From limit cycles to strange attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.Comment: 27 page

On the critical solutions in coating and rimming flow on a uniformly rotating horizontal cylinder

We use a combination of analytical and numerical techniques to re-examine the question posed by Moffatt [Journal de Mæ#169;canique 16 (1977) 651-673] of determining the critical weights of fluid that can be maintained per unit length in a steady, smoothly varying, two-dimensional film on either the outside ('coating flow') or the inside ('rimming flow') of a rotating horizontal cylinder. We use a pseudospectral method to obtain highly accurate numerical solutions for steady Stokes flow on a cylinder and hence to calculate the critical weights. In particular, these calculations reveal that the behaviour of the critical solutions in the thin-film limit 0 (where is the aspect ratio of the film) in an inner region near the horizontal on the ascending side of the cylinder (where Moffatt's leading-order outer solution has a corner) are not captured by naive outer asymptotic solutions in integer powers of . Motivated by these numerical results we obtain the uniformly valid critical asymptotic solutions in the thin-film limit to sufficient accuracy to enable us to calculate the critical fluxes and weights to accuracies o(4/3 (log )-3) and o(4/3 (log )-2) relative to Moffatt's leading-order values, respectively. We find that our asymptotic solutions for the critical weights are in good agreement with the numerically calculated results over a wide range of values of . In particular, our numerical and asymptotic calculations show that, even in the absence of surface-tension effects, the corner predicted by Moffatt's leading-order outer solution never actually occurs. In practice the higher-order terms obtained in the present work dominate the formally lower-order term that can be obtained straightforwardly without a detailed knowledge of the solution in the inner region, and so these higher-order terms must be included in order to obtain accurate corrections to Moffatt's leading-order value of the critical weight. In particular, in practice the critical weights in both coating and rimming flow always exceed Moffatt's value

A slender rivulet of a power-law fluid driven by either gravity or a constant shear stress at the free surface

Similarity solutions that describe the flow of a slender non-uniform rivulet of non-Newtonian power-law fluid down an inclined plane are obtained. Rivulets driven by either gravity or a constant shear stress at the free surface are investigated, and in both cases solutions are obtained for both weak and strong surface-tension effects. We find that, despite the rather different physical mechanisms driving the flow, the solutions for gravity-driven and shear-stress-driven rivulets are qualitatively similar. When surface-tension effects are weak there is a unique similarity solution in which the transverse rivulet profile has a single global maximum. This solution represents both a diverging and shallowing sessile rivulet and a converging and deepening pendent rivulet. On the other hand, when surface-tension effects are strong there is a one-parameter family of similarity solutions in which the transverse profile of a diverging and shallowing rivulet has one global maximum, while that of a converging and deepening rivulet has either one global maximum or two equal global maxima. We also show how the present similarity solutions can be modified to accommodate a fixed-contact-angle condition at the contact line by incorporating sufficiently strong slip at the solid/fluid interface into the model

Fractal Properties of Robust Strange Nonchaotic Attractors in Maps of Two or More Dimensions

We consider the existence of robust strange nonchaotic attractors (SNA's) in a simple class of quasiperiodically forced systems. Rigorous results are presented demonstrating that the resulting attractors are strange in the sense that their box-counting dimension is N+1 while their information dimension is N. We also show how these properties are manifested in numerical experiments.Comment: 9 pages, 14 figure

Detectability of non-differentiable generalized synchrony

Generalized synchronization of chaos is a type of cooperative behavior in directionally-coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is non-differentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases, in which a differentiable synchronization function exists, are considered to make sense in practice. We show that this viewpoint is not always correct and the non-differentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of non-differentiable generalized synchronization.Comment: 8 pages, 8 figures, submitted to PR

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $\sigma$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $\Pi$ the set of all Borel probabilities $\pi$ on $X\times X$, such that, both its $x$ and $y$ marginal are $\sigma$-invariant probabilities. We are interested in the optimal plan $\pi$ which minimizes $\int c d \pi$ among the probabilities on $\Pi$. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on $c$. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs $c$ the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan $\pi$ by convex combinations of plans such that the support projects in periodic orbits

Changing patterns of eastern Mediterranean shellfish exploitation in the Late Glacial and Early Holocene: Oxygen isotope evidence from gastropod in Epipaleolithic to Neolithic human occupation layers at the Haua Fteah cave, Libya

The seasonal pattern of shellfish foraging at the archaeological site of Haua Fteah in the Gebel Akhdar, Libya was investigated from the Epipaleolithic to the Neolithic via oxygen isotope (d18O) analyses of the topshell Phorcus (Osilinus) turbinatus. To validate this species as faithful year-round palaeoenvironmental recorder, the intra-annual variability of d18O in modern shells and sea water was analysed and compared with measured sea surface temperature (SST). The shells were found to be good candidates for seasonal shellfish forging studies as they preserve nearly the complete annual SST cycle in their shell d18O with minimal slowing or stoppage of growth. During the terminal Pleistocene Early Epipaleolithic (locally known as the Oranian, with modeled dates of 17.2-12.5 ka at 2sigma probability, Douka et al., 2014), analysis of archaeological specimens indicates that shellfish were foraged year-round. This complements other evidence from the archaeological record that shows that the cave was more intensively occupied in this period than before or afterwards. This finding is significant as the period of the Oranian was the coldest and driest phase of the last glacial cycle in the Gebel Akhdar, adding weight to the theory that the Gebel Akhdar may have served as a refugium for humans in North Africa during times of global climatic extremes. Mollusc exploitation in the Latest Pleistocene and Early Holocene, during the Late Epipaleolithic (locally known as the Capsian, c. 12.7 to 9 ka) and the Neolithic (c. 8.5 to 5.4 ka), occurred predominantly during winter. Other evidence from these archaeological phases shows that hunting activities occurred during the warmer months. Therefore, the timing of Holocene shellfish exploitation in the Gebel Akhdar may have been influenced by the seasonal availability of other resources at these times and possibly shellfish were used as a dietary supplement when other foods were less abundant

Quantum Tomography under Prior Information

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d-4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log_2(d) smaller than this upper bound.Comment: v3: There was a mistake in the derived finer upper bound in Theorem 3. The corrected upper bound is +1 to the earlier versio