Related papers: Horocyclic Orbits on $\Gamma(1)\frontslash\mathcal…
The topological dynamics of the horocyclic flow $h_{\mathbb{R}}$ on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular, on such a surface, the flow $h_{\mathbb{R}}$ is minimal, or the minimal…
We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…
Suppose X/Gamma is an arithmetic locally symmetric space of noncompact type (with the natural metric induced by the Killing form of the isometry group of X), and let p be a point on the visual boundary of X. It was shown by T.Hattori that…
We propose a new theory to explain the formation of spiral arms and of all types of outer rings in barred galaxies. We have extended and applied the technique used in celestial mechanics to compute transfer orbits. Thus, our theory is based…
Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by some recent bifurcation results for homoclinic trajectories of nonautonomous vectorfield,…
A non-autonomous version of the standard map with a periodic variation of the parameter is introduced and studied. Symmetry properties in the variables and parameters of the map are found and used to find relations between rotation numbers…
Starting from a semiclassical quantization condition based on the trace formula, we derive a periodic-orbit formula for the distribution of spacings of eigenvalues with k intermediate levels. Numerical tests verify the validity of this…
We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be $C^1$ approximated by vector fields with orbit-flip homoclinic orbits.
The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical…
We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of…
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the…
We investigate the resonance spectrum of the H\'enon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating…
This paper generalizes the result of Sarnak and Ubis \cite{sarnak-ubis} about non-concentration of primes in horocycle orbits on $PSL_2(\mathbb{Z}) \backslash PSL_2(\mathbb{R})$ to any lattice in $PSL_2(\mathbb{R})$. The proof combines the…
Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the simplest model of child's swing. Melnikov's analysis is carried out to find bifurcations of homoclinic,…
We study in detail the form of the orbits in integrable generalized H\'enon-Heiles systems with Hamiltonians of the form $H = \frac{1}{2}(\dot{x}^2 + Ax^2 + \dot{y}^2 + By^2) + \epsilon(xy^2 + \alpha x^3).$ In particular, we focus on the…
We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit…
In a 2D conservative Hamiltonian system there is a formal integral $\Phi$ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we…
We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish…
We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex…
In this paper, we study the possible bifurcations of periodic orbits by reducing them to graphs. The aforementioned allows to study the genericity of routes to chaos, as well as to analyze their possible complexity. In particular, our…