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.
Influence of electric fields on friction electrification between metals and borosilicate glass
Abstract In a cylindrical geometry, electric fields were applied through the entire contact area between the sliding metal and the glass surface. The maximum electric charges developed on the metal were measured and plotted against the applied voltage producing the field. The metals were high-purity nickel and gold. The borosilicate glass was Pyrex, brand 7740. The experiments with nickel, which has a work function greater than that of the glass, were performed in 10 −7 mm Hg vacuum. The experiments with gold, with a work function smaller than that of glass, were carried out in air at atmospheric pressure. The results show that: 1. 1) There is both asymmetry and a reversed slope portion in the charge vs. voltage plot. 2. 2) For a certain range of reversing fields, the electric charge developed on the nickel is always negative and the charge on gold is always positive.