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At least 127 records · Page 7

Chaotic instability in the three-body problem

The existence of global exponential instability leading to chaos in plane (nonrigid) motions of the three-body system is demonstrated. In the presence of Newtonian attracting forces the trajectories of the three-body system in the configuration space will no longer be geodesic, and their convergence will depend on the geodesic curvature in addition to the Gaussian curvature. It is noted that the global exponential instability occurs in the n-body problem for n greater than 3 if at least two angular coordinates are disturbed, and that the chaotic instability is not necessarily accompanied by an exponential increase of the distance between the mass-points, but is associated with the exponential divergence of trajectories in the configuration space.

Zak, M.↗

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

Bartels, Robert E.↗

Fomalhaut's Debris Disk and Planet: Constraining the Mass of Formalhaut B from Disk Morphology

Following the optical imaging of exoplanet candidate Fomalhaut b (Fom b), we present a numerical model of how Fomalhaut's debris disk is gravitationally shaped by a single interior planet. The model is simple, adaptable to other debris disks, and can be extended to accommodate multiple planets. If Fom b is the dominant perturber of the belt, then to produce the observed disk morphology it must have a mass M(sub pl) < 3M(sub J), an orbital semimajor axis a(sub pl) > 101.5AU, and an orbital eccentricity e(sub pl) = 0.11 - 0.13. These conclusions are independent of Fom b's photometry. To not disrupt the disk, a greater mass for Fom b demands a smaller orbit farther removed from the disk; thus, future astrometric measurement of Fom b's orbit, combined with our model of planet-disk interaction, can be used to determine the mass more precisely. The inner edge of the debris disk at a approximately equals 133AU lies at the periphery of Fom b's chaotic zone, and the mean disk eccentricity of e approximately equals 0.11 is secularly forced by the planet, supporting predictions made prior to the discovery of Fom b. However, previous mass constraints based on disk morphology rely on several oversimplifications. We explain why our constraint is more reliable. It is based on a global model of the disk that is not restricted to the planet's chaotic zone boundary. Moreover, we screen disk parent bodies for dynamical stability over the system age of approximately 100 Myr, and model them separately from their dust grain progeny; the latter's orbits are strongly affected by radiation pressure and their lifetimes are limited to approximately 0.1 Myr by destructive grain-grain collisions. The single planet model predicts that planet and disk orbits be apsidally aligned. Fomalhaut b's nominal space velocity does not bear this out, but the astrometric uncertainties are difficult to quantify. Even if the apsidal misalignment proves real, our calculated upper mass limit of 3 M(sub J) still holds. Parent bodies are evacuated from mean-motion resonances with Fom b; these empty resonances are akin to the Kirkwood gaps opened by Jupiter. The belt contains at least 3M(sub Earth) of solids that are grinding down to dust, their velocity dispersions stirred so strongly by Fom b that collisions are destructive. Such a large mass in solids is consistent with Fom b having formed in situ.

Chiang, E.↗

Periodicity and chaos from switched flow systems - Contrasting examples of discretely controlled continuous systems

We analyze two examples of the discrete control of a continuous variable system. These examples exhibit what may be regarded as the two extremes of complexity of the closed-loop behavior: one is eventually periodic, the other is chaotic. Our examples are derived from sampled deterministic flow models. These are of interest in their own right but have also been used as models for certain aspects of manufacturing systems. In each case, we give a precise characterization of the closed-loop behavior.

Chase, Christopher↗

Relative Time-scale for Channeling Events Within Chaotic Terrains, Margaritifer Sinus, Mars

A relative time scale for ordering channel and chaos forming events was constructed for areas within the Margaritifer Sinus region of Mars. Transection and superposition relationships of channels, chaotic terrain, and the surfaces surrounding them were used to create the relative time scale; crater density studies were not used. Channels and chaos in contact with one another were treated as systems. These systems were in turn treated both separately (in order to understand internal relationships) and as members of the suite of Martian erosional forms (in order to produce a combined, master time scale). Channeling events associated with chaotic terrain development occurred over an extended geomorphic period. The channels can be divided into three convenient groups: those that pre-date intercrater plains development post-plains, pre-chasma systems; and those associated with the development of the Vallis Marineris chasmata. No correlations with cyclic climatic changes, major geologic events in other regions on Mars, or triggering phenomena (for example, specific impact events) were found.

Janke, D.↗

Origin of Martian outflow channels - The eolian hypothesis

It has been recognized for many years that wind injects dust into the atmosphere on Mars and moves around thin layers of surface material. An investigation is conducted regarding the rates of eolian transport on Mars in the context of eolian channel formation. A model is developed for the formation of an erosional channel by sandblasting with materials transported by wind out of chaotic terrains. This model is applied to the interpretation of topographic measurements made in the chaotic terrain source area and channel of Aromatum Vallis, a small well-preserved channel system southwest of the Chryse basin, and constraints are developed on the eolian mechanism.

Cutts, J. A.↗

Topography of Valles Marineris: Implications for erosional and structural history

Compilation of a simplified geologic/geomorphic map onto digital terrain models of the Valles Marineris permitted an evaluation of elevations in the vicinity of the troughs and the calculation of depth of troughs below surrounding plateaus, thickness of deposits inside the troughs, volumes of void spaces above geologic/geomorphic units, and volumes of deposits. The central troughs north Ophir, north and central Candor, and north Melas Chasmata lie as much as 11 km below the adjacent plateaus. In Ophir and Candor chasmata, interior layered deposits reach 8 km in elevation. If the deposits are lacustrine and if all troughs were interconnected, lake waters standing 8 km high would have spilled out of Coprates Chasma onto the surrounding plateaus having surface elevations of only 4-5 km. On the other hand, the troughs may not have been interconnected at the time of interior-deposit emplacement; they may have formed isolated ancestral basins. The existence of such basins is supported by independent structural and stratigraphic evidence. The ancestral basins may have eventually merged, perhaps through renewed faulting, to form northern subsidiary troughs in Ophir and Candor Chasmata and the Coprates/north Melas/Ius graben system. The peripheral troughs are only 2-5 km deep, shallower than the central troughs. Chaotic terrain is seen in the peripheral troughs near a common contour level of about 4 km on the adjacent plateaus, which supports the idea of release of water under artesian pressure from confined aquifers. The layered deposits in the peripheral troughs may have formed in isolated depressions that harbored lakes and predated the formation of the deep outflow channels. (If these layered deposits are of volcanic origin, they may have been emplaced beneath ice in the manner of table mountains.) Areal and volumetric computations show that erosion widened the troughs by about one-third and that deposits occupy one-sixth of the interior space. Even though the volume eroded is larger than the volume deposited, topographic and geologic considerations imply that material eroded from trough walls was probably part of the interior layered deposits but not their sole source. Additional material may have come from subterranean piping, from reworking of local disintegration products on the floors, such as chaotic materials, or from eolian influx. But overall it is likely that the additional material is volcanic and that it forms mostly the upper, more diversely bedded layers of the interior deposits.

Lucchitta, B. K.↗

Computing Sensitivities in Evolutionary Systems: A Real-time Reduced Order Modeling Strategy

We present a new methodology for computing sensitivities in evolutionary systems using a model-driven low-rank approximation. To this end, we formulate a variational principle that seeks to minimize the distance between the time derivative of the reduced approximation and sensitivity dynamics. The first order optimality condition of the variational principle leads to a system of closed form evolution equations for an orthonormal basis and corresponding sensitivity coefficients. This approach allows for the computation of sensitivities with respect to a large number of parameters in an accurate and tractable manner by extracting correlations between different sensitivities on the fly. The presented method requires solving forward evolution equations, sidestepping the restrictions imposed by the forward/backward workflow of adjoint sensitivities. For example, the presented method, unlike the adjoint equation, does not impose any input/output load and can be used in applications in which real-time sensitivities are of interest. We demonstrate the utility of the method for three test cases: (1) computing sensitivity with respect to model parameters in the Rössler system, (2) computing sensitivity with respect to an infinite-dimensional forcing parameter in the chaotic Kuramoto--Sivashinsky equation, and (3) computing sensitivity with respect to reaction parameters for species transport in a turbulent reacting flow.

Reduced order model↗

Tidal interactions and the formation of Magellanic spiral galaxies

The closest, brightest and best resolved galaxy in the sky, the Large Magellanic Cloud, is a prototype of the late-type disk systems referred to collectively as barred Magellanic spiral galaxies (SBm). These systems occupy a pivotal stage in the Hubble sequence since they represent the transition from spiral disk galaxies to chaotic, irregular galaxies and are characterized by an asymmetric spiral arm which emanates from one end of a high surface brightness bar. We present evidence that the formation of the one-armed spiral morphology involves the tidal interaction with a companion galaxy. In summary the major points of our observational investigation include the following: (1) The vast majority of one-armed spiral galaxies possess bright nearby neighbors which appear to be physically associated. (2) The HI gas distribution and kinematics of one pair of SBm galaxies clearly reveal this system to be tidally interacting. (3) The infrared properties of the bars in two SBm galaxies support optical studies of bar luminosity profiles in late-type systems, and also suggest that the formation of compact star forming regions in these bars may not be uncommon.

Odewahn, S. C.↗

Approximate constants of motion for classically chaotic vibrational dynamics - Vague tori, semiclassical quantization, and classical intramolecular energy flow

Substantial short time regularity, even in the chaotic regions of phase space, is found for what is seen as a large class of systems. This regularity manifests itself through the behavior of approximate constants of motion calculated by Pade summation of the Birkhoff-Gustavson normal form expansion; it is attributed to remnants of destroyed invariant tori in phase space. The remnant torus-like manifold structures are used to justify Einstein-Brillouin-Keller semiclassical quantization procedures for obtaining quantum energy levels, even in the absence of complete tori. They also provide a theoretical basis for the calculation of rate constants for intramolecular mode-mode energy transfer. These results are illustrated by means of a thorough analysis of the Henon-Heiles oscillator problem. Possible generality of the analysis is demonstrated by brief consideration of classical dynamics for the Barbanis Hamiltonian, Zeeman effect in hydrogen and recent results of Wolf and Hase (1980) for the H-C-C fragment.

Shirts, R. B.↗

Atmospheric dynamics of the outer planets

Despite major differences in the solar and internal energy inputs, the atmospheres of the four Jovian planets all exhibit latitudinal banding and high-speed jet streams. Neptune and Saturn are the windiest planets, Jupiter is the most active, and Uranus is a tipped-over version of the others. Large oval storm systems exhibit complicated time-dependent behavior that can be simulated in numerical models and laboratory experiments. The largest storm system, the Great Red Spot of Jupiter, has survived for more than 300 years in a chaotic shear zone where smaller structures appear and dissipate every few days. Future space missions will add to the understanding of small-scale processes, chemical composition, and vertical structure. Theoretical hypotheses about the interiors provide input for fluid dynamical models that reproduce many observed features of the winds, temperatures, and cloud patterns. In one set of models the winds are confined to the thin layer where clouds form. In other models, the winds extend deep into the planetary fluid interiors. Hypotheses will be tested further as observations and theories become more exact and detailed comparisons are made.

Ingersoll, Andrew P.↗

Chaos in the sunspot cycle - Analysis and prediction

The variability of solar activity over long time scales, given semiquantitatively by measurements of sunspot numbers, is examined as a nonlinear dynamical system. First, a discussion of the data set used and the techniques utilized to reduce the noise and capture the long-term dynamics inherent in the data is presented. Subsequently, an attractor is reconstructed from the data set using the method of time delays. The reconstructed attractor is then used to determine both the dimension of the underlying system and also the largest Lyapunov exponent, which together indicate that the sunspot cycle is indeed chaotic and also low dimensional. In addition, recent techniques of exploiting chaotic dynamics to provide accurate, short-term predictions are utilized in order to improve upon current forecasting methods and also to place theoretical limits on predictability extent. The results are compared to chaotic solar-dynamo models as a possible physically motivated source of this chaotic behavior.

Mundt, Michael D.↗

Upper Limits of Predictability in Long-Range Climate/Hydrologic Forecasts

The accurate forecasting of el nino or la nina conditions in the tropical Pacific can potentially lead to valuable predictions of hydrological anomalies over land at seasonal to interannual timescales. Even with highly accurate earth system models, though, our ability to generate these continental forecasts will always be limited by the chaotic nature of the atmospheric circulation. The nature of this fundamental limitation is explored through the use of 16-member ensembles of multi-decade GCM simulations. In each simulation of the first ensemble, sea surface temperatures (SSTs) are given the same realistic interannual variations over a 45-year period, and land surface state is allowed to evolve with that of the atmosphere. Analysis of the results shows that the SSTs control the temporal organization of continental precipitation anomalies to a significant extent in the tropics and to a much smaller extent in midlatitudes. In each simulation of the second ensemble, we prescribe SSTs as before, but we also prescribe interannual variations in the low frequency component of evaporation efficiency over land. Thus, in the second ensemble, we effectively make the extreme assumption that surface boundary conditions across the globe are perfectly predictable, and we quantify the consistency with which the atmosphere (particularly precipitation) responds to these boundary conditions. The resulting "absolute upper limit" on the predictability of precipitation is found to be quite high in the tropics yet only moderate in many midlatitude regions.

Koster, R. D.↗

Flutter Analysis of the Thermal Protection Layer on the NASA HIAD

A combination of classical plate theory and a supersonic aerodynamic model is used to study the aeroelastic flutter behavior of a proposed thermal protection system (TPS) for the NASA HIAD. The analysis pertains to the rectangular configurations currently being tested in a NASA wind-tunnel facility, and may explain why oscillations of the articles could be observed. An analysis using a linear flat plate model indicated that flutter was possible well within the supersonic flow regime of the wind tunnel tests. A more complex nonlinear analysis of the TPS, taking into account any material curvature present due to the restraint system or substructure, indicated that significantly greater aerodynamic forcing is required for the onset of flutter. Chaotic and periodic limit cycle oscillations (LCOs) of the TPS are possible depending on how the curvature is imposed. When the pressure from the base substructure on the bottom of the TPS is used as the source of curvature, the flutter boundary increases rapidly and chaotic behavior is eliminated.

