855 research outputs found
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
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