On the Symmetry-Breaking Bifurcation of Chaotic Attractors
Symmetry breaking is a common phenomenon in nature. (1) In the field of dynamical systems, the symmetry-breaking bifurcation of attractors is a subject of special interest. For limit cycles this phenomenon has been studied extensively, (a) especially in the Lorenz model. (3) Less is known about the symmetry breaking of chaotic attractors. A symmetry-recovering crisis has been discovered experimentally and modeled in ref. 4. Here we consider the symmetry breaking of strange attractors from another point of view in a more general class of systems and we show that a new type of crisis (5) may exist leading to an anomalous band splitting (6) or to a symmetry-breaking transition of the strange attractor, depending on the actual values of the parameters. We investigate Lorenz-type systems characterized by the property that trajectories on the attractor pass close to a saddle point. This saddle has a two-dimensional stable and a one-dimensional unstable invariant manifold. Therefore, trajectories approaching the stable manifold may stay for an
Riddled Basins and Coupled Dynamical Systems
The collective dynamics of groups of coupled dynamical systems is of great interest for understanding spontaneous pattern formation in biological and many other systems; see for example [58]. One can learn a lot about such systems by first studying idealized cases where the systems are perfectly identical; this approach has been very successful in understanding general properties of synchronization as well as particular applications; see for example [77]. In this chapter we consider how this can lead to the appearance of attractors with riddled basins. These basins appear because symmetries of dynamical systems force the presence of invariant submanifolds; the attractors within invariant manifolds may be only weakly attracting transverse to the invariant manifold and this leads to a basin structure that is, roughly speaking, full of holes. From a theoretical point of view, this behaviour is of interest because it seems strange or pathological but is in some sense common. From a practical point of view, this behaviour points towards the presence of extreme sensitivity of the dynamics to noise, also called ‘bubbling’ of attractors. Most interestingly, if we consider generic dynamics within a class of symmetric systems, riddled basins can appear as a robust phenomenon; they can be persistent for open sets of parameters of the system. For the remainder of this section we briefly discuss basins of attraction and a motivating example of a piecewise linear map with an explicitly computable riddled basin attractor. More general properties of riddled sets and basins are discussed in Sect. 2 including their noise sensitivity. This is followed in Sect. 3 by a discussion of the use of symmetries, ergodic measures and Lyapunov exponents tools for identifying riddled basins; we also discuss anisotropic riddling in Sect. 4 along the lines of [7]. Finally in Sect. 5 we outline a few open problems related to riddling phenomena.
From the World Literature
LIST OF ILLUSTRATIONS ix ACKNOWLEDGMENTS xi INTRODUCTION: Goethe Coins a Phrase 1 PART ONE: CIRCULATION Chapter 1: Gilgamesh's Quest 39 Chapter 2: The Pope's Blowgun 78 Chapter 3: From the Old World to the Whole World 110 PART TWO: TRANSLATION Chapter 4: Love in the Necropolis 147 Chapter 5: The Afterlife of Mechthild von Magdeburg 170 Chapter 6: Kafka Comes Home 187 PART THREE: PRODUCTION Chapter 7: English in the World 209 Chapter 8: Rigoberta Menchu in Print 231 Chapter 9: The Poisoned Book 260 CONCLUSION: World Enough and Time 281 BIBLIOGRAPHY 305 INDEX 319