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.

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...

. 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 ...

For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient `bursts'. The time between the transients increases as the trajectory approaches the cycle. This behaviour for a cycle connecting symmetrically related chaotic sets has been called `cycling chaos' by Dellnitz et al. (1995). We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in th...

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The synchronization of chaotic systems has received a great deal of attention. However, most of the literature has focused on systems that possess invariant manifolds that persist as the coupling is varied. In this paper, we describe the process whereby synchronization is lost in systems of nonidentical coupled chaotic oscillators without special symmetries. We qualitatively and quantitatively analyze such systems in terms of the evolution of the unstable periodic orbit structure. Our results are illustrated with data from physical experiments. 1.

In a previous paper (Comput. Methods Appl. Mech. Engrg 170, 223-237, 1999) we introduced a new method for computing the dominant Lyapunov exponent of a chaotic map by using spatial integration involving a matrix norm. We conjectured that this sequence of integrals decayed proportional to 1/n. We now prove this conjecture and derive a bound on the next term in the asymptotic expansion of the terms in the sequence. The Henon map and a system of coupled Duffing oscillators are explored in detail in the light of these theoretical results.

We consider examples of loss of stability of chaotic attractors in invariant subspaces (blowouts) that occur on varying two parameters, i.e. codimension two blowout bifurcations. Such bifurcations act as organising centres for nearby codimension one behaviour, analogous to the case for codimension two bifurcations of equilibria. We consider examples of blowout bifurcations showing change of criticality, blowouts that occur into two different invariant subspaces and interact, blowouts that occur with onset of hyperchaos, interaction of blowout and symmetry increasing bifurcations and collision of blowout bifurcations. As in the case of bifurcation of equilibria, there are many cases in which one can infer the presence and form of secondary bifurcations and associated branches of attractors. There is presently no generic theory of such higher codimension blowouts (there is not even such a theory for codimension one blowouts). We want to present a number of examples that would need to be ...

This paper examines an electronic oscillator forced by a pseudo-random noise signal. We give evidence of the existence of one or more random attractors for the system depending on noise amplitude and system parameters. These random attractors may appear to be random fixed points or random chaotic attractors. In the latter case, we observe a form of intermittent synchronization of the response of the system to the noise signal. We show how this can be understood as on-off intermittency in an extended system. Keywords: Chaotic dynamics, electronic oscillator, randomly forced system. 1 Introduction Although one would like to model real systems in terms of deterministic systems, inevitably noise will be present from a number of sources. One can classify many influences that cannot be modelled in a deterministic way as random fluctuations; for example, thermal noise in electronic systems can be explained in terms of kinetic noise of electrons, but modelling a circuit by the motion of indi...

n! R n is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. We study the orbit stability and stability of mixing of T (x; y) = (f (x); g(x; y)) under deterministic and random perturbation of g. We show that such systems are stable in the sense that for any ffl? 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance ffl of each other except for a fraction ffl of the time. We also show that such systems preserve ergodicity and higher order mixing properties under deterministic perturbation of g and perturbation of g by i.i.d. additive noise. Furthermore we show that the invariant measure for the perturbed system is continuous (in the Hutchinson metric) as a function of the size of the perturbation to g (Lipschitz topology) and the noise distribution. Our results have applications to the stability of Iterated Function Systems which "contract on average". 1 Introduction In this paper we address various questions to do with the stability of dynamically invariant subspaces. An obvious and topical application of our results is in the area of chaotic synchronisation [27] and its use in communication systems [12, 19]. Consider, for example, a dynamical system which has been divided into 2 parts: d dt