Results for “chaos” — Page 7

NASA NTRS1985

Fluvial Drainage Basins and Valley Networks: Eastern Margaritifer Sinus, Mars

The fluvial drainage of the eastern Margaritifer Sinus (MC-19NE, SE) and northeastern Argyre (MC-26NE) Quadrangles is dominated by two major longitudinal valley networks, the Parana/Loire system on the east, and the Samara Himera system to the west. It is believed that both of these drainages are through-going to the northwest and debouch into Margaritifer Chaos (general location: 12S, 22.5W). The Parana/Loire drainage is bounded on the east in part by an ancient multi-ringed impact basin. The Parana multi-digitate network drains northwest into a depositional basin, and impact basin floor, characterized by positive relief chaos. It is believed that Loire Vallis heads in the basin; thus Parana and Loire Valles may be treated as one system. Samara Valles heads in the northeastern Argyre Quadrangle and extends as a major truck valley to the northwest. Samara Valles cuts through the hills forming one of the concentric rings of the Ladon impact basin and joins the Himera drainage to trend in a more northerly direction to Margaritifer Chaos. The downstream portion of Himera is considered to be part of the Samara

Boothroyd, J. C.↗

NASA NTRS1988

Recent insights into instability and transition to turbulence in open-flow systems

Roads to turbulence in open-flow shear layers are interpreted as sequences of often competing instabilities. These correspond to primary and higher order restructurings of vorticity distributions which culminate in convected spatial disorder (with some spatial coherence on the scale of the shear layer) traditionally called turbulence. Attempts are made to interpret these phenomena in terms of concepts of convective and global instabilities on one hand, and of chaos and strange attractors on the other. The first is fruitful, and together with a review of mechanisms of receptivity provides a unifying approach to understanding and estimating transition to turbulence. In contrast, current evidence indicates that concepts of chaos are unlikely to help in predicting transition in open-flow systems. Furthermore, a distinction should apparently be made between temporal chaos and the convected spatial disorder of turbulence past Reynolds numbers where boundary layers and separated shear layers are formed.

Morkovin, Mark V.↗

NASA NTRS1987

Nonlinear dynamics, fractals, cardiac physiology and sudden death

The authors propose a diametrically opposite viewpoint to the generally accepted tendency of equating healthy function with order and disease with chaos. With regard to the question of sudden cardiac death and chaos, it is suggested that certain features of dynamical chaos related to fractal structure and fractal dynamics may be important organizing principles in normal physiology and that certain pathologies, including ventricular fibrillation, represent a class of 'pathological periodicities'. Some laboratory work bearing on the relation of nonlinear analysis to physiological and pathophysiological data is briefly reviewed, with tentative theories and models described in reference to the mechanism of ventricular fibrillation.

Goldberger, Ary L.↗

NASA NTRS1990

Studies in astronomical time series analysis. IV - Modeling chaotic and random processes with linear filters

While chaos arises only in nonlinear systems, standard linear time series models are nevertheless useful for analyzing data from chaotic processes. This paper introduces such a model, the chaotic moving average. This time-domain model is based on the theorem that any chaotic process can be represented as the convolution of a linear filter with an uncorrelated process called the chaotic innovation. A technique, minimum phase-volume deconvolution, is introduced to estimate the filter and innovation. The algorithm measures the quality of a model using the volume covered by the phase-portrait of the innovation process. Experiments on synthetic data demonstrate that the algorithm accurately recovers the parameters of simple chaotic processes. Though tailored for chaos, the algorithm can detect both chaos and randomness, distinguish them from each other, and separate them if both are present. It can also recover nonminimum-delay pulse shapes in non-Gaussian processes, both random and chaotic.

Scargle, Jeffrey D.↗

NASA NTRS1990

Period and phase of the 88-year solar cycle and the Maunder minimum - Evidence for a chaotic sun

The problem of whether the solar dynamo is quasi-periodic or chaotic is addressed by examining 1500 years of sunspot, geomagnetic, and auroral activity cycles. Sub-harmonics were found of the fundamental solar cycle period during the years preceding the Maunder minimum and loss of phase of the subharmonic on emergence from it. These phenomena are indicative of chaos. They indicate that the solar dynamo is chaotic and is operating in a region close to the transition between period doubling and chaos. Since Maunder-type minima reoccur irregularly for millennia, it appears that the sun remains close to this transition to and from chaos. This is postulated to be a universal characteristic of solar type stars caused by feedback in the dynamo number.

