Hans Thunberg

Periodicity versus Chaos in One-Dimensional Dynamics

We survey recent results in one-dimensional dynamics and, as an application, we derive rigorous basic dynamical facts for two standard models in population dynamics, the Ricker and the Hassell families. We also informally discuss the concept of chaos in the context of one-dimensional discrete time models. First we use the model case of the quadratic family for an informal exposition. We then review precise generic results before turning to the population models. Our focus is on typical asymptotic behavior, seen for most initial conditions and for large sets of maps. Parameter sets corresponding to different types of attractors are described. In particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models. Natural measures (asymptotic distributions) and their stability properties are considered.

The Widening Gap - A Swedish Perspective

Transition problems from secondary to tertiary level in mathematics have been a recurrent issue in Sweden. This paper summarises the development during the last decades. Results from two recent research studies that illuminate the transition problem are presented. The first one, based on empirical data from a major Swedish technical university, characterises the widening gap, in content and in approach, between secondary school and first year university courses. The second study deals with students’ encounters with mathematical proof and is based on a large investigation at another main Swedish university. We discuss the influence on the current transition problems of school reforms and of the great expansion of higher education in Sweden during the last 10 – 15 years in view of the results from the research studies.

Unfolding of chaotic unimodal maps and the parameter dependence of natural measures

We consider one-parameter families fa of interval maps, and discuss the structure in parameter space and the (dis)continuity properties of the natural measure as a function of the parameter near certain strongly chaotic maps (post-critically finite Misiurewicz maps and Benedicks–Carleson maps). In particular, it is shown that the mapping a � µa (the natural measure of fa )i s severely discontinuous at these strongly chaotic maps and is not continuous on any full measure set of parameters in full, generic families. Going in the other direction, it is also shown that if such a chaotic map has a measure for which the critical point is generic, then this measure can be approximated with measures supported on periodic attractors of nearby maps. The main idea is to construct cascades of post-critically finite Misiurewicz map and cascades of maps with periodic attractors, whose critical orbits reproduce various invariant sets of the unperturbed map. In the special case of the quadratic family, generalizations can be made to any non-renormalizable maps.

An elementary approach to dynamics and bifurcations of skew tent maps

In this paper, the dynamics of skew tent maps are classified in terms of two bifurcation parameters. In time series analysis such maps are usually referred to as continuous threshold autoregressive models (TAR(1)-models) after Tong (Non-Linear Time Series, Clarendon Press, Oxford, UK, 1990). This study contains results simplifying the use of TAR(1)-models considerably, e.g. if a periodic attractor exists it is unique. On the other hand, we also claim that care must be exercised when TAR models are used. In fact, they possess a very special type of dynamical pattern with respect to the bifurcation parameters and their transition to chaos is far from standard.

A RECYCLED CHARACTERIZATION OF KNEADING SEQUENCES

For any infinite sequence E on two symbols one can define two sequences of positive integers S(E) (the splitting times) and T(E) (the cosplitting times), which each describe the self-replicative structure of E. If E is the kneading sequence of a unimodal map, it is known that S(E) and T(E) carry a lot of information on the dynamics, and that they are disjoint. We show the reverse implication: A nonperiodic sequence E is the kneading sequence of some unimodal map if the sequences S(E) and T(E) are disjoint.

Periodicity versus chaos in certain population models

Using recent results in one-dimensional dynamics, many of the well-known properties of the logistic family can be proved for more general families. We investigate two standard models in population dynamics. Each system under consideration has a unique attractor, and we describe parameter sets corresponding to different types of attractors. Although systems with periodic attractors are dense in parameter space, strongly chaotic systems still appear with positive probability. Natural measures (asymptotic distributions) and their stability properties are considered. We also discuss the concept of chaos in the context of one-dimensional modeling.

Feigenbaum Numbers for Certain Flat-Top Families

We report on numerical results for certain families of S-unimodal maps with flat critical point. For four one-parameter families, differing in their amount of flatness, we study the Feigenbaum limits α and δ. There seems to be a finite δ and a finite α associated with each period doubling cascade in each family. Some rough numerical estimates are obtained, and our upper bound on δ is smaller than the corresponding supremum for families with nonflat critical point. One would expect that these numbers should only depend on the nature (flatness) of the maximum, and thus be constant in each family. Our data support this hypothesis for α, but are inconclusive when it comes to δ.