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.
Observations of nearshore infragravity waves: Seaward and shoreward propagating components
[1] The variation of seaward and shoreward infragravity energy fluxes across the shoaling and surf zones of a gently sloping sandy beach is estimated from field observations and related to forcing by groups of sea and swell, dissipation, and shoreline reflection. Data from collocated pressure and velocity sensors deployed between 1 and 6 m water depth are combined, using the assumption of cross-shore propagation, to decompose the infragravity wave field into shoreward and seaward propagating components. Seaward of the surf zone, shoreward propagating infragravity waves are amplified by nonlinear interactions with groups of sea and swell, and the shoreward infragravity energy flux increases in the onshore direction. In the surf zone, nonlinear phase coupling between infragravity waves and groups of sea and swell decreases, as does the shoreward infragravity energy flux, consistent with the cessation of nonlinear forcing and the increased importance of infragravity wave dissipation. Seaward propagating infragravity waves are not phase coupled to incident wave groups, and their energy levels suggest strong infragravity wave reflection near the shoreline. The cross-shore variation of the seaward energy flux is weaker than that of the shoreward flux, resulting in cross-shore variation of the squared infragravity reflection coefficient (ratio of seaward to shoreward energy flux) between about 0.4 and 1.5.