Michael Schanz

On multi-parametric bifurcations in a scalar piecewise-linear map

In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that the structure of the 2D and 3D parameter spaces can be simply described using the concept of multi-parametric bifurcations. It is demonstrated that an infinite number of two-parametric bifurcation lines starts at the origin of the 3D parameter space. Along each of these lines an infinite number of bifurcation planes starts, whereas the origin represents a three-parametric bifurcation.

Multi-parametric bifurcations in a piecewise-linear discontinuous map

In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained.

BORDER-COLLISION BIFURCATIONS IN 1D PIECEWISE-LINEAR MAPS AND LEONOV'S APPROACH

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonovs approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.

On the fully developed bandcount adding scenario

In this paper we report a new bifurcation phenomenon induced by interior crises, which explains the structure of multi-band chaotic attractors in the case of a piecewise-linear discontinuous map. This phenomenon, denoted as a fully developed bandcount adding scenario, leads to a self-similar structure of the chaotic region in parameter space.

CALCULATION OF BIFURCATION CURVES BY MAP REPLACEMENT

The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonovs technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonovs technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.

Delayed complex systems: an overview

There does not exist a generally accepted definition for the notion of complex systems in science, but it is a common belief that complex systems show features that cannot be explained by just looking at their constituents. Thus, a complex system normally involves interaction of subunits, and

APPLICATION OF ORDER PARAMETER EQUATIONS FOR THE ANALYSIS AND THE CONTROL OF NONLINEAR TIME DISCRETE DYNAMICAL SYSTEMS

This work is based on the concept of order parameters of synergetics. The order parameter equations describe the behavior of a system in the vicinity of an instability and are used here not only for the analysis but also for the control of nonlinear time discrete dynamical systems. Usually, the dimensionality of the evolution equations of the order parameters is less than the dimensionality of the original evolution equations. It is, therefore, convenient to introduce control mechanisms, first in the order parameter equations, and then to use the obtained results for the control of the original system. The aim of the control in this case is to avoid chaotic behavior of the system. This is achieved by shifting appropriate bifurcation points of a period-doubling cascade. In this work we concentrate on the shifting of only the first bifurcation point. The used control mechanisms are delayed feedback schemes. As an example the well-known Henon map is investigated. The order parameter equation is calculated using both the adiabatic elimination procedure and the center manifold theory. Using the order parameter concept two types of control mechanisms are constructed, analyzed and compared.

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.

Treatment of combinatorial optimization problems using selection equations with cost terms. Part II. NP -hard three-dimensional assignment problems

Abstract This paper is the first of two parts (Part II, Physica D 134 (1999) 242–252) which present a novel approach to the solution of assignment problems using time continuous dynamical systems. This first part concentrates on the two-dimensional assignment problem of combinatorial optimization, while in the second part the NP -hard three-dimensional problem is treated. The proposed dynamical system approach works by using coupled selection equations with cost terms. This is a combination of two basic ideas, namely first, coupled selection processes with suitably chosen initial values, where the dynamical system has suitable asymptotically stable points which represent the feasible solutions of the assignment problem. It is an important fact that there are feasible solutions only and no spurious states in this system. The second idea is based on the appropriate distortion of the basins of attraction of the asymptotically stable points by cost terms similar to classical penalty methods.

The bandcount increment scenario. I. Basic structures

Bifurcation structures in two-dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In the first part, we report a novel bifurcation scenario formed by crises bifurcations, which includes multi-band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain.