Jens Starke

Equation-free detection and continuation of a Hopf bifurcation point in a particle model of pedestrian flow

Using an equation-free analysis approach we identify a Hopf bifurcation point and perform a two-parameter continuation of the Hopf point for the macroscopic dynamical behavior of an interacting particle model. Due to the nature of systems with a moderate number of particles and noise, the quality of the available numerical information requires the use of very robust numerical algorithms for each of the building blocks of the equation-free methodology. As an example, we consider a particle model of a crowd of pedestrians where particles interact through pairwise “social forces.” The pedestrians move along a corridor where they are constrained by the walls of the corridor, and two crowds are aiming, from opposite directions, to pass through a narrowing doorway perpendicular to the corridor. We focus our investigation on the collective behavior of the model. As the width of the doorway is increased, we observe an onset of oscillations of the net pedestrian flux through the doorway, described by a Hopf bifurc...

Analytical solutions of jam pattern formation on a ring for a class of optimal velocity traffic models

A follow-the-leader model of traffic flow on a closed loop is considered in the framework of the extended optimal velocity (OV) model where the driver reacts to both the following and the preceding car. Periodic wave train solutions that describe the formation of traffic congestion patterns are found analytically. Their velocity and amplitude are determined from a perturbation approach based on collective coordinates with the discrete modified Korteweg–de Vries equation as the zero order equation. This contains the standard OV model as a special case. The analytical results are in excellent agreement with numerical solutions.

Evaluating Coupled Selection Equations for Dynamic Task Assignment Using a Behavior Framework

In this paper we focus on methods for a reliable and robust mechanism to distribute roles among a team of cooperating robots. In previous work, we showed the principal applicability of a novel approach based on self organization using coupled selection equations. To show the applicability in the robocup scenario we used a simple scenario to assign the roles attacker and defender. In this paper we present the application of the novel approach to more realistic and complex scenarios like kick-off or pass play. One of the critical parts in this method is the parameterization of utility and activation functions used to determine the additional parameters.

An Iterative Method for the Approximation of Fibers in Slow-Fast Systems

In this paper we extend a method for iteratively improving slow manifolds so that it also can be used to approximate the fiber directions. The extended method is applied to general finite-dimensional real analytic systems where we obtain exponential estimates of the tangent spaces to the fibers. The method is demonstrated on the Michaelis--Menten--Henri model and the Lindemann mechanism. The latter example also serves to demonstrate the method on a slow-fast system in nonstandard slow-fast form. Finally, we extend the method further so that it also approximates the curvature of the fibers.

Visualization of Individual Growth-Related Craniofacial Changes Based on Cephalometric Landmark Data: A Pilot Study

OBJECTIVE An approach based on Euclidean distances between cephalometric landmarks is presented (1) to visualize and localize the individual shape changes of the complex craniofacial skeleton during growth and (2) to depict the individual dynamic behavior of developmental size and shape changes. PATIENTS AND METHOD Growth-related craniofacial changes were investigated exemplarily for two male orthodontically untreated subjects from the Belfast Growth Study on the basis of lateral cephalograms at 7, 9, 11, 13, and 15 years. The interlandmark distances among seven skeletal cephalometric landmarks served as a database for the study. A modified Karhunen-Loève decomposition based on orthogonal modes and time-dependent scalar amplitudes was used to describe the growth process. The individual shape changes of the various craniofacial regions were visualized by allocation of colors to the respective distances, and overdrawn representations were reconstructed by means of multidimensional scaling. RESULTS AND CONCLUSIONS This visualization technique allows anatomical regions to be characterized with respect to reduced or strengthened growth, compared with pure size changes. The clinically relevant mechanisms of craniofacial changes are visualized (e.g., shifts in the anteroposterior or vertical dimensions of the jaws in relation to cranial base and structural imbalances during development). In addition, overdrawing the effects of shape change on the skeletal structures gives a more readily comprehensible impression of the growth process. Taking account of the methodical limitations of this approach (e.g., the restrictions concerning the number of landmarks), the clinician may take advantage of this technique in orthodontic or surgical diagnostics to gain additional insight into the individual complex size and shape changes during development along with their dynamic behavior.

Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow

Equation-free methods make it possible to analyze the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behavior depends on parameter values including an understanding of transitions between different types of qualitative behavior. These methods are introduced and explained for traffic jam formation and emergence of oscillatory pedestrian counter flow in a corridor with a narrow door.

Multijam Solutions in Traffic Models with Velocity-Dependent Driver Strategies

The optimal-velocity follow-the-leader model is augmented with an equation that allows each driver to adjust their target headway according to the velocity difference between the driver and the car in front. In this more detailed model, which is investigated on a ring, stable and unstable multipulse or multijam solutions emerge. Analytical investigations using truncated Fourier analysis are confirmed and complemented by a detailed numerical bifurcation analysis. In addition to standard rotating waves, time-modulated waves are found.

Bifurcation of learning and structure formation in neuronal maps

Most learning processes in neuronal networks happen on a much longer time scale than that of the underlying neuronal dynamics. It is therefore useful to analyze slowly varying macroscopic order parameters to explore a networks learning ability. We study the synaptic learning process giving rise to map formation in the laminar nucleus of the barn owls auditory system. Using equation-free methods, we perform a bifurcation analysis of spatio-temporal structure formation in the associated synaptic-weight matrix. This enables us to analyze learning as a bifurcation process and follow the unstable states as well. A simple time translation of the learning window function shifts the bifurcation point of structure formation and goes along with traveling waves in the map, without changing the animals sound localization performance.

Experimental Bifurcation Analysis by Control-Based Continuation: Determining Stability

The newly developed control-based continuation technique has made it possible to perform experimental bifurcation analysis, e.g. to track stable as well as unstable branches of frequency responses directly in experiments. The method bypasses mathematical models, and systematically explores how vibration characteristics of dynamical systems change under variation of parameters. The method employs a control scheme to modify the response stability. While this facilitates exploration of the unstable branches of a bifurcation diagram, it unfortunately makes it impossible to distinguish previously stable and unstable equilibrium states. We present the ongoing work of developing and applying the control-based continuation method to an experimental mechanical test-rig, consisting of a harmonically forced nonlinear impact oscillator controlled by electromagnetic actuators. Furthermore we propose and test ideas on how to determine the stability of equilibria states during continuation.Copyright

Interacting Many-Particle Systems of Different Particle Types Converge to a Sorted State

We consider a model class of interacting many-particle systems consisting of different types of particles defined by a gradient flow. The corresponding potential expresses attractive and repulsive interactions between particles of the same type and different types, respectively. The introduced system converges by self-organized pattern formation to a sorted state where particles of the same type share a common position and those of different types are separated from each other. This is proved in the sense that we show that the property of being sorted is asymptotically stable and all other states are unstable. The models are motivated from physics, chemistry, and biology, and the principal investigations can be useful for many systems with interacting particles or agents. The models match particularly well a system in neuroscience, namely the axonal pathfinding and sorting in the olfactory system of vertebrates.