Coupling of bouncing-ball modes to the chaotic sea and their counting function

We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_bb(E) ~ E^\delta. For the stadium billiard we find agreement with the previous result \delta = 3/4. For the cosine billiard we derive \delta = 5/8, which is confirmed numerically and is well below the previously predicted upper bound \delta=9/10.Comment: 10 pages, 6 figure

Visualization and comparison of classical structures and quantum states of 4D maps

For generic 4D symplectic maps we propose the use of 3D phase-space slices which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods like 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the first comparison of Husimi functions of eigenstates of 4D maps with classical phase space structures. This confirms the semi-classical eigenfunction hypothesis for 4D maps.Comment: For videos with rotated view of the 3D phase-space slices in high resolution see http://www.comp-phys.tu-dresden.de/supp

Partial Weyl Law for Billiards

For two-dimensional quantum billiards we derive the partial Weyl law, i.e. the average density of states, for a subset of eigenstates concentrating on an invariant region $\Gamma$ of phase space. The leading term is proportional to the area of the billiard times the phase-space fraction of $\Gamma$. The boundary term is proportional to the fraction of the boundary where parallel trajectories belong to $\Gamma$. Our result is numerically confirmed for the mushroom billiard and the generic cosine billiard, where we count the number of chaotic and regular states, and for the elliptical billiard, where we consider rotating and oscillating states.Comment: 5 pages, 3 figures, derivation extended, cosine billiard adde

Bifurcations of families of 1D-tori in 4D symplectic maps

The regular structures of a generic 4D symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1D-tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example we study two coupled standard maps by visualizing the elliptic and hyperbolic 1D-tori in a 3D phase-space slice, local 2D projections, and frequency space. The observed bifurcations are consistent with analytical predictions previously obtained for quasi-periodically forced oscillators. Moreover, the new families emerging from such a bifurcation form the skeleton of the corresponding resonance channel.Comment: 14 pages, 10 figures. For videos of 3D phase-space slices see http://www.comp-phys.tu-dresden.de/supp

3D billiards: visualization of regular structures and trapping of chaotic trajectories

The dynamics in three-dimensional billiards leads, using a Poincar\'e section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincar\'e recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.Comment: 20 pages, 11 figures. For videos of 3D phase-space slices and time-resolved animations see http://www.comp-phys.tu-dresden.de/supp

Aortic volume determines global end-diastolic volume measured by transpulmonary thermodilution

BACKGROUND: Global end-diastolic volume (GEDV) measured by transpulmonary thermodilution is regarded as indicator of cardiac preload. A bolus of cold saline injected in a central vein travels through the heart and lung, but also the aorta until detection in a femoral artery. While it is well accepted that injection in the inferior vena cava results in higher values, the impact of the aortic volume on GEDV is unknown. In this study, we hypothesized that a larger aortic volume directly translates to a numerically higher GEDV measurement. METHODS: We retrospectively analyzed data from 88 critically ill patients with thermodilution monitoring and who did require a contrast-enhanced thoraco-abdominal computed tomography scan. Aortic volumes derived from imaging were compared with GEDV measurements in temporal proximity. RESULTS: Median aortic volume was 194 ml (interquartile range 147 to 249 ml). Per milliliter increase of the aortic volume, we found a GEDV increase by 3.0 ml (95% CI 2.0 to 4.1 ml, p < 0.001). In case a femoral central venous line was used for saline bolus injection, GEDV raised additionally by 2.1 ml (95% CI 0.5 to 3.7 ml, p = 0.01) per ml volume of the vena cava inferior. Aortic volume explained 59.3% of the variance of thermodilution-derived GEDV. When aortic volume was included in multivariate regression, GEDV variance was unaffected by sex, age, body height, and weight. CONCLUSIONS: Our results suggest that the aortic volume is a substantial confounding variable for GEDV measurements performed with transpulmonary thermodilution. As the aorta is anatomically located after the heart, GEDV should not be considered to reflect cardiac preload. Guiding volume management by raw or indexed reference ranges of GEDV may be misleading

Regular-to-Chaotic Tunneling Rates: From the Quantum to the Semiclassical Regime

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given $\hbar$-regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.Comment: 4 pages, 3 figures, reference added, small text change

Experimental Observation of Resonance-Assisted Tunneling

We present the first experimental observation of resonance-assisted tunneling, a wave phenomenon, where regular-to-chaotic tunneling is strongly enhanced by the presence of a classical nonlinear resonance chain. For this we use a microwave cavity made of oxygen free copper with the shape of a desymmetrized cosine billiard designed with a large nonlinear resonance chain in the regular region. It is opened in a region, where only chaotic dynamics takes place, such that the tunneling rate of a regular mode to the chaotic region increases the line width of the mode. Resonance-assisted tunneling is demonstrated by (i) a parametric variation and (ii) the characteristic plateau and peak structure towards the semiclassical limit.Comment: 5 pages, 2 figure

Mechanisms of eye development and evolution of the arthropod visual system: The lateral eyes of myriapoda are not modified insect ommatidia

AbstractThe lateral eyes of Crustacea and Insecta consist of many single optical units, the ommatidia, that are composed of a small, strictly determined and evolutionarily conserved set of cells. In contrast, the eyes of Myriapoda (millipedes and centipedes) are fields of optical units, the lateral ocelli, each of which is composed of up to several hundreds of cells. For many years these striking differences between the lateral eyes of Crustacea/Insecta versus Myriapoda have puzzled evolutionary biologists, as the Myriapoda are traditionally considered to be closely related to the Insecta. The prevailing hypothesis to explain this paradox has been that the myriapod fields of lateral ocelli derive from insect compound eyes by disintegration of the latter into single ommatidia and subsequent fusion of several ommatidia to form multicellular ocelli. To provide a fresh view on this problem, we counted and mapped the arrangement of ocelli during postembryonic development of a diplopod. Furthermore, the arrangement of proliferating cells in the eyes of another diplopod and two chilopods was monitored by labelling with the mitosis marker bromodeoxyuridine. Our results confirm that during eye growth in Myriapoda new elements are added to the side of the eye field, which extend the rows of earlier-generated optical units. This pattern closely resembles that in horseshoe crabs (Chelicerata) and Trilobita. We conclude that the trilobite, xiphosuran, diplopod and chilopod mechanism of eye growth represents the ancestral euarthropod mode of visual-system formation, which raises the possibility that the eyes of Diplopoda and Chilopoda may not be secondarily reconstructed insect eyes