4 research outputs found
Regimes of stability of accelerator modes
The phase diagram of a simple area-preserving map, which was motivated by the
quantum dynamics of cold atoms, is explored analytically and numerically.
Periodic orbits of a given winding ratio are found to exist within wedge-shaped
regions in the phase diagrams, which are analogous to the Arnol'd tongues which
have been extensively studied for a variety of dynamical systems, mostly
dissipative ones. A rich variety of bifurcations of various types are observed,
as well as period doubling cascades. Stability of periodic orbits is analyzed
in detail.Comment: Submitted to Physica
Scaling and Universality of the Complexity of Analog Computation
We apply a probabilistic approach to study the computational complexity of
analog computers which solve linear programming problems. We analyze
numerically various ensembles of linear programming problems and obtain, for
each of these ensembles, the probability distribution functions of certain
quantities which measure the computational complexity, known as the convergence
rate, the barrier and the computation time. We find that in the limit of very
large problems these probability distributions are universal scaling functions.
In other words, the probability distribution function for each of these three
quantities becomes, in the limit of large problem size, a function of a single
scaling variable, which is a certain composition of the quantity in question
and the size of the system. Moreover, various ensembles studied seem to lead
essentially to the same scaling functions, which depend only on the variance of
the ensemble. These results extend analytical and numerical results obtained
recently for the Gaussian ensemble, and support the conjecture that these
scaling functions are universal.Comment: 22 pages, latex, 12 eps fig