Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.

: This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.

Abstract. A distortion theory is developed for S−unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S−unimodal maps is classified according to a distortion property, called the Markov-property. 1.

Abstract. We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive 3-dimensional Lebesgue measure whose points of Lebesgue density are dense in the whole manifold. This situation is stable under small perturbations. 1.

Abstract. We give an intrinsic definition of a heteroclinic network as a flow invariant set that is indecomposable but not recurrent. Our definition covers many previously discussed examples of heteroclinic behavior. In addition, it provides a natural framework for discussing cycles between invariant sets more complicated than equilibria or limit cycles. We allow for cycles that connect chaotic sets (cycling chaos) or heteroclinic cycles (cycling cycles). Both phenomena can occur robustly in systems with symmetry. We analyze the structure of a heteroclinic network as well as dynamics on and near the network. In particular, we introduce a notion of ‘depth ’ for a heteroclinic network (simple cycles between equilibria have depth one), characterize the connections and discuss issues of attraction, robustness and asymptotic behavior near a network. We consider in detail a system of nine coupled cells where one can find a variety of complicated, yet robust, dynamics in simple polynomial vector fields that possess symmetries. For this model system, we find and prove the existence of depth

In this paper we shall show that there exists a polynomial unimodal map f : [0; 1] ! [0; 1] which is non-renormalizable (therefore for each x from a residual set, !(x) is equal to an interval), for which !(c) is a Cantor set and for which !(x) = !(c) for Lebesgue almost all x. So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor [Mil]. 1 Introduction One of the central themes in the theory of dynamical systems is the concept of attractors. However, there is no complete consensus about the `correct' denition of this notion. In particular it is not clear whether an attractor should attract a topologically big set or a set which is large in a metric sense. So, if f : M !M is a dynamical system dened on a manifold M , then we could dene a closed forward invariant set X to be a topological respectively a metric attractor if 1. its basin B(X) = fx ; !(x) Xg contains a residual subset of an open neigh...

Suppose a chaotic attractor A in an invariant subspace loses stability on varying a parameter. At the point of loss of stability, the most positive Lyapunov exponent of the natural measure on A crosses zero at what has been called a `blowout' bifurcation. We introduce the notion of an essential basin of an attractor with an invariant measure ¯. This is the set of points such that the set of measures defined by the sequence of measures 1 n P n\Gamma1 k=0 ffi f k (x) has an accumulation point in the support of ¯. We characterise supercritical and subcritical scenarios according to whether the Lebesgue measure of the essential basin of A is positive or zero. We study a drift-diffusion model and a model class of piecewise linear mappings of the plane. In the supercritical case, we find examples where a Lyapunov exponent of the branch of attractors may be positive (`hyperchaos') or negative, depending purely on the dynamics far from the invariant subspace. For the mappings we find asymp...

We study statistical properties of a family of maps acting in the space of integer valued sequences, which model dynamics of simple deterministic traffic flows. We obtain asymptotic (as time goes to infinity) properties of trajectories of those maps corresponding to arbitrary initial configurations in terms of statistics of densities of various patterns and describe weak attractors of these systems and the rate of convergence to them. Previously only the so called regular initial configurations (having a density with only finite fluctuations of partial sums around it) in the case of a slow particles model (with the maximal velocity 1) have been studied rigorously. Applying ideas borrowed from substitution dynamics we are able to reduce the analysis of the traffic flow models corresponding to the multi-lane traffic and to the flow with fast particles (with velocities greater than 1) to the simplest case of the flow with the one-lane traffic and slow particles, where the crucial technical step is the derivation of the exact life-time for a given cluster of particles. Applications to the optimal redirection of the multi-lane traffic flow are discussed as well.

This paper deals with strange attractors of S-unimodal maps f . It generalizes results from [BKNS] in the sense that very general topological conditions are given that either i) guarantee the existence of an absorbing Cantor set provided the critical point of f is su#ciently degenerate, or ii) prohibit the existence of an absorbing Cantor set altogether. As a byproduct we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for f . 1.

. Suppose a smooth dynamical system has an invariant subspace and a parameter that leaves the dynamics in the invariant subspace invariant while changing the normal dynamics. Then we say the parameter is a normal parameter, and much is understood of how attractors can change with normal parameters. Unfortunately, normal parameters do not arise very often in practise. We consider the behaviour of attractors near invariant subspaces on varying a parameter that does not preserve the dynamics in the invariant subspace but is otherwise generic, in a smooth dynamical system. We refer to such a parameter as "non-normal". If there is chaos in the invariant subspace that is not structurally stable, this has the effect of "blurring out" blowout bifurcations over a range of parameter values that we show can have positive measure in parameter space. Associated with such blowout bifurcations are bifurcations to attractors displaying a new type of intermittency that is phenomenologically similar to ...