Integrals of motion and the shape of the attractor for the Lorenz model

In this paper, we consider three-dimensional dynamical systems, as for example the Lorenz model. For these systems, we introduce a method for obtaining families of two-dimensional surfaces such that trajectories cross each surface of the family in the same direction. For obtaining these surfaces, we are guided by the integrals of motion that exist for particular values of the parameters of the system. Nonetheless families of surfaces are obtained for arbitrary values of these parameters. Only a bounded region of the phase space is not filled by these surfaces. The global attractor of the system must be contained in this region. In this way, we obtain information on the shape and location of the global attractor. These results are more restrictive than similar bounds that have been recently found by the method of Lyapunov functions.Comment: 17 pages,12 figures. PACS numbers : 05.45.+b / 02.30.Hq Accepted for publication in Physics Letters A. e-mails : [email protected] & [email protected]

Aubry transition studied by direct evaluation of the modulation functions of infinite incommensurate systems

Incommensurate structures can be described by the Frenkel Kontorova model. Aubry has shown that, at a critical value K_c of the coupling of the harmonic chain to an incommensurate periodic potential, the system displays the analyticity breaking transition between a sliding and pinned state. The ground state equations coincide with the standard map in non-linear dynamics, with smooth or chaotic orbits below and above K_c respectively. For the standard map, Greene and MacKay have calculated the value K_c=.971635. Conversely, evaluations based on the analyticity breaking of the modulation function have been performed for high commensurate approximants. Here we show how the modulation function of the infinite system can be calculated without using approximants but by Taylor expansions of increasing order. This approach leads to a value K_c'=.97978, implying the existence of a golden invariant circle up to K_c' > K_c.Comment: 7 pages, 5 figures, file 'epl.cls' necessary for compilation provided; Revised version, accepted for publication in Europhysics Letter

Prepontine non-giant neurons drive flexible escape behavior in zebrafish

Many species execute ballistic escape reactions to avoid imminent danger. Despite fast reaction times, responses are often highly regulated, reflecting a trade-off between costly motor actions and perceived threat level. However, how sensory cues are integrated within premotor escape circuits remains poorly understood. Here, we show that in zebrafish, less precipitous threats elicit a delayed escape, characterized by flexible trajectories, which are driven by a cluster of 38 prepontine neurons that are completely separate from the fast escape pathway. Whereas neurons that initiate rapid escapes receive direct auditory input and drive motor neurons, input and output pathways for delayed escapes are indirect, facilitating integration of cross-modal sensory information. These results show that rapid decision-making in the escape system is enabled by parallel pathways for ballistic responses and flexible delayed actions and defines a neuronal substrate for hierarchical choice in the vertebrate nervous system

Observations of Hoppel Minima in CCN Spectra in Oklahoma

Aerosols are one of the most fundamental keys to understanding the future state of the climate. Aerosols impact the radiation budget of the Earth in numerous ways and are poorly understood. Some aerosols can act as cloud condensation nuclei (CCN) and can significantly change the properties of clouds; this is known as the Indirect Aerosol Effect (IAE) and it remains the largest climate change uncertainty. Most studies concerning CCN and the impacts of the CCN distributions occur over the ocean, leaving questions about the processing occurring over the continents. Eleven days of measurements from the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) site were taken from an Aerosol Intensive Operational Period (IOP) during May 2003. A ground based CCN spectrometer and differential mobility analyzer (DMA) were deployed to study the distributions of the CCN spectra and dry aerosol size distributions. 268 measurement periods were sorted by their spectral shapes by using two rating systems. Case studies of the characteristics of the spectra observed during specific times of day or particular meteorological conditions were created and it was shown that meteorological conditions have a significant impact on the shapes of the CCN distributions. Back trajectories were also analyzed and shown to have an even larger impact on the observations of the Hoppel Minima, a minima located between the processed and unprocessed CCN modes. Using vertical velocity and back trajectories along with numerous meteorological measurements it can be shown that cloud processing is not only occurring over the continent but transport of the cloud processed air to the surface is also occurring. The Hoppel Minima during this Oklahoma project had a mean critical supersaturation (Sc) of 0.68%

