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.
Ahlers and Cannell respond.
Ahlers and Cannell Respond: Folse and Mead make the interesting observation that the patterns observed by Ahlers, Cannell, and Steinberg^ rotate at a uniform rate for t ̂ 60. Here ris time measured in units of the horizontal thermal diffusion time L^/^, with t^^SJK and L=^ Rl d^ where d\s the vertical dimension of the container of radius i?, and K is the thermal diffusivity. The original authors were aware of the fact that their patterns rotated (see Ref. 10 of Ref. 1), but they had not established the uniformity of that rotation. With the orientation B of the pattern in radians and the time I'm units of t^, we obtain 0-= d9/dl^2.0x lO"""* from the analysis of Folse and Mead. Rotation of a convective flow pattern in a circular container had been studied previously by Steinberg, Ahlers, and Cannell.^ Those authors found that rotation occurred at the uniform rate ^ = LOxlO""^ for their case, but rotation ceased when a small change occurred in the pattern. This change yielded a pattern with a plane of mirror symmetry. Although more extensive experimental and theoretical studies of this problem are clearly desirable, it appears that the rotation is ''driven" by departures of the pattern from mirror symmetry. On the basis of the observations of Steinberg, Ahlers, and Cannell, we assume that a pattern with mirror symmetry consisting of two identical grain boundaries and otherwise straight rolls in a cylindrical cell does not rotate.^ Thus we also presume that the rotation of the patterns of Ref. 1 was not strictly uniform. Whenever two nearly identical grain boundaries were established, rotation probably ceased. However, this situation prevailed only for a small fraction of the time of the experiment reported in Ref. 1. We also presume that the instantaneous rotation rate depended upon the extent of departures from mirror symmetry. However these short-time variations in 0 are not resolved by the data of Ref. 1 or the analysis of Folse and Mead. Rather, the analysis shows that the coarse-grained value of 9 which can be inferred from the experiment is constant. We therefore presume that the extent of deviations of the patterns from mirror symmetry, on the same coarse-grained scale, is time independent. We believe that this lends further support to the idea that the process involved in the time dependence reported in Ref. 1 is stationary and not relaxational.