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At least 127 records · Page 7
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NASA NTRS1992

A fundamental theorem for the model reduction of nonlinear systems

A simple, but fundamental, theorem is given on the extent to which a nonlinear system model can have its order reduced. Essentially, the result is that the order, or the dimension of the state space representation, cannot be reduced to, or below, the dimension of the system's attractor. Several examples are given to illustrate this point. The result is especially applicable to higher order systems such as the infinite dimensional systems arising from the modeling of distributed parameter systems.

Mossayebi, Faramarz↗
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 NTRS1992

Combustion noise and combustion instabilities in propulsion systems

This paper is concerned with some aspects of non-linear behavior of unsteady motions in combustion chambers. The emphasis is on conditions under which organized oscillations having discrete frequencies may exist in the presence of random motions. In order to treat the two types of motions together, and particularly to investigate coupling between noise and combustion instabilities, the unsteady field is represented as a synthesis of acoustic modes having time-varying amplitudes. Each of the amplitudes are written as the sum of two parts, one associated with the random field and the remainder representing the organized oscillations. After spatial averaging, the general problem is reduced to solution of a set of second-order ordinary differential equations whose structure depends on the sorts of nonlinear processes accounted for. This formulation accommodates any physical process; in particular, terms are included to represent noise sources, although only limited modeling is discussed. Our results suggest that random sources of noise have only small effects on combustion instabilities and seem not to be a cause of unstable motions. However, the coupling between the two sorts of unsteady motions may be important as an essential process in a proposed scheme for noise control. It is now a familiar observation that many nonlinear deterministic systems are capable of exhibiting apparently random motions called 'chaos.' This is a particularly interesting possibility for systems which also executed non-deterministic random motions. In combustion chambers, a nonlinear deterministic system (acoustical motions) exists in the presence of noise produced by flow separation, turbulent motions, and energy released by combustion processes. The last part of the paper is directed to the matter of discovering whether or not chaotic motions exist in combustion systems. Analysis has not progressed sufficiently far to answer the question. We report here recent results of processing data taken in one combustor to determine the dimensions of any attractors in the motions. No evidence has been found for chaos in the strict sense, but the method seems to be an important means of investigating the nonlinear behavior of combustion systems.

F. E. C. Culick↗
NASA NTRS1993

The structure of the nearby universe traced by theIRAS galaxies

One of the most important discoveries of the Infrared Astronomical Satellite (IRAS) has been the detection of about 20,000 galaxies with 60 microns fluxes above 0.5 Jy. From the observational point of view, the IRAS galaxies are ideal tracers of density, since they are homogeneously detected over most of the sky, and their fluxes are unaffected by galactic extinction. The nearby universe was mapped by the IRAS galaxies to a distance of approximately 200 h(exp -1) Mpc for the absolute value of b less than 5 deg. The ability to map down to such low galactic latitudes has proven to be particularly imporant, since some of the most important nearby large-scale structures, such as the Great Attractor, the Perseus-Pisces region, and the Shapley concentration, all lie there. Two major results of the U.S. IRAS redshift survey are discussed.

Yahil, Amos↗
NASA NTRS1992

Non-scalar uncertainty: Uncertainty in dynamic systems

The following point is stated throughout the paper: dynamic systems are usually subject to uncertainty, be it the unavoidable quantic uncertainty when working with sufficiently small scales or when working in large scales uncertainty can be allowed by the researcher in order to simplify the problem, or it can be introduced by nonlinear interactions. Even though non-quantic uncertainty can generally be dealt with by using the ordinary probability formalisms, it can also be studied with the proposed non-scalar formalism. Thus, non-scalar uncertainty is a more general theoretical framework giving insight into the nature of uncertainty and providing a practical tool in those cases in which scalar uncertainty is not enough, such as when studying highly nonlinear dynamic systems. This paper's specific contribution is the general concept of non-scalar uncertainty and a first proposal for a methodology. Applications should be based upon this methodology. The advantage of this approach is to provide simpler mathematical models for prediction of the system states. Present conventional tools for dealing with uncertainty prove insufficient for an effective description of some dynamic systems. The main limitations are overcome abandoning ordinary scalar algebra in the real interval (0, 1) in favor of a tensor field with a much richer structure and generality. This approach gives insight into the interpretation of Quantum Mechanics and will have its most profound consequences in the fields of elementary particle physics and nonlinear dynamic systems. Concepts like 'interfering alternatives' and 'discrete states' have an elegant explanation in this framework in terms of properties of dynamic systems such as strange attractors and chaos. The tensor formalism proves especially useful to describe the mechanics of representing dynamic systems with models that are closer to reality and have relatively much simpler solutions. It was found to be wise to get an approximate solution to an accurate model than to get a precise solution to a model constrained by simplifying assumptions. Precision has a very heavy cost in present physical models, but this formalism allows the trade between uncertainty and simplicity. It was found that modeling reality sometimes requires that state transition probabilities should be manipulated as nonscalar quantities, finding at the end that there is always a transformation to get back to scalar probability.

Martinez, Salvador Gutierrez↗
NASA NTRS1993

Quantifying predictability variations in a low-order ocean-atmosphere model - A dynamical systems approach

The predictability of the weather and climatic states of a low-order moist general