Symbolic Dynamics of Magnetic Bumps

For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.Comment: 11 pages, 4 figure

Contact variational integrators

We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz' variational principle. First we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle. Then we discuss the backward error analysis of our variational integrators, including the construction of a modified Lagrangian. Throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues

OwlDE:Making ODEs first-class Owl citizens

Source code of [OwlDE](https://github.com/owlbarn/owl_ode), a suite of ordinary differential equation integrators for [Owl](https://github.com/owlbarn/owl), the OCaml scientific framework

Recurrence for quenched random Lorentz tubes

We consider the billiard dynamics in a strip-like set that is tessellated by countably many translated copies of the same polygon. A random configuration of semidispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global choice of scatterers, is called `quenched random Lorentz tube'. We prove that, under general conditions, almost every system in the ensemble is recurrent.Comment: 23 pages, 8 figures. Version published on Chaos, vol. 20 (2010) + correction of small erratum in condition (A3

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

Starting from a contact Hamiltonian description of Li\'enard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime

Numerical integration in celestial mechanics:A case for contact geometry

Several dynamical systems of interest in Celestial Mechanics can be written in the form of a Newton equation with time-dependent damping, linear in the velocities. For instance, the modified Kepler problem, the spin–orbit model and the Lane–Emden equation all belong to such class. In this work, we start an investigation of these models from the point of view of contact geometry. In particular, we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators

Accumulation of complex eigenvalues of an indefinite Sturm-Liouville operator with a shifted Coulomb potential

Abstract For a particular family of long-range potentials V , we prove that the eigenvalues of the indefinite Sturm-Liouville operator A = sign(x)(−∆ + V (x)) accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators