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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Pyoverdine binding aptamers and label-free electrochemical detection of pseudomonads

Pyoverdines are iron-chelating siderophores employed by various pseudomonads to promote their growth in iron-limited environments, facilitating both beneficial and detrimental interactions with co-inhabiting microbes or hosts, including plants and animals. The fluorescent pseudomonads produce fluorescent pyoverdines comprised of a conserved central chromophore and a unique strain-specific peptidic side chain produced by non-ribosomal peptide synthetases. Pyoverdine Pf5 (PVD-Pf5) is produced by Pseudomonas protegens Pf-5, a species known for supporting plant growth and its involvement in plant pathogen control. To develop a means of exploring the dynamics of P. protegens activity in soil and in the rhizosphere, we selected DNA aptamers that specifically recognize PVD-Pf5 with high affinities. Two selected aptamers with only 16% identity in sequence were examined for structure and function. We found evidence that both aptamers form structures in their apo-forms and one aptamer has structural features suggesting the presence of a G-quadruplex. Although their tertiary structures are predicted to be different, both aptamers bind the target PVD-Pf5 with similar affinities and do not bind other siderophores, including the related pyoverdine, pseudobactin, produced by Pseudomonas sp. B10. One aptamer binds the pyoverdine peptide component and may also interact with the chromophore. This aptamer was integrated into a nanoporous aluminum oxide biosensor and demonstrated to successfully detect PVD-Pf5 and not to detect other siderophores that do not bind to the aptamer when evaluated in solution. This sensor provides a future opportunity to track the locations of P. protegens around plant roots and to monitor PVD-Pf5 production and movement through the soil.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1989

Test facility and preliminary performance of a 100 kW class MPD thruster

A 260 kW magnetoplasmadynamic (MPD) thruster test facility was assembled and used to characterize thrusters at power levels up to 130 kW using argon and helium propellants. Sensitivities of discharge characteristics to arc current, mass flow rate, and applied magnetic field were investigated. A thermal efficiency correlation developed by others for low power MPD thrusters defined parametric guidelines to minimize electrode losses in MPD thrusters. Argon and helium results suggest that a parameter defined as the product of arc voltage and the square root of the mass flow rate must exceed .7 V-kg(1/2)-s(-1/2) in order to obtain thermal efficiencies in excess of 60 percent.

Sovey, James S.↗
NASA NTRS1989

Test Facility and Preliminary Performance of a 100 kW Class MPD Thruster

A 260 kW magnetoplasmadynamic (MPD) thruster test facility was assembled and used to characterize thrusters at power levels up to 130 kW using argon and helium propellants. Sensitivities of discharge characteristics to arc current, mass flow rate, and applied magnetic field were investigated. A thermal efficiency correlation developed by others for low power MPD thrusters defined parametric guidelines to minimize electrode losses in MPD thrusters. Argon and helium results suggest that a parameter defined as the product of arc voltage and the square root of the mass flow rate must exceed 0.7 V/kg(sup 1/2)/sec(sup 1/2) in order to obtain thermal efficiencies in excess of 60 percent.

Sovey, J. S.↗
NASA NTRS1993

Requirements for the early atmosphere of Mars from nitrogen isotope ratios

The N escape models of Fox and Dalgarno and Fox required the presence of a dense, early CO2 atmosphere to inhibit fractionation of the N isotopes N-15 and N-14. The computed photochemical escape fluxes are so large at the present that the isotope ratio measured by Viking (about 1.62x terrestrial) can be produced in about 1.5 b.y. This model was refined in several ways. It was updated to incorporate the variation of the escape fluxes with increases in the solar fluxes at earlier times according to the model of Zahnle and Walker. As expected, this exacerbates the problem with overfractionation, but not greatly. Most of the escape and fractionation of the N occurs in the last 1.5 b.y., when the solar flux was only slightly different from the present. The dense early atmosphere must persist only a bit longer in order to reproduce the measured isotope ratio. The model was also modified to take into account changes in the O mixing ratio with time in the past, assuming that the O abundance is proportional to the square root of the solar flux. Although the production rate of O from photodissociation of CO2 scales as the solar flux, the strength of the winds and other mixing processes also increases with the solar flux, resulting in possibly more effective transport of O to the lower atmosphere where it is destroyed by catalytic and three-body recombination mechanisms.

