is very interesting. Furthermore, many exoplanets seem to be locked in a mean-motion resonance (MMR),…
In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the…
A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex…
In Hamiltonian systems subjected to periodic perturbations the stable and unstable manifolds of the unstable periodic orbits provide the dynamical "skeleton" that drives the mixing process and bounds the chaotic regions of the phase space.…
In this manuscript, we consider finitely many maps, all of which are defined on a smooth compact measure space, with at least one map in the collection having degree strictly bigger than 1. Working with random dynamics generated by this…
In this paper, we propose POTATOES (Partitioning OverfiTting AuTOencoder EnSemble), a new method for unsupervised outlier detection (UOD). More precisely, given any autoencoder for UOD, this technique can be used to improve its accuracy…
Using the Lorenz equations, we have investigated whether unstable periodic orbits (UPOs) associated with a strange attractor may predict the occurrence of the robust sharp peaks in histograms of some experimental chaotic time series.…
We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial…
A general, variational approach to derive low-order reduced systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes the more classical notions of invariant or slow manifold when…
In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the…