Three Dimensional Integrable Mappings

We derive three-dimensional integrable mappings which have two invariants.Comment: 6 pages, part of this paper is taken from the author's PhD thesis, not submitted for publicatio

On the nonlinear stability of symplectic integrators

The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially for nonlinear oscillators. We give conditions under which an initial condition on a compact energy surface will remain bounded for exponentially long times for sufficiently small time steps. For example, the implicit midpoint rule achieves this for the critical energy surface of the H´enon- Heiles system, while the leapfrog method does not. We construct explicit methods which are nonlinearly stable for all simple mechanical systems for exponentially long times. We also address questions of topological stability, finding conditions under which the original and modified energy surfaces are topologically equivalent

Geometric Numerical Integration Applied to The Elastic Pendulum at Higher Order Resonance

In this paper we study the performance of a symplectic numerical integrator based on the splitting method. This method is applied to a subtle problem i.e. higher order resonance of the elastic pendulum. In order to numerically study the phase space of the elastic pendulum at higher order resonance, a numerical integrator which preserves qualitative features after long integration times is needed. We show by means of an example that our symplectic method offers a relatively cheap and accurate numerical integrator.Comment: 15 pages, 6 figure

An ultradiscrete integrable map arising from a pair of tropical elliptic pencils

We present a tropical geometric description of a piecewise linear map whose invariant curve is a concave polygon. In contrast to convex polygons, a concave one is not directly related to tropical geometry; nevertheless the description is given in terms of the addition formula of a tropical elliptic curve. We show that the map is arising from a pair of tropical elliptic pencils each member of which is the invariant curve of the ultradiscrete QRT map.Comment: 14 pages, 4 figure