Related papers: Discrete cyclic systems and circle congruences
We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we…
We consider quadratic three-dimensional differential systems having a Hopf singular point. We study the cyclicity when the singular point is a center on the center manifold using higher order developments of the Lyapunov constants. As a…
The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works \cite{X-Z DCS1, X-Z DCS2}, we established, respectively, a…
Cluster integrable systems are a broad class of integrable systems modelled on bipartite dimer models on the torus. Many discrete integrable dynamics arise by applying sequences of local transformations, which form the cluster modular group…
The dynamical properties, especially the symmetric orbits, of the 2-parameter family of circle maps called off-center reflection is studied.
This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space $\mathbb{E}_3$ in quantum mechanics. In contrast to the growing interest in complex…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a…
An overview is given of basic models combining discreteness in their linear parts (i.e. the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the…
A contractible simplicial complex is constructed that parametrizes different ways of representing a fixed one-dimensional homology class in a closed orientable surface by isotopy classes of systems of disjoint oriented simple closed curves.…
In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2pi/p with p no smaller than 2. We will mainly concentrate on the groups where some elements are…
Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the…
We introduce discrete multivortex solitons in a ring of nonlinear oscillators coupled to a central site. Regular clusters of discrete vortices appear as a result of mode collisions and we show that their stability is determined by global…
A dynamical system with discrete time is studied by means of algebraic geometry. The system admits a reduction that is interpreted as a classical field theory in 2+1-dimensional wholly discrete space-time. The integrals of motion of a…
With every family of finitely many subsets of a finite-dimensional vector space over the Galois-field with two elements we associate a cyclic transversal polytope. It turns out that those polytopes generalize several well-known polytopes…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We study the discretization of Darboux integrable systems. The discretization is done using $x$-, $y$-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an…
We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the…