Fox, J. L.↗
NASA NTRS2018

GRACE-Assimilated Drought Indicators for the U.S. Drought Monitor

The Gravity Recovery and Climate Experiment (GRACE) mission detects changes in Earth's gravity field by precisely monitoring the changes in distance between two satellites orbiting the Earth in tandem. Scientists at NASA's Goddard Space Flight Center generate GRACE-assimilated groundwater and soil moisture drought indicators each week, for drought monitor-related studies and applications. The GRACE-assimilated Drought Indicator Version 2.0 data product (GRACE-DA-DM V2.0) is archived at, and distributed by, the NASA GES DISC (Goddard Earth Sciences Data and Information Services Center). More information about the data and data access is available on the data product landing page at https://disc.gsfc.nasa.gov/datasets /GRACEDADM_CLSM0125US_7D_2.0/summary. The GRACE-DA-DM V2.0 data product contains three drought indicators: Groundwater Percentile, Root Zone Soil Moisture Percentile, and Surface Soil Moisture Percentile. The drought indicators are of wet or dry conditions, expressed as a percentile, indicating the probability of occurrence within the period of record from 1948 to 2012. These GRACE-assimilated drought indicators, with improved spatial and temporal resolutions, should provide a more comprehensive and objective identification of drought conditions. This presentation describes the basic characteristics of the data and data services at NASA GES DISC and collaborative organizations, and uses a few examples to demonstrate the simple ways to explore the GRACE-assimilated drought indicator data.

hydrology↗
NASA NTRS2018

The SMAP Mission Combined Active-Passive Soil Moisture Product at 9 Km and 3 Km Spatial Resolutions

The NASA Soil Moisture Active Passive (SMAP) mission was launched on January 31st, 2015. The spacecraft was to provide high-resolution (3 km and 9 km) global soil moisture estimates at regular intervals by combining for the first time L-band radiometer and radar observations. On July 7th, 2015, a component of the SMAP radar failed and the radar ceased operation. However, before this occurred the mission was able to collect and process ~2.5 months of the SMAP high-resolution active-passive soil moisture data (L2SMAP) that coincided with the Northern Hemisphere's vegetation green-up and crop growth season. In this study, we evaluate the SMAP high-resolution soil moisture product derived from several alternative algorithms against in situ data from core calibration and validation sites (CVS), and sparse networks. The baseline algorithm had the best comparison statistics against the CVS and sparse networks. The overall unbiased root-mean-square-difference is close to the 0.04 cu. m/cu. m the SMAP mission requirement. A 3 km spatial resolution soil moisture product was also examined. This product had an unbiased root-mean-square-difference of ~0.053 cu. m/cu. m. The SMAP L2SMAP product for ~2.5 months is now validated for use in geophysical applications and research and available to the public through the NASA Distributed Active Archive Center (DAAC) at the National Snow and Ice Data Center (NSIDC). The L2SMAP product is packaged with the geo-coordinates, acquisition times, and all requisite ancillary information. Although limited in duration, SMAP has clearly demonstrated the potential of using a combined L-band radar-radiometer for proving high spatial resolution and accurate global soil moisture.

high resolution soil moisture↗
NASA NTRS1983

Gas flow through rotameters

Using data available for small rotameters that use spherical floats in gas flow, a linear relationship is derived. It is noted that the relationship provides a good fit for variable volumetric flow, density, and viscosity at constant flow height. With low Reynolds numbers (Re being less than 1), the product of the variable volumetric flow and the viscosity becomes constant; at high Reynolds numbers (Re being greater than 2000), the product of the variable volumetric flow and the square root of the density becomes constant. It is pointed out that the equation given here can be used to obtain an indirect calibration with any gas of known density and viscosity. The constancy of the product of the variable volumetric flow and viscosity at low variable volumetric flows is seen as suggesting the development of simple, inexpensive gas viscometers using rotameter technology.

Levin, H.↗
NASA NTRS1984

Analysis of the quality of image data acquired by the LANDSAT-4 Thematic Mapper and Multispectral Scanners

The geometric quality of TM film and digital products is evaluated by making selective photomeasurements and by measuring the coordinates of known features on both the TM products and map products. These paired observations are related using a standard linear least squares regression approach. Using regression equations and coefficients developed from 225 (TM film product) and 20 (TM digital product) control points, map coordinates of test points are predicted. The residual error vectors and analysis of variance (ANOVA) were performed on the east and north residual using nine image segments (blocks) as treatments. Based on the root mean square error of the 223 (TM film product) and 22 (TM digital product) test points, users of TM data expect the planimetric accuracy of mapped points to be within 91 meters and within 117 meters for the film products, and to be within 12 meters and within 14 meters for the digital products.

Colwell, R. N.↗
DOE OSTI2025

Deep Roots Supply Reactivity and Enhance Silicate Weathering in the Bedrock Vadose Zone

In upland environments, roots commonly extend deep below soil into partially saturated bedrock. This Bedrock Vadose Zone (BVZ) has been shown to store and circulate water, host organic carbon respiration and serve as a critical source of rock-derived nutrients. However, the extent to which deep roots influence chemical weathering rates remains poorly understood. Here, we report 4 years of depth-resolved major ion chemistry over a 16-m thick BVZ hosting a deep rhizosphere in a catchment subject to a Mediterranean climate. These data allow development and validation of a reactive transport model (RTM), revealing that the timescales of water storage and drainage in the BVZ are sufficient to facilitate substantial chemical weathering of the shale bedrock. However, observed solute concentrations are only reproduced by the RTM when we explicitly include measured rates of CO 2(g) production meters below soil driven by the deeply rooted forest. By combining direct observations and a process-based RTM we conclude that the carbon respiration promoted by deep roots significantly enhances chemical weathering rates in the BVZ, constituting 43% ± 3% of total solute flux from the base of the BVZ to the water table.

Osorio‐Leon, Ivan D. [University of Illinois Urban↗
NASA NTRS2011

Operational Evaluation of VEGGIE Food Production System in the Habitat Demonstration Unit

The 2010 Desert Research and Technology Studies (DRATS) of the VEGGIE Food Production System in the Habitat Demonstration Unit (HDU) Pressurized Excursion Module (PEM) was the first operational evaluation of salad crop production technology in a NASA analog test. A systematic evaluation of rooting media and nutrient delivery systems were evaluated for three lettuce cultivars that have shown promise as candidates for a surface based food production system. The VEGGIE nutrient delivery system worked well, was able to be maintained by multiple operators with a minimum of training, and supported excellent lettuce growth for the duration of the test. A Hazard Analysis and Critical Control Point (HACCP) evaluation was performed using ProSan(tm) as sanitation agent prior to consumption was approved, and the crew was allowed to consume the lettuce grown using the VEGGIE light cap and gravity based nutrient delivery system at the completion of the 14-day DRAT field test. The DRAT field test validated the crew operations; Growth of all lettuce cultivars was excellent. The operational DRAT field testing in the HDU identified light quality issues related to morphology and pigment development that will need to be addressed through additional testing. Feedback from the crew, ground support personnel, and human factors leads was uniformly positive on the psychological value of having the crop production system in the excursion module. A number of areas have been identified for future work, to minimize the "footprint" of the Food Production system through creative use of unused wall and floor space in the unit.

Stutte, Gary W.↗
NASA NTRS2011

Operation Evaluation of the VEGGIE Food Production System in the Habitat Demonstration Unit

The 2010 Desert Research and Technology Studies (DRATS) of the VEGGIE Food Production System in the Habitat Demonstration Unit (HDU) Pressurized Excursion Module (PEM) was the first operational evaluation of salad crop production technology in a NASA analog test. A systematic evaluation of rooting media and nutrient delivery systems were evaluated for three lettuce cultivars that have shown promise as candidates for a surface based food production system. The VEGGIE nutrient delivery system worked well, was able to be maintained by multiple operators with a minimum of training, and supported excellent lettuce growth for the duration of the test. A Hazard Analysis and Critical Control Point (HACCP) evaluation was performed using ProSantm as sanitation agent prior to consumption was approved, and the crew was allowed to consume the lettuce grown using the VEGGIE light cap and gravity based nutrient delivery system at the completion of the 14-day DRAT field test. The DRAT field test validated the crew operations; Growth of all lettuce cultivars was excellent. The operational DRAT field testing in the HDU identified light quality issues related to morphology and pigment development that will need to be addressed through additional testing. Feedback from the crew, ground support personnel, and human factors leads was uniformly positive on the psychological value of having the crop production system in the excursion module. A number of areas have been identified for future work, to minimize the "footprint" of the Food Production system through creative use of unused wall and floor space in the unit.

Stutte, Gary W.↗
DOE OSTI2025

Root genetics in the field to understand drought adaptation and carbon sequestration (Final Scientific/Technical Report)

For all crop plants, roots play a critical role in growth. Roots anchor the plants, and are the primary site of nutrient and water uptake. Roots are also the main source of C to soil in the form of root tissues and exudates, and thus greatly influence SOM stocks. To perform these functions, primary roots extend into soil, producing a network of branching roots of characteristic form, known as its root system architecture (RSA). RSA varies among species, and among varieties within a species that are adapted to different environments. Root traits are major targets for the second green revolution because of their potential to improve crop productivity, increase drought tolerance and nutrient acquisition, and increase C capture of soil. Improving the quality of roots in maize will be particularly valuable, since this crop is planted on over 92 million acres annually in the US. The future sustainability of agricultural systems relies on their ability to enhance soil organic matter (SOM) storage and reduce GHG emissions, while maintaining or enhancing productivity. This program had two components, Sensors and Models. For the first component, we designed and built a high-throughput phenotyping platform for root pulling of maize plants. This eliminated the physical labor of manually pulling up plants and reduced the number of personnel required down to one. The standardized pulling mechanism allowed recording force curves during the pulling process, providing additional information. We validated that the maximum force for pulling the root system was well-correlated with the root system mass and provided root crowns for further RSA analysis. These root crowns identified significant correlations with 2D root area and root depth, along with 3D root volume, total root length and number of root tips. We then used this system for field-based studies in maize on the genetics of root system architecture and its relation to nitrogen-use efficiency (NUE), including using lines relevant to the Corteva breeding program. Varieties were also evaluated at Corteva sites in the cornbelt and Danforth farm in Missouri, to establish responses across sites. From these studies we have identified genetic loci associated with root traits and created mutant lines for these loci and correlations of root traits with NUE. For the Models component, we worked to incorporate root and soil characteristics into the MEMS 2.0 soil and ecosystem biogeochemical model. Existing soil C models, such as Century, are unable to represent specific root trait interactions with the soil environment and therefore to accurately forecast the potential C sequestration benefits of root breeding under different climatic and soil type conditions. We have developed the MEMS 2.0 ecosystem biogeochemical model to improve quantification of farm-scale soil carbon and greenhouse gas emissions. The new knowledge and large datasets produced by this project will be used to develop and drive an innovative model capable of forecasting the impacts on soil C stocks and nutrient dynamics. An innovation was to use the empirical data from the field studies (in 1, above) to model genetic variation in nitrogen use efficiencies and soil C input. Our work demonstrated that maize root-derived C rapidly replaces existing soil C and after 3 years of continuous maize, up to 20% of soil organic C in the topsoil (0-15cm) and 3% in the subsoil (15-30cm) was contributed by maize. However, this contribution did not entirely represent a net increase. Root C contribution to soil was affected by maize genetics. We have analyzed soils derived from the CSU field trials for C and N stocks, in the different soil physical fractions represented by the MEMS model, using both physical fractionation with elemental analyses, and Fourier transformed infrared spectroscopy. Data will be used to link crop nitrogen use efficiencies with soil C sequestration and provide data to bridge the field trials with the model development, for verification of model predictions. The project had a number of successful outcomes: we have used the new phenotyping platform to identify new genetic loci that can enhance root phenotypes; we have partnered with multiple maize seed companies phenotype varieties in their breeding programs; we have developed the MEMS model that can help inform industry on the potential for carbon sequestration in the agricultural sector, and which is now available at the CSU Soil Carbon Solutions Center for use.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2020

Impact of Gauge-Based Precipitation Corrections on the Skill of SMAP Level-4 Soil Moisture Estimates

The NASA Soil Moisture Active Passive (SMAP) mission provides observations of L-band (1.4 GHz) passive microwave brightness temperature (Tb) observations at a resolution of ~40 km globally every 2-3 days. These observations are routinely assimilated into the NASA Catchment land surface model to generate the Level-4 Soil Moisture (L4_SM) product, which provides global estimates of surface and root-zone soil moisture, soil temperature, and surface fluxes (among others) at 9-km, 3-hourly resolution with ~2.5-day latency. The Catchment land surface model in the L4_SM algorithm is driven with 0.25°, hourly surface meteorological forcing data from the NASA Goddard Earth Observing System (GEOS) "forward-processing" product. Outside of Africa and the high latitudes, the GEOS precipitation forcing is corrected using the Climate Prediction Center Unified (CPCU) gauge-based, 0.5°, daily precipitation product.Soil moisture estimates from the L4_SM product were previously shown to improve over land model-only estimates that do not benefit from the assimilation of Tb observations, thereby demonstrating the value of assimilating SMAP observations for soil moisture estimation. In this presentation, we further isolate the contribution of the gauge-based precipitation corrections to the skill of the L4_SM soil moisture estimates. Specifically, we compare the skill of the L4_SM soil moisture to that of separate model-only and assimilation estimates obtained without the benefit of the gauge-based precipitation corrections.Preliminary results suggest that the soil moisture skill added by the CPCU-based precipitation corrections primarily depends on the quality of the CPCU precipitation product and is greatest in regions where the CPCU gauge network is dense and reliable. Conversely, in regions where the CPCU product is known to be of poor quality, for example in central Australia, the assimilation of SMAP Tb observations provides the most benefit. The presentation will provide an in-depth evaluation of the soil moisture skill of the model-only and assimilation estimates vs. independent in situ and satellite measurements.

Reichle, Rolf↗
NASA NTRS2002

Molecular Genetics of Root Thigmoresponsiveness in Arabidopsis thaliana

The molecular mechanisms that allow plant roots to use gravity and touch as growth guides are investigated. We are using a molecular genetic strategy in Arabidopsis thaliana to study these processes. When Arabidopsis thaliana seedlings grow on tilted hard-agar surfaces, their roots develop a wavy pattern of growth which appears to derive from a succession of left-handed and right-handed circumnutation-like processes triggered by gravity and touch stimulation (Okada and Shimura, 1990; Rutherford et al., 1998; Rutherford and Masson, 1996). Interestingly, mutations that affect root waving on tilted hard-agar surfaces can be identified and characterized. Some of these mutations affect root gravitropism, while others appear to be responsible for the production of abnormal waves (no waves, compressed or square waves, coils) without affecting gravitropism. The specific objectives of this project were to functionally characterize two genes (WVD2 and WVD6) which are required for root waving on tilted agar surfaces, but not for root gravitropism. Specific objectives included a physiological and cytological analysis of the mutants, and molecular cloning and characterization of the corresponding genes. As summarized in this paper, we have reached these objectives. We have also identified and partially characterized other mutations that affect root skewing on hard-agar surfaces (sku5-1 and ago1), and have completed our work on the root-wave phenotype associated with mutations in genes of the tryptophan biosynthesis pathway (Lynn et al., 1999; Rutherford et al., 1998; Sedbrook et al., 2000, 2002). We briefly describe our progress on the cloning and characterization of WVD6, WVD2 and SKU5, and provide a list of papers (published, or in preparation) that derived from this grant. We also discuss the biological implications of our findings, with special emphasis on the analysis of WVD2.

Masson, Patrick H.↗
NASA NTRS1990

A porous stainless steel membrane system for extraterrestrial crop production

A system was developed in which nutrient flow to plant roots is controlled by a thin (0.98 or 1.18 mm) porous (0.2 or 0.5 microns) stainless steel sheet membrane. The flow of nutrient solution through the membrane is controlled by adjusting the relative negative pressure on the nutrient solution side of the membrane. Thus, the nutrient solution is contained by the membrane and cannot escape from the compartment even under microgravity conditions if the appropriate pressure gradient across the membrane is maintained. Plant roots grow directly on the top surface of the membrane and pull the nutrient solution through this membrane interface. The volume of nutrient solution required by this system for plant growth is relatively small, since the plenum, which contains the nutrient solution in contact with the membrane, needs only to be of sufficient size to provide for uniform flow to all parts of the membrane. Solution not passing through the membrane to the root zone is recirculated through a reservoir where pH and nutrient levels are controlled. The size of the solution reservoir depends on the sophistication of the replenishment system. The roots on the surface of the membrane are covered with a polyethylene film (white on top, black on bottom) to maintain a high relative humidity and also limit light to prevent algal growth. Seeds are sown directly on the stainless steel membrane under the holes in the polyethylene film that allow a pathway for the shoots.

NASA Discipline Number 61-20↗
NASA NTRS2003

Demonstration of a Porous Tube Hydroponic System to Control Plant Moisture and Growth

Accurate remote detection of plant health indicators such as moisture, plant pigment concentrations, photosynthetic flux, and other biochemicals in canopies is a major goal in plant research. Influencing factors include complex interactions between wavelength dependent absorbing and scattering features from backgrounds as well as canopy biochemical and biophysical constituents. Accurately controlling these factors in outdoor field studies is difficult. Early testing of a porous tube plant culture system has indicated that plant biomass production, biomass partitioning, and leaf moisture of plants can be controlled by precisely managing the root water potential. Managing nutrient solution chemistry can also control plant pigments, biochemical concentrations, plant biomass production, and photosynthetic rates. A test bed was developed which utilized the porous tube technology with the intent of evaluating remote sensing systems, spectral analyses procedures, gas-exchange, and fluorescence measurements for their ability to detect small differences in plant water status. Spectral analysis was able to detect small differences in the mean leaf water content between the treatments. However these small differences were not detectable in the gas-exchange or fluorescence measurements.

Dreschel, T. W.↗
NASA NTRS1995

Effects of microgravity and clinorotation on stress ethylene production in two starchless mutants of Arabidopsis thaliana

Starch filled plastids termed amyloplasts, contained within columella cells of the root caps of higher plant roots, are believed to play a statolith-like role in the gravitropic response of roots. Plants having amyloplasts containing less starch exhibit a corresponding reduction in gravitropic response. We have observed enhanced ethylene production by sweet clover (Melilotus alba L.) seedlings grown in the altered gravity condition of a slow rotating clinostat, and have suggested that this is a stress response resulting from continuous gravistimulation rather than as a result of the simulation of a microgravity condition. If so, we expect that plants deficient in starch accumulation in amyloplasts may produce less stress ethylene when grown on a clinostat. Therefore, we have grown Arabidopsis thaliana in the small, closed environment of the Fluid Processing Apparatus (FPA). In this preliminary report we compare stationary plants with clinorotated and those grown in microgravity aboard Discovery during the STS-63 flight in February 1995. In addition to wildtype, two mutants deficient in starch biosynthesis, mutants TC7 and TL25, which are, respectively, deficient in the activity of amyloplast phosphoglucomutase and ADP-glucose pyrophosphorylase, were grown for three days before being fixed within the FPA. Gas samples were aspirated from the growth chambers and carbon dioxide and ethylene concentations were measured using a gas chromatograph. The fixed tissue is currently undergoing further morphologic and microscopic characterization.

Gallegos, Gregory L.↗
DOE OSTI2025

Order conditions for nonlinearly partitioned Runge-Kutta methods

Recently, a new class of nonlinearly partitioned Runge–Kutta (NPRK) methods was proposed for nonlinearly partitioned systems of autonomous ordinary differential equations y' = F(y, y). The target class of problems are those in which different scales, stiffnesses, or physics are coupled in a nonlinear way, wherein the desired partition cannot be written in a classical additive or component-wise fashion. Here we use a rooted-tree analysis to derive full-order conditions for NPRKM methods, where M denotes the number of nonlinear partitions. Due to the nonlinear coupling and thereby the mixed product differentials, it turns out that the standard node-colored rooted tree analysis used in analyzing ODE integrators does not naturally apply. Instead we develop a new edge-colored rooted-tree framework to address the nonlinear coupling. The resulting order conditions are enumerated, are provided directly for up to fourth order with M = 2 and third order with M = 3, and are related to existing order conditions of additive and partitioned RK methods. We conclude with an example that shows how the nonlinear order conditions can be used to obtain an embedded estimate of the state-dependent nonlinear coupling strength in a dynamical system.

97 MATHEMATICS AND COMPUTING↗