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
Chaotic mixing in crystalline granular media
We study the Lagrangian kinematics of steady three-dimensional Stokes flow over simple cubic (SC) and body-centred cubic (BCC) lattices of close-packed spheres, and uncover the mechanisms governing chaotic mixing in these crystalline structures. Due to the cusp-shaped sphere contacts, the topology of the skin friction field is fundamentally different to that of continuous (non-granular) media, such as open pore networks, with significant implications for fluid mixing. Weak symmetry breaking of the flow orientation with respect to the lattice symmetries imparts a transition from regular to strong chaotic mixing in the BCC lattice, whereas the SC lattice only exhibits weak mixing. Whilst the SC and BCC lattices posses the same symmetry point group, these differences are explained in terms of their space groups. This insight is used to develop accurate predictions of the Lyapunov exponent distribution over the parameter space of mean flow orientation. These results point to a general theory of mixing and dispersion based upon the inherent symmetries of arbitrary crystalline structures.
Rotation of time-dependent flow patterns near convective threshold.
In a recent Letter, Ahlers, Cannell, and Steinberg^ report shadowgraphs of time-dependent convective flow patterns in a cylindrical chamber (aspect ratio L = 15.0) at Rayleigh number above convective threshold, /? = l,141i?cThe purpose of this Comment is to point out a rotation of their observed straight-line roll patterns that obeys a simple power law for times greater than sixty horizontal diffusion times ( / ; i -3 .33h) . Ahlers, Cannell, and Steinberg showed shadowgraphs of flow patterns for 0 ^ ^^200/), , where t is the time since their initial pattern was rendered unstable. After a transient stage, a time-dependent pattern was established for t^^Ot^ and they observed that ' 'a characteristic feature of this late-time evolution is the repeated establishment (albeit at random intervals) of patterns consisting primarily of nearly straight rolls terminating almost perpendicular to the boundary and two defect combinations nearly on opposite sides of the cell." More complicated patterns with an additional grain boundary appeared between each roll pattern occurrence. Figure 1 shows two representative examples of their observed patterns at r = 89.4r/̂ and 100.6/),. We have measured the angle 9 between the parallel lines in the pattern and a fixed vertical line for the eighteen patterns in Figs. 2, 3, and 4 of Ref. 1. We assumed that the angle 9 was monotonically increasing with time; whenever the measured angle decreased at successive times we added TT radians to the angle (the lines in Fig. 1 are not directed and their an-