Carsten Knudsen

Bifurcations and chaos in a model of a rolling railway wheelset

In this paper we present the results of a numerical investigation of the dynamics of a model of a suspended railway wheelset in the speed range between 0 and 180 km h-1. The wheel rolls on a straight and horizontal track unaffected by external torques. A nonlinear relation between the creepage and the creep forces in the ideal wheel rail contact point is used. The effect of flange contact is modelled by a very stiff spring with a dead band. The suspension elements have linear characteristics, and the wheel profile is assumed to be conical. All other parameters than the speed are kept constant. Both symmetric and asymmetric oscillations and chaotic motion are found. The results are presented as bifurcation diagrams, time series and Poincaré section plots. We apply bifurcation and path following routines to obtain the results. In the last chapter we examine one of the chaotic regions with the help of symbolic dynamics.

Chaos Without Nonperiodicity

QUESTIONS. (i) Give formulae or meaningful bounds for the preperiod io. (As for the period p, it turns out that for the case of complete sequences, p < 2 except that p = 3 for the example in the right column of Table 1. A similar result holds for sequences that are not complete.) (ii) Is there an infinite sequence S0 for which every successor is defined, such that Si+1 differs from Si in infinitely many elements for some i? (iii) Is there an infinite sequence S0 such that every successor is defined and 9 is not ultimately periodic?

Dynamics of a model of a railway wheelset

In this paper we continue a numerical study of the dynamical behavior of a model of a suspended railway wheelset. We investigate the effect of speed and suspension and flange stiffnesses on the dynamics. Numerical bifurcation analysis is applied and one- and two-dimensional bifurcation diagrams are constructed. The onset of chaos as a function of speed, spring stiffness, and flange forces is investigated through the calculation of Lyapunov exponents with adiabatically varying parameters. The different transitions to chaos in the system are discussed and analyzed using symbolic dynamics. Finally, we discuss the change in orbit structure as stochastic perturbations are taken into account.

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.

Phase-locking regions in a forced model of slow insulin and glucose oscillations.

We present a detailed numerical investigation of the phase-locking regions in a forced model of slow oscillations in human insulin secretion and blood glucose concentration. The bifurcation structures of period 2pi and 4pi tongues are mapped out and found to be qualitatively identical to those of several other periodically forced self-oscillating systems operating across a Hopf-bifurcation point. The numerical analyses are supplemented by clinical experiments. (c) 1995 American Institute of Physics.

Bifurcations in a vocal fold model

An autonomous fourth order model of vocal fold vibrations is proposed. Each fold is represented by a lower and upper mass, and the aerodynamic forces are derived from a modified Bernoulli equation. The model exhibits many features of normal phonation in a wide parameter region. At the borderlines of this region coexistence of limit cycles, period-doubling and chaos are observed. Implications for an understanding of pathological voices are discussed.

Events Maps in a Stick-Slip System

This paper describes a one-dimensional map generated by a two degree-of-freedom mechanical system that undergoes self-sustained oscillations induced by dry friction. The iterated map allows a much simpler representation and a better understanding of some dynamic features of the system. Some applications of the map are illustrated and its behaviour is simulated by means of an analytically defined one-dimensional map. A method of reconstructing one-dimensional maps from experimental data from the system is introduced. The method uses cubic splines to approximate the iterated mappings. From a sequence of such time series the parameter dependent bifurcation behaviour is analysed by interpolating between the defined mappings. Similarities and differences between the bifurcation behaviour of the exact iterated mapping and the reconstructed mapping are discussed.

Phase diagrams for periodically driven Gunn diodes

Abstract A simulation model of the electron transfer mechanism for bulk negative differential conductivity in n-GaAs is applied to investigate the formation and propagation of subsequent high field domains in the presence of an external microwave field. Phase diagrams for the distribution of modes are drawn for two different values of the non-equivalent intervalley relaxation time. The domain mode is found to entrain with the microwave signal in a variety of different frequency-locked solutions. At higher microwave amplitudes, period-doubling and other forms of mode-converting bifurcations take place. In this region, spatio-temporal chaos may also be observed, as indicated by a positive value of the largest Lyapunov exponent. At still higher amplitudes, transitions to delayed, quenched, and limited space charge accumulation modes occur. The D q -curve characterizing the multifractal structure is calculated for a typical chaotic solution.

CATASTROPHE THEORETIC CLASSIFICATION OF NONLINEAR OSCILLATORS

Catastrophe theory is employed to classify different types of nonlinear oscillators, basing on the complication of their potentials. By using Thoms catastrophe unfoldings as oscillator potentials, we have introduced more general models to describe the dynamics of nonlinear oscillators, differing from each other by the form of their potential wells and by the possibility of escape. Spreading the investigation in the space of the parameters of the potential function, we have revealed that our examples defined via Thoms catastrophe unfoldings have some type of universal properties in the context of forced oscillations. For oscillators with nonescaping solutions, we have detected such typical bifurcation structures as crossroad areas and spring areas, and have described the universal scenario of their evolution under the forcing amplitude variation. On increasing the potential function degree, the complexity of the charts of the dynamical regimes results from the repetition of the described bifurcation scenario. For oscillators with escaping solutions, such general properties were investigated, as dependence of the charts of the dynamical regimes and the basins on the parameters of the potential function. We have observed that these properties are typical in a broad range of the control parameters.

Information Dynamics of Self-Programmable Matter

Using the simple observation that programs are identical to data, programs alter data, and thus programs alter programs, we have constructed a self-programming system based on a parallel von Neumann architecture. This system has the same fundamental property as living systems have: the ability to evolve new properties. We demonstrate how this constructive dynamical system is able to develop complex cooperative structures with adaptive responses to external perturbations. The experiments with this system are discussed with special emphasis on the relation between information theoretical measures (entropy and mutual information functions) and on the emergence of complex functional properties. Decay and scaling of long-range correlations are studied by calculation of mutual information functions.