Related papers: Discretized rotation has infinitely many periodic …
We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $\mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each…
We investigate the problem of the typical rotation states of the small planetary satellites from the viewpoint of the dynamical stability of their rotation. We show that the majority of the discovered satellites with unknown rotation…
We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo an incomplete periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show…
The spatial Kepler problem with a perturbation satisfying the rotational symmetry w.r.t. the $z$-axis and the reflection symmetry w.r.t. the $(x, y)$-plane, can be reduced to an Hamiltonian system with 2 degrees of freedom after fixing the…
A discrete Hubbard-Stratonovich transformation is presented for systems with an orbital degeneracy $N$ and a Hubbard Coulomb interaction without multiplet effects. An exact transformation is obtained by introducing an external field which…
For controlling periodic orbits with delayed feedback methods the periodicity has to be known a priori. We propose a simple scheme, how to detect the period of orbits from properties of the control signal, at least if a periodic but…
Periodic orbits and cycles, respectively, play a significant role in discrete- and continuous-time dynamical systems (i.e. maps and flows). To succinctly describe their shifts when the system is applied perturbation, the notions of…
We investigate a rotated, orthogonal gravitational wedge billiard - a special case of the asymmetric wedge billiard - in which the dynamics are integrable. We derive equations and conditions under which periodic orbits may be constructed…
As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation.…
In this paper we consider oscillating non-exact magnetic fields on surfaces with genus at least two and show that for almost every energy level $k$ below a certain value $\tau_+^*(g,\sigma)$ less than or equal to the "Ma\~n\'e critical…
We develop reductions for classifications of singularities of orbit closures in module varieties. Then we show that the orbit closures for representations of Dynkin quivers are regular in codimension two.
For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach…
We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the…
One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg…
We provide sufficient conditions on a positive function so that its associated special flow over any irrational rotation is either weak mixing or $L^2$-conjugate to a suspension flow.
Let $G:= (C^*)^k\times SL_2(C)$ act linearly on a vector space or its projectivisation. We obtain an effective criterion to detect whether a number of orbits in an orbit-closure is finite or not.
We consider an exact magnetic flow on the tangent bundle of a closed surface. We prove that for almost every energy level $\kappa$ below the Ma\~n\'e critical value of the universal cover there are infinitely many periodic orbits with…
We study discrete orderings in the real spectrum of a commutative ring by defining discrete prime cones and give an algebro-geometric meaning to some kind of diophantine problems over discretely ordered rings. Also for a discretely ordered…
We use symplectic tools to establish a smooth variant of Franks theorem for a closed orientable surface of positive genus $g$; it implies that a symplectic diffeomorphism isotopic to the identity with more than $2g-2$ fixed points, counted…
Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a…