Arne Wunderlin

An introduction to synergetics

We introduce basic concepts and principles of synergetics in an elementary way and work out the interdisciplinary aims of this young scientific discipline. To some detail the laser as a system of physics and an application of synergetics to a problem from economics will be presented. A generalization of the methods to a class of functional differential equations will be outlined.

Analytical approach for the Floquet theory of delay differential equations.

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincaré-Lindstedt and the Shohat expansions, which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows us to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.

A theoretical model of sinusoidal forearm tracking with delayed visual feedback

We present a phenomenological model to an experiment, where a person is systematically confronted with a delayed effect of her or his reaction to a time-periodic external signal. The model equations are derived from purely macroscopic considerations. Applying methods developed in the realm of synergetics we can analyze the first instability in the persons behaviour semi-analytically. A careful numerical study is devoted to the higher order instabilities and a comparison between experiment and the results obtained from our model is performed in detail.

Generalized Ginzburg-Landau equations, slaving principle and center manifold theorem

The Generalized Ginzburg-Landau equations, introduced by one of us (H.H.), are considered in a simplified version to clarify their relation to the center manifold theorem.

Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps

We first treat multidimensional nonlinear noisy maps. We assume that the variables can be split into two classes of variablesu ands so that the linearized equations would give rise to growth or decay foru ands, respectively. We show how the slaved variabless can be explicitly expressed by the order parametersu by making use of the fully nonlinear equations. By taking the limit of vanishing time steps and using a Wiener process and the Îto calculus we derive the corresponding formulas for stochastic differential equations (including multiplicative noise). In this way a high-dimensional problem can be reduced to a problem of much lower dimensions described again by stochastic equations of theÎto type. A similar procedure holds for theStratonovich calculus.

New interpretation and size of strange attractor of the Lorenz model of turbulence

Abstract Using the equivalence of the Lorenz equations with laser equations we show how the irregular jumping from one segment of the variable space to the other one can be understood. We also estimate the size of the Lorenz attractor.

Scaling theory for nonequilibrium systems

We develop a general scaling theory of one-dimensional systems withN components having applications to disorder-order-transitions or order-order transitions of non-equilibrium systems, such as lasers, hydrodynamical systems and non-equilibrium chemical reactions. We include both cases of soft and hard modes. Since fluctuations play a decisive role at the transition point, we take fully account of them. We start from general equations of motion which contain nonlinear forces (or rates), diffusion terms and fluctuating forces. These equations depend on external parameters. When linearized around their steady state solutions, the equations allow for stable, marginal or unstable solutions. The solutions near critical points are represented as superpositions of marginal solutions, whose amplitudes are determined by comparing the coefficients of the scaling parameter up to third order. The scaling of the fluctuating forces and, in the case of chemical reactions, their correlation functions are derived in detail.

Analytical treatment of delayed feedback control

Abstract The stabilization of unstable periodic orbits by means of time-delayed feedback mechanisms is treated analytically on the basis of new methods which appear as a generalization of results from the theory of ordinary differential equations. We derive explicit expressions for the Floquet exponents which govern the linear stability of the controlled orbit. The analytical method is applied to the Lorenz-Haken equations describing single mode laser dynamics. We compare our analytical results with numerical simulations and present a significant extension of the original time-delayed control mechanism.

Computer simulations as a tool for understanding the evolution of water transport systems in land plants: a review and new data

Abstract The analyses of the form–function relationship in plant water transport systems are severely restricted due to several factors: (1) It is almost impossible to directly observe the water flow in single components of the xylem: tracheids or vessels. (2) The water relations of tissues or whole plant organs are difficult to measure experimentally. (3) Systematic comparative studies of various xylem constructions are severely hampered by individual variations in morphology as well as by the complex pattern of physiological reactions of living plants. (4) Finally, direct experimental work on fossil plants is obviously impossible. Here, numerical modelling is provided as an alternative to experimental studies. Computer simulations have the following advantages and possibilities. (1) The different levels of xylem organization, conduits or conducting bundles and systems of conducting bundles, can be modelled by appropriate physical models. (2) Systematic parameter variations are easily accomplished. The consideration of arbitrary architectures is possible and, thus, a direct comparison of various xylem architectures. (3) High resolution results on the spatial as well as the temporal level can be obtained. (4) The evolutionary process can be simulated with the approach of synergetics (a mathematical theory which describes self organization processes) allowing form–function relationships to be tested. In this paper some numerical studies of the water transport properties of telomes with various stele types to provide a functional explanation for observed patterns of stelar evolution are presented. It is demonstrated that water transporting function is the crucial factor in the evolution of the various steles. Three geometric parameters which determine the water transport performance and the water transport efficiency of transpiring plant axes are identified. These geometric parameters provide the basis for a functional explanation of the evolution of the stele. Using the approach of phenomenological synergetics the stelar evolution itself is simulated as system state transitions induced by water supply demands of the axis.

Some Applications of Basic Ideas and Models of Synergetics to Sociology

We shall propose a general procedure how to apply synergetics to social processes, i.e. in which way one may translate social behaviour into mathematical structures. This is achieved by merely using macroscopic quantities like order parameters. This may at least partly remove the difficulties connected with the problem of measurability in social systems. Our ideas are exemplified by using models on the formation of public opinion. Furthermore we shall construct a minimal model for social processes which summarizes all ingredients of a synergetic system.