Skylab investigation of the upwelling off the Northwest coast of Africa

The upwelling off the NW coast of Africa in the vicinity of Cape Blanc was studied in February - March 1974 from aircraft and in September 1973 from Skylab. The aircraft study was designed to determine the effectiveness of a differential radiometer in quantifying surface chlorophyll concentrations. Photographic images of the S190A Multispectral Camera and the S190B Earth Terrain Camera from Skylab were used to study distributional patterns of suspended material and to locate ocean color boundaries. The thermal channel of the S192 Multispectral Scanner was used to map sea-surface temperature distributions offshore of Cape Blanc. Correlating ocean color changes with temperature gradients is an effective method of qualitatively estimating biological productivity in the upwelling region off Africa

Engineering stochasticity in gene expression

Stochastic fluctuations (noise) in gene expression can cause members of otherwise genetically identical populations to display drastically different phenotypes. An understanding of the sources of noise and the strategies cells employ to function reliably despite noise is proving to be increasingly important in describing the behavior of natural organisms and will be essential for the engineering of synthetic biological systems. Here we describe the design of synthetic constructs, termed ribosome competing RNAs (rcRNAs), as a means to rationally perturb noise in cellular gene expression. We find that noise in gene expression increases in a manner proportional to the ability of an rcRNA to compete for the cellular ribosome pool. We then demonstrate that operons significantly buffer noise between coexpressed genes in a natural cellular background and can even reduce the level of rcRNA enhanced noise. These results demonstrate that synthetic genetic constructs can significantly affect the noise profile of a living cell and, importantly, that operons are a facile genetic strategy for buffering against noise

Observable Properties of Orbits in Exact Bumpy Spacetimes

We explore the properties of test-particle orbits in "bumpy" spacetimes - stationary, reflection-symmetric, asymptotically flat solutions of Einstein equations that have a non-Kerr (anomalous) higher-order multipole-moment structure but can be tuned arbitrarily close to the Kerr metric. Future detectors should observe gravitational waves generated during inspirals of compact objects into supermassive central bodies. If the central body deviates from the Kerr metric, this will manifest itself in the emitted waves. Here, we explore some of the features of orbits in non-Kerr spacetimes that might lead to observable signatures. As a basis for this analysis, we use a family of exact solutions proposed by Manko & Novikov which deviate from the Kerr metric in the quadrupole and higher moments, but we also compare our results to other work in the literature. We examine isolating integrals of the orbits and find that the majority of geodesic orbits have an approximate fourth constant of the motion (in addition to the energy, angular momentum and rest mass) and the resulting orbits are tri-periodic to high precision. We also find that this fourth integral can be lost for certain orbits in some oblately deformed Manko-Novikov spacetimes. However, compact objects will probably not end up on these chaotic orbits in nature. We compute the location of the innermost stable circular orbit (ISCO) and find that the behavior of orbtis near the ISCO can be qualitatively different depending on whether the ISCO is determined by the onset of an instability in the radial or vertical direction. Finally, we compute periapsis and orbital-plane precessions for nearly circular and nearly equatorial orbits in both the strong and weak field, and discuss weak-field precessions for eccentric equatorial orbits.Comment: 42 pages, 20 figures, accepted by Phys. Rev. D, v2 has minor changes to make it consistent with published versio

The role of quantum fluctuations in the optomechanical properties of a Bose-Einstein condensate in a ring cavity

We analyze a detailed model of a Bose-Einstein condensate trapped in a ring optical resonator and contrast its classical and quantum properties to those of a Fabry-P{\'e}rot geometry. The inclusion of two counter-propagating light fields and three matter field modes leads to important differences between the two situations. Specifically, we identify an experimentally realizable region where the system's behavior differs strongly from that of a BEC in a Fabry-P\'{e}rot cavity, and also where quantum corrections become significant. The classical dynamics are rich, and near bifurcation points in the mean-field classical system, the quantum fluctuations have a major impact on the system's dynamics.Comment: 11 pages, 11 figures, submitted to PR

Spacetime Encodings II - Pictures of Integrability

I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a two degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers' intuition about how integrability arises, and to summarize some of the known facts about two degree of freedom systems. Evidence is given, in the form of orbit-crossing structure, that geodesics in SAV spacetimes might admit, a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).Comment: 11 pages, 10 figure