Mathias Bode

Soft nearest prototype classification

We propose a new method for the construction of nearest prototype classifiers which is based on a Gaussian mixture ansatz and which can be interpreted as an annealed version of learning vector quantization (LVQ). The algorithm performs a gradient descent on a cost-function minimizing the classification error on the training set. We investigate the properties of the algorithm and assess its performance for several toy data sets and for an optical letter classification task. Results show 1) that annealing in the dispersion parameter of the Gaussian kernels improves classification accuracy; 2) that classification results are better than those obtained with standard learning vector quantization (LVQ 2.1, LVQ 3) for equal numbers of prototypes; and 3) that annealing of the width parameter improved the classification capability. Additionally, the principled approach provides an explanation of a number of features of the (heuristic) LVQ methods.

Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component reaction-diffusion system

We investigate the static and dynamic behaviour of well localized solitary solutions of a three-component reaction-diffusion model in two- and three-dimensional space. These solutions behave like particles under many aspects and we refer to them as dissipative solitons. These objects may interact with each other, and are influenced by boundaries and inhomogeneities of the parameters. Depending on parameters and initial conditions they may be generated or annihilated. Reflection, scattering and the formation of bound states is commonly observed and dissipative solitons essentially retain their identity under such interactions if these are sufficiently weak. For parameters near to the onset of propagation, the field equations are reduced to a set of ordinary differential equations describing rather well the dynamical behaviour of many aspects of isolated and interacting dissipative solitons using their center coordinates and amplitudes of certain propagator modes. The work demonstrates that dissipative solitons are a generic self-organized pattern of reaction-diffusion systems, that they are rather robust under interaction and in many circumstances can be considered as elementary constituents of patterns of higher complexity. The reduced description can be looked upon as a theoretical foundation of the concept of dissipative solitons exhibiting particle-like behaviour. In addition, for the first time these equations allow a numerical investigation of systems with large number of dissipative solitons as they are observed experimentally. It is pointed out that many of the described properties of dissipative solitons are observed in experimental systems of reaction-diffusion type.

Pattern formation in reaction-diffusion systems—dissipative solitons in physical systems

Abstract Two-component reaction-diffusion systems are discussed. These systems may have dissipative solitions as solutions. This is demonstrated by reporting recent experimental results obtained from physical systems. Generation of dissipative solitons and their behaviour is discussed theoretically in terms of interacting fronts. Various patterns as there are stationary and moving fronts, cascades of stationary dissipative solitons and travelling, swinging and breathing solitons are the natural outcome of the theory.

Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities

Abstract The effect of spatial parameter inhomogeneities on front propagation is analysed for spatially one dimensional reaction-diffusion systems with two components. Close to pinned front solutions the dynamics can be reduced to two degrees of freedom. Local bifurcation points can be given explicitly. For special localized degenerate situations a Takens-Bogdanov respectively symmetric Takens-Bogdanov bifurcation describes the local dynamics and is a key element for the understanding of the global behaviour.

Measurement of the transition from uni- to bi-directional front propagation in a reaction-diffusion system

Abstract The transition from uni- to bi-directional front propagation, i.e. the breakdown of the strict concept of strong and weak domains, is studied experimentally using a chain of 128 coupled nonlinear electrical oscillators as a two component reaction-diffusion system. We compare the experimentally obtained transition points with recently derived theoretical criteria and find good agreement.

On the influence of inhomogeneities in a reaction-diffusion system

Abstract We compare experimental and analytical results concerning the influence of spatial inhomogeneities on front dynamics in the case of a one-component reaction-diffusion system represented by a chain of bistable electrical relaxation elements. Spatial variation of the front velocity, especially the pinning behavior, is in quantitative agreement with predictions of multiple scale perturbation theory.

Parallel hardware implementation of Kohonen's algorithm with an active medium

We present a hardware realization of Kohonens algorithm using many simple, linearly coupled active elements, namely a reaction-diffusion medium, that is a spatially extended active element. This medium also controls the learning-process. Both classification and learning are realized in a way that only next-neighbor coupling is necessary. We exemplify the learning process by means of some simple tasks.

The Generation of Dissipative Quasi-Particles near Turing’s Bifurcation in Three-Dimensional Reacting Diffusion Systems

In order to model pattern formation processes in a dc driven semiconductor-gas discharge system on a phenomenological level, we investigate a three-component reaction-diffusion system of 1-activator-2-inhibitor-type. The solutions of this system show localized moving and stationary structures which interact by scattering, annihilation or more complex scenarios. Because of this particle-like behaviour the structures are called dissipative quasi-particles. This work deals with the generation mechanism of dissipative quasi-particles related to Turing’s destabilisation of homogeneous states. Two- and three-dimensional simulations are shown and in the two-dimensional case compared with experimental results.

Oscillations of fronts and front pairs in two- and three-component reaction-diffusion systems

Abstract We investigate multi-co,ponent reaction-diffusion systems on a finite one-dimensional domain. The first component is assumed both to react on a short time-scale, and to have a small diffusion length leading to patterns with sharp fronts. To analyze stability, we apply the SLEP method by Nishiura and Fujii. This method is superior to standard perturbation methods. We illustrate this using a two-component system. In the three-component system, in a situation where both the influence of the first component on the other components and the interaction of the latter is only weak, we show that there is a unique Hopf destabilization of stationary fronts when changing time constants. As an example we treat a system with global coupling used e.g. for semi-conductor devices. For filaments (front pairs) we find two types of Hopf destabilizations: breathing and swinging. The global coupling controls which type occurs. This corresponds to recent experimental results and is confirmed via numerical calculations.

Quasi-Particles in a Three-Dimensional Three-Component Reaction-Diffusion System

We investigate a reaction-diffusion system which consists of a set of three partial differential equations. Due to the reaction kinetics the system can be referred to as a 1-activator-2-inhibitor system. We show, that such systems axe capable of supporting localized moving structures, so called quasi-particles. For certain parameters it is possible to predict the propagation speed of these solutions as well as their behaviour in scattering processes. In more general cases we have carried out simulations which reveal different scattering results depending on the parameters. We find annihilation, reflection and merging of particles.