Feynman, J.↗

NASA NTRS1992

Self-organized rock textures and multiring structure in the Duolun crater

The Duolun impact crater is a multiring basin located 200 km north of Beijing. From the center to the edge of the crater there are innermost rim, inner ring, outer rim, and outermost ring. Recently, we have found some self-organized textures or chaos phenomena in shock-metamorphic rocks from the Duolun impact crater, such as turbulence in matrices of impact glass, oscillatory zoning, or chemical chaos of spherulites in spherulitic splashed breccia, fractal wavy textures or self-similar wavy textures with varied scaling in impact glass, and crystallite beams shaped like Lorentz strange attractors. The rare phenomena indicate that the shock-metamorphic rocks from Duolun crater are formed far from equilibrium. If impact cratering generates momentarily under high-pressure and superhigh-temperature, occurrence of those chaos phenomena in shock-metamorphic rock is not surprising.

Wu, Siben↗

NASA NTRS1994

A comparative survey of current and proposed tropospheric refraction-delay models for DSN radio metric data calibration

The standard tropospheric calibration model implemented in the operational Orbit Determination Program is the seasonal model developed by C. C. Chao in the early 1970's. The seasonal model has seen only slight modification since its release, particularly in the format and content of the zenith delay calibrations. Chao's most recent standard mapping tables, which are used to project the zenith delay calibrations along the station-to-spacecraft line of sight, have not been modified since they were first published in late 1972. This report focuses principally on proposed upgrades to the zenith delay mapping process, although modeling improvements to the zenith delay calibration process are also discussed. A number of candidate approximation models for the tropospheric mapping are evaluated, including the semi-analytic mapping function of Lanyi, and the semi-empirical mapping functions of Davis, et. al.('CfA-2.2'), of Ifadis (global solution model), of Herring ('MTT'), and of Niell ('NMF'). All of the candidate mapping functions are superior to the Chao standard mapping tables and approximation formulas when evaluated against the current Deep Space Network Mark 3 intercontinental very long baselines interferometry database.

Estefan, J. A.↗

NASA NTRS1995

The circum-Chryse region as a possible example of a hydrologic cycle on Mars: Geologic observations and theoretical evaluation

The transection and superposition relationships among channels, chaos, surface materials units, and other features in the circum-Chryse region of Mars were used to evaluate relative age relationships and evolution of flood events. Channels and chaos in contact (with one another) were treated as single discrete flood-carved systems. Some outflow channel systems form networks and are inferred to have been created by multiple flood events. Within some outflow channel networks, several separate individual channel systems can be traced to a specific chaos which acted as flood-source area to that specific flood channel. Individual flood-carved systems were related to widespread materials units or other surface features that served as stratigraphic horizons. Chryse outflow channels are inferred to have formed over most of the perceivable history of Mars. Outflow channels are inferred to become younger with increasing proximity to the Chryse basin. The relationship of subsequent outflow channel sources to the sources of earlier floods is inferred to disfavor episodic flooding due to the progresssive tapping of a juvenile near-surface water supply. Instead, we propose the circum-Chryse region as a candidate site of past hydrological recycling. The discharge rates necessary to carve the circum-Chryse outflow channels would have inevitably formed temporary standing bodies of H2O on the Martian surface where the flood-waters stagnated and pooled (the Chryse basin is topographically enclosed). These observations and inferences have led us to formulate and evaluate two hypotheses. Our numerical evaluations indicate that of these two hypotheses formulated, the groundwater seep cycle seems by far the more viable. Further observations from forthcoming missions may permit the determination of which mechanisms may have operated to recycle the Chryse flood-waters.

Moore, Jeffrey M.↗

NASA NTRS1996

Regular and Chaotic Quantum Dynamics of Two-Level Atoms in a Selfconsistent Radiation Field

Dynamics of two-level atoms interacting with their own radiation field in a single-mode high-quality resonator is considered. The dynamical system consists of two second-order differential equations, one for the atomic SU(2) dynamical-group parameter and another for the field strength. With the help of the maximal Lyapunov exponent for this set, we numerically investigate transitions from regularity to deterministic quantum chaos in such a simple model. Increasing the collective coupling constant b is identical with 8(pi)N(sub 0)(d(exp 2))/hw, we observed for initially unexcited atoms a usual sharp transition to chaos at b(sub c) approx. equal to 1. If we take the dimensionless individual Rabi frequency a = Omega/2w as a control parameter, then a sequence of order-to-chaos transitions has been observed starting with the critical value a(sub c) approx. equal to 0.25 at the same initial conditions.

Konkov, L. E.↗

NASA NTRS1995

The circum-Chryse region as a possible example of a hydrologic cycle on Mars: Geologic observations and theoretical evaluation

The transection and superposition relationships among channels, chaos, surface materials units, and other features in the circum-Chryse region of Mars were used to evaluate relative age relationships and evolution of flood events. Channels and chaos in contact (with one another) were treated as single discrete flood-carved systems. Some outflow channel systems form networks and are inferred to have been created by multiple flood events. Within some outflow channel networks, several separate individual channel systems can be traced to a specific chaos which acted as flood-source area to that specific flood channel. Individual flood-carved systems were related to widespread materials units or other surface features that served as stratigraphic horizons. Chryse outflow channels are inferred to have formed over most of the perceivable history of Mars. Outflow channels are inferred to become younger with increasing proximity to the Chryse basin. In addition, outflow channels closer to the basin show a greater diversity in age. The relationship of subsequent outflow channel sources to the sources of earlier floods is inferred to disfavor episodic flooding due to the progressive tapping of a juvenile near-surface water supply. Instead, we propose the circum-Chryse region as a candidate site of past hydrological recycling. The discharge rates necessary to carve the circum-Chryse outflow channels would have inevitably formed temporary standing bodies of H2O on the Martian surface where the flood-waters stagnated and pooled (the Chryse basin is topographically enclosed). These observations and inferences have led us to formulate and evaluate two hypotheses: (1) large amounts of the sublimated H2O off the Chryse basin flood lakes precipitated (snowed) onto the flood-source highlands and this H2O was incorporated into the near surface, recharging the H2O sources, making possible subsequent deluges; and (2) ponded flood-water in Chryse basin drained back down an anti basinward dipping subsurface layer accessed along the southern edge of the lake, recharging the flood-source aquifers. H2O not redeposited in the flood-source region was largely lost to the hydrologic cycle. This loss progressively lowered the vitality of the cycle, probably by now killing it. Our numerical evaluations indicate that of the two hypotheses we formulated, the groundwater seep cycle seems by far the more viable. Optimally, approx. 3/4 of the original mass of an ice-covered cylindrical lake (albedo 0.5, 1 km deep, 100-km radius, draining along its rim for one quarter of its circumference into substrata with a permeability of 3000 darcies) can be modeled to have moved underground (on timescales of the order of 10(exp 3) years) before the competing mechanisms of sublimation and freeze down choked off further water removal. Once underground, this water can travel distances equal to the separation between Chryse basin and flood-source sites in geologically short (approx. 10(exp 6) year-scale) times. Conversely, we calculate that optimally only approx. 40% of the H2O carried from Chryse can condense at the highlands, and most of the precipitate would either collect at the base of the highlands/lowlands scarp or sublimate at rates greater than it would accumulate over the flood-source sites. Further observations from forthcoming missions may permit the determination of which mechanisms may have operated to recycle the Chryse flood-waters.

Moore, Jeffrey M.↗

NASA NTRS1995

The Circum-Chryse Region as a Possible Example of a Hydrologic Cycle on Mars: Geologic Observations and Theoretical Evaluation

The transection and superposition relationships among channels, chaos, surface materials units, and other features in the circum-Chryse region of Mars were used to evaluate relative age relationships and evolution of flood events. Channels and chaos in contact (with one another) were treated as single discrete flood-carved systems. Some outflow channel systems form networks and are inferred to have been created by multiple flood events. Within some outflow channel networks, several separate individual channel systems can be traced to a specific chaos which acted as flood-source area to that specific flood channel. Individual flood-carved systems were related to widespread materials units or other surface features that served as stratigraphic horizons. Chryse outflow channels are inferred to have formed over most of the perceivable history of Mars. Outflow channels are inferred to become younger with increasing proximity to the Chryse basin. In addition, outflow channels closer to the basin show a greater diversity in age. The relationship of subsequent outflow channel sources to the sources of earlier floods is inferred to disfavor episodic flooding due to the progressive tapping of a juvenile near-surface water supply. Instead, we propose the circum-Chryse region as a candidate site of past hydrological recycling. The discharge rates necessary to carve the circum-Chryse outflow channels would have inevitably formed temporary standing bodies of H2O on the Martian surface where the flood-waters stagnated and pooled (the Chryse basin is topographically enclosed). These observations and inferences have led us to formulate and evaluate two hypotheses: (1) large amounts of the sublimated H2O off the Chryse basin flood lakes precipitated (snowed) onto the flood-source highlands and this H2O was incorporated into the near surface, recharging the H2O sources, making possible subsequent deluges; and (2) ponded flood-water in Chryse basin drained back down an anti basinward dipping subsurface layer accessed along the southern edge of the lake, recharging the flood-source aquifers. H2O not redeposited in the flood-source region was largely lost to the hydrologic cycle. This loss progressively lowered the vitality of the cycle, probably by now killing it. Our numerical evaluations indicate that of the two hypotheses we formulated, the groundwater seep cycle seems by far the more viable. Optimally, approximately 3/4 of the original mass of an ice-covered cylindrical lake (albedo 0.5, 1 km deep, 100-km radius, draining along its rim for one quarter of its circumference into substrata with a permeability of 3000 darcies) can be modeled to have moved underground (on timescales of the order of 10(exp 3) years) before the competing mechanisms of sublimation and freeze down choked off further water removal. Once underground, this water can travel distances equal to the separation between Chryse basin and flood-source sites in geologically short (approximately 10(exp 6) year-scale) times. Conversely, we calculate that optimally only approximately 40% of the H2O carried from Chryse can condense at the highlands, and most of the precipitate would either collect at the base of the highlands/lowlands scrap or sublimate at rates greater than it would accumulate over the flood-source sites. Further observations from forthcoming missions may permit the determination of which mechanisms may have operated to recycle the Chryse flood-waters.

Moore, Jeffrey M.↗

NASA NTRS2008

The Origin of the Terra Meridiani Sediments: Volatile Transport and the Formation of Sulfate Bearing Layered Deposits on Mars

The chemistry, sedimentology, and geology of the Meridiani sedimentary deposits are best explained by eolian reworking of the sublimation residue of a large scale ice/dust deposit. This large ice deposit was located in close proximity to Terra Meridiani and incorporated large amounts of dust, sand, and SO2 aerosols generated by impacts and volcanism during early martian history. Sulfate formation and chemical weathering of the initial igneous material is hypothesized to have occurred inside of the ice when the darker mineral grains were heated by solar radiant energy. This created conditions in which small films of liquid water were created in and around the mineral grains. This water dissolved the SO2 and reacted with the mineral grains forming an acidic environment under low water/rock conditions. Subsequent sublimation of this ice deposit left behind large amounts of weathered sublimation residue which became the source material for the eolian process that deposited the Terra Meridiani deposit. The following features of the Meridiani sediments are best explained by this model: The large scale of the deposit, its mineralogic similarity across large distances, the cation-conservative nature of the weathering processes, the presence of acidic groundwaters on a basaltic planet, the accumulation of a thick sedimentary sequence outside of a topographic basin, and the low water/rock ratio needed to explain the presence of very soluble minerals and elements in the deposit. Remote sensing studies have linked the Meridiani deposits to a number of other martian surface features through mineralogic similarities, geomorphic similarities, and regional associations. These include layered deposits in Arabia Terra, interior layered deposits in the Valles Marineris system, southern Elysium/Aeolis, Amazonis Planitia, and the Hellas basin, Aram Chaos, Aureum Chaos, and Ioni Chaos. The common properties shared by these deposits suggest that all of these deposits share a common formation process which must have acted over a large area of Mars. The results of this study suggest a mechanism for volatile transport on Mars without invoking an early greenhouse. They also imply a common formation mechanism for most of the sulfate minerals and layered deposits on Mars, which explains their common occurrence.

Niles, P.B.↗

NASA NTRS2009

Extending Newtonian Dynamics to Include Stochastic Processes

A paper presents further results of continuing research reported in several previous NASA Tech Briefs articles, the two most recent being Stochastic Representations of Chaos Using Terminal Attractors (NPO-41519), [Vol. 30, No. 5 (May 2006), page 57] and Physical Principle for Generation of Randomness (NPO-43822) [Vol. 33, No. 5 (May 2009), page 56]. This research focuses upon a mathematical formalism for describing post-instability motions of a dynamical system characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism involves fictitious control forces that couple the equations of motion of the system with a Liouville equation that describes the evolution of the probability density of errors in initial conditions. These stabilizing forces create a powerful terminal attractor in probability space that corresponds to occurrence of a target trajectory with probability one. The effect in configuration space (ordinary three-dimensional space as commonly perceived) is to suppress exponential divergences of neighboring trajectories without affecting the target trajectory. As a result, the post-instability motion is represented by a set of functions describing the evolution of such statistical quantities as expectations and higher moments, and this representation is stable.

Zak, Michail↗

NASA NTRS2018

Multifidelity Uncertainty Quantification of a Commercial Supersonic Transport

The objective of this work was to develop a multifidelity uncertainty quantification approach for efficient analysis of a commercial supersonic transport. An approach based on non-intrusive polynomial chaos was formulated in which a low-fidelity model could be corrected by any number of high-fidelity models. The formulation and methodology also allows for the addition of uncertainty sources not present in the lower fidelity models. To demonstrate the applicability of the multifidelity polynomial chaos approach, two model problems were explored. The first was supersonic airfoil with three levels of modeling fidelity, each capturing an additional level of physics. The second problem was a commercial supersonic transport. This model had three levels of fidelity that included two different modeling approaches and the addition of physics between the fidelity levels. Both problems illustrate the applicability and significant computational savings of the multifidelity polynomial chaos method.

West, Thomas K., IV↗

NASA NTRSDate not supplied

Efficient Parametric Uncertainty Analysis of an Earth Entry Vehicle Concept Using Least Angle Regression

The objective of this work was to outline and apply an efficient and accurate parametric un-certainty propagation approach to the analysis of convective heating on an Earth entry vehicle concept. The described approach was based on Least Angle Regression used to solve a sparse and underdetermined linear system in the point-collocation non-intrusive polynomial chaos surrogate method. This approach involved an iterative process to computing the non-zero terms of the underlying polynomial chaos model using only enough samples to converge uncertainty interval predictions and Sobol index values based global nonlinear sensitivity estimates. The Earth entry vehicle was analyzed at three points along a representative trajectory for a Mars return mission. 329 sources of uncertainty were identified in the computational fluid dynamics model used to predict the forebody convective heating. These included uncertainty in flow field chemical rates, collision integrals, heats of formation, surface finite rate char model reaction rates, wall roughness height, and the turbulent Schmidt number. Results from this study showed that convective heating uncertainty as high as 50% of the nominal was predicted with only about 50 evaluations of the computational model. This was far fewer than would be required for a sampling-based approach or a full basis polynomial chaos model, which would have required over 50,000 samples. Additionally, results showed that over 90% of the total convective heating uncertainty was due to uncertainty in the N2catalytic rate on the surface, while the remainder of the uncertainty was attributed to the turbulent Schmidt number and the wall roughness uncertainties.

Thomas K West IV↗

NASA NTRSDate not supplied

End-To-End Uncertainty Quantification with Analytical Derivatives for Design Under Uncertainty

Uncertainty quantification (UQ) is a rapidly growing and evolving discipline, especially within the aerospace community. Performing analysis with UQ can provide decision makers with a wealth of information about a candidate design. However, the value of UQ is fully realized when the information gained during UQ analysis is leveraged in a feedback loop of a design optimization process, often referred to as design under uncertainty. Although design under uncertainty can be a powerful risk mitigation technique, there are a number of roadblocks that prevent its implementation. Two primary factors are computational costs and added complexity of the analysis. High fidelity simulations on the order tens of uncertain variables quickly become computationally infeasible. Also, implementing UQ into an existing multidisciplinary design and optimization (MDO) process often requires extensive knowledge of the UQ methods and careful treatment of the problem formulation. The objective of this work is to address these two primary roadblocks and enable practitioners to efficiently perform design under uncertainty with limited knowledge of the UQ discipline. Methods outlined in this paper demonstrate MDO incorporating UQ into the design process, leveraging an analytic derivative tool chain through the entire optimization. The proposed approach leverages machine learning techniques to generate a differentiable confidence interval output from polynomial chaos models. This technique, coupled with the incorporation of analytical derivatives through the Polynomial Chaos Expansion (PCE) process, eliminates the need to estimate derivatives which are usually obtained from finite difference, complex step, or similar methods. Developing a differentiable confidence interval allows mixed uncertainty problems (both epistemic and aleatory) to be modeled. Without such modeling, these problems cannot accurately predict objective functions containing statistical quantities such as mean and variance. The addition of analytic derivatives to a polynomial chaos-based UQ method decreases the computational costs of performing design under uncertainty by orders of magnitude in comparison with methods such as complex step. The method and codes developed are modular in nature and are a drop-in solution for design under uncertainty within existing MDO problems. A low-fidelity analytical multidisciplinary optimization under uncertainty for a wing design in OpenMDAO is detailed in this paper. This demonstration case will include both objective functions and constraints which are influenced by uncertain parameters.

Ben D Phillips↗

NASA NTRSDate not supplied

End-To-End Uncertainty Quantification with Analytical Derivatives for Design Under Uncertainty

Uncertainty quantification (UQ) is a rapidly growing and evolving discipline, especially within the aerospace community. Performing analysis with UQ can provide decision makers with a wealth of information about a candidate design. However, the value of UQ is fully realized when the information gained during UQ analysis is leveraged in a feedback loop of a design optimization process, often referred to as design under uncertainty. Although design under uncertainty can be a powerful risk mitigation technique, there are a number of roadblocks that prevent its implementation. Two primary factors are computational costs and added complexity of the analysis. High fidelity simulations on the order tens of uncertain variables quickly become computationally infeasible. Also, implementing UQ into an existing multidisciplinary design and optimization (MDO) process often requires extensive knowledge of the UQ methods and careful treatment of the problem formulation. The objective of this work is to address these two primary roadblocks and enable practitioners to efficiently perform design under uncertainty with limited knowledge of the UQ discipline. Methods outlined in this paper demonstrate MDO incorporating UQ into the design process, leveraging an analytic derivative tool chain through the entire optimization. The proposed approach leverages machine learning techniques to generate a differentiable confidence interval output from polynomial chaos models. This technique, coupled with the incorporation of analytical derivatives through the Polynomial Chaos Expansion (PCE) process, eliminates the need to estimate derivatives which are usually obtained from finite difference, complex step, or similar methods. Developing a differentiable confidence interval allows mixed uncertainty problems (both epistemic and aleatory) to be modeled. Without such modeling, these problems cannot accurately predict objective functions containing statistical quantities such as mean and variance. The addition of analytic derivatives to a polynomial chaos-based UQ method decreases the computational costs of performing design under uncertainty by orders of magnitude in comparison with methods such as complex step. The method and codes developed are modular in nature and are a drop-in solution for design under uncertainty within existing MDO problems. A low-fidelity analytical multidisciplinary optimization under uncertainty for a wing design in OpenMDAO is detailed in this paper. This demonstration case will include both objective functions and constraints which are influenced by uncertain parameters.

Ben Phillips↗

NASA NTRS1983

On error detection in the dynamics part of primitive equation models

Rossby-Haurwitz waves have been commonly employed as initial conditions for testing the dynamics part of global atmospheric models. Difficulties arise in connection with the primitive equation (PE) model. Chao and Geller (1982) have suggested that it would be better if normal mode solutions of the linearized primitive equations would be used as initial conditions. However, tests involving normal mode conditions are not adequate for the detection of errors in the nonlinear terms. The present investigation has the objective to review the study conducted by Chao and Geller and to provide tests for the nonlinear terms.

Chao, W. C.↗