Consider what is called the classical (Birkhoff) billiard problem: a particle moves in the planar convex domain D along the straight lines interacting with the boundary ∂D according to the Fermat’s law“ angle of reflection is equal to the angle of incidence”. In mathematical physics it is important to understand how many periodic orbits can be present in a planar billiard. Periodic orbits are t...

The orbital evolution of about 26,000 Jupiter-crossing objects and 1,500 resonant asteroids under the gravitational influence of planets was investigated. The rate of their collisions with terrestrial planets was estimated by computing the probabilities of collisions based on random-phase approximations and the orbital elements sampled with a 500 yr step. The Bulirsh-Stoer and a symplectic orbi...

This paper explores orbits in extended mass distributions and develops an analytic approximation scheme based on epicycloids (spirograph patterns). We focus on the Hernquist potential ψ = 1/(1 + ξ), which provides a good model for many astrophysical systems, including elliptical galaxies (with an R law), dark matter halos (where N-body simulations indicate a nearly universal density profile), a...

Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R×M → M, where M is a locally compact manifold without boundary, typically R . In particular, we investigate F (μ, ·) for μ ∈ J = [μ1, μ2], when F (μ1, ·) has only finitely many periodic orbits while F (μ2, ·) has exponential growth of the number of periodic orbits as a function of ...

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory...

Let G be a complex reductive group. We study the problem of associating Dixmier algebras to nilpotent (co)adjoint orbits of G, or, more generally, to orbit data for G. If g = 0 + n + in is a triangular decomposition of g and 0 is a nilpotent orbit, we consider the irreducible components of 0 n n, which are Lagrangian subvarieties of 0. The main idea is to construct, starting with certain "good"...