Goldman, Benjamin D.↗

Mission analysis for the EM-1 CubeSats EQUULEUS and OMOTENASHI

EQUULEUS is a Lunar L2 orbiter and a 6-Unit CubeSat by JAXA and the University of Tokyo. OMOTENASHI is a 6-Unit CubeSat by JAXA, the world’s smallest Lunar lander. EQUULEUS and OMOTENASHI are among the 13 secondary payloads selected by NASA to be launched with Exploration Mission-1 in 2019. Despite their limited size and cost, EQUULEUS and OMOTENASHI are challenging missions, especially in terms of trajectory design and control. EQUULEUS exploits the Earth-Sun-Moon chaotic dynamics and enter a libration point orbit around the L2 of the Earth-Moon system, using a new water propulsion system with low thrust and little propellant. This “Orbit Control Experiment” is one of the main objectives of the mission. OMOTENASHI executes a semi-hard landing that requires breaking the spacecraft to a stop just a few-hundred meters above the Moon’s surface. Both missions present new and unique challenges, where the design of the nominal trajectory is mainly driven by the constrains on orbital control capabilities, and operational and robustness considerations. This paper presents the current baselines, and give an overview of the new techniques developed for their design.

Oshima, Kenta↗

Hydrothermal systems on Mars: an assessment of present evidence

Hydrothermal processes have been suggested to explain a number of observations for Mars, including D/H ratios of water extracted from Martian meteorites, as a means for removing CO2 from the Martian atmosphere and sequestering it in the crust as carbonates, and as a possible origin for iron oxide-rich spectral units on the floors of some rifted basins (chasmata). There are numerous examples of Martian channels formed by discharges of subsurface water near potential magmatic heat sources, and hydrothermal processes have also been proposed as a mechanism for aquifer recharge needed to sustain long term erosion of sapping channels. The following geological settings have been identified as targets for ancient hydrothermal systems on Mars: channels located along the margins of impact crater melt sheets and on the slopes of ancient volcanoes; chaotic and fretted terranes where shallow subsurface heat sources are thought to have interacted with ground ice; and the floors of calderas and rifted basins (e.g. chasmata). On Earth, such geological environments are often a locus for hydrothermal mineralization. But we presently lack the mineralogical information needed for a definitive evaluation of hypotheses. A preferred tool for identifying minerals by remote sensing methods on Earth is high spatial resolution, hyperspectral, near-infrared spectroscopy, a technique that has been extensively developed by mineral explorationists. Future efforts to explore Mars for ancient hydrothermal systems would benefit from the application of methods developed by the mining industry to look for similar deposits on Earth. But Earth-based exploration models must be adapted to account for the large differences in the climatic and geological history of Mars. For example, it is likely that the early surface environment of Mars was cool, perhaps consistently below freezing, with the shallow portions of hydrothermal systems being dominated by magma-cryosphere interactions. Given the smaller gravitational field, declining atmospheric pressure, and widespread, permeable megaregolith on Mars, volatile outgassing and magmatic cooling would have been more effective than on Earth. Thus, hydrothermal systems are likely to have had much lower average surface temperatures than comparable geological settings on Earth. The likely predominance of basaltic crust on Mars suggests that hydrothermal fluids and associated deposits should be enriched in Fe, Mg, Si and Ca, with surficial deposits being dominated by lower temperature, mixed iron oxide and carbonate mineralogies.

Review↗

Detection of "noisy" chaos in a time series

Time series from biological system often displays fluctuations in the measured variables. Much effort has been directed at determining whether this variability reflects deterministic chaos, or whether it is merely "noise". The output from most biological systems is probably the result of both the internal dynamics of the systems, and the input to the system from the surroundings. This implies that the system should be viewed as a mixed system with both stochastic and deterministic components. We present a method that appears to be useful in deciding whether determinism is present in a time series, and if this determinism has chaotic attributes. The method relies on fitting a nonlinear autoregressive model to the time series followed by an estimation of the characteristic exponents of the model over the observed probability distribution of states for the system. The method is tested by computer simulations, and applied to heart rate variability data.

Non-NASA Center↗

Is the normal heartbeat chaotic or homeostatic?

Limits to the usefulness of homeostasis as a guiding physiological principle are revealed by new mechanisms derived from study of nonlinear systems that generate a type of variability called chaos. Loss of complex physiological variability may occur in certain pathological conditions including heart rate dynamics before sudden death and with aging.

NASA Program Space Physiology and Countermeasures↗