Embedding dynamics for round-off errors near a periodic orbit

Mixing and the fractal geometry of piecewise isometries

Mathematical concepts often have applicability in areas that may have surprised their original developers. This is the case with piecewise isometries (PWIs), which transform an object by cutting it into pieces that are then rearranged to reconstruct the original object, and which also provide a paradigm to study mixing via cutting and shuffling in physical sciences and engineering. Every PWI is characterized by a geometric structure called the exceptional set, E, whose complement comprises nonmixing regions in the domain. Varying the parameters that define the PWI changes both the structure of E as well as the degree of mixing the PWI produces, which begs the question of how to determine which parameters produce the best mixing. Motivated by mixing of yield stress materials, for example granular media, in physical systems, we use numerical simulations of PWIs on a hemispherical shell and examine how the fat fractal properties of E relate to the degree of mixing for any particular PWI. We present numerical evidence that the fractional coverage of E negatively correlates with the intensity of segregation, a standard measure for the degree of mixing, which suggests that fundamental properties of E such as fractional coverage can be used to predict the effectiveness of a particular PWI as a mixing mechanism.

Mixing with piecewise isometries on a hemispherical shell

We introduce mixing with piecewise isometries (PWIs) on a hemispherical shell, which mimics features of mixing by cutting and shuffling in spherical shells half-filled with granular media. For each PWI, there is an inherent structure on the hemispherical shell known as the exceptional set E, and a particular subset of E, E+, provides insight into how the structure affects mixing. Computer simulations of PWIs are used to visualize mixing and approximations of E+ to demonstrate their connection. While initial conditions of unmixed materials add a layer of complexity, the inherent structure of E+ defines fundamental aspects of mixing by cutting and shuffling.

Unstable periodic orbits and discretization cycles

Limit cycles that arise from discretizing the variable(s) of a nonlinear map are generally found to shadow individual unstable periodic orbits (UPOs) of the corresponding continuous map. In a few cases the discretization cycles can only be explained with other mechanisms, such as the near-occurrence of an UPO, or crossover between two or more UPOs.

Coarse-grained probabilistic automata mimicking chaotic systems

Discretization of phase space usually nullifies chaos in dynamical systems. We show that if randomness is associated with discretization dynamical chaos may survive and be indistinguishable from that of the original chaotic system, when an entropic, coarse-grained analysis is performed. Relevance of this phenomenon to the problem of quantum chaos is discussed.

Numerical analysis for a discontinuous rotation of the torus

In this paper, we study a class of piecewise rotations on the square. While few theoretical results are known about them, we numerically compute box-counting dimensions, correlation dimensions and complexity of the symbolic language produced by the system. Our results seem to confirm a conjecture that the fractal dimension of the exceptional set is two, as well as indicate that the dynamics on it is not ergodic. We also explore a relationship between the piecewise rotations and discretized rotations on lattices Z(2n).