Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size restrictions]; v2 - added Section 9 on Kramers' formula, additional computations, corrected typos, improved explanation

Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.Comment: 22 page

A Self-Consistent Dynamical Model for the COBE Detected Galactic Bar

A 3D steady state stellar dynamical model for the Galactic bar is constructed with 485 orbit building blocks using an extension of Schwarzschild technique. The weights of the orbits are assigned using non-negative least square method. The model fits the density profile of the COBE light distribution, the observed solid body stellar rotation curve, the fall-off of minor axis velocity dispersion and the velocity ellipsoid at Baade's window. We show that the model is stable. Maps and tables of observable velocity moments are made for easy comparisons with observation. The model can also be used to set up equilibrium initial conditions for N-body simulations to study stability. The technique used here can be applied to interpret high quality velocity data of external bulges/bars and galactic nuclei.Comment: submitted to MNRAS; 37 page AAS latex file with 2 tables and no figures; complete uuencoded compressed PS file with 9 figs is available at ftp://ibm-1.mpa-garching.mpg.de/pub/hsz/cobe_bar_dynamics.uu Hardcopies are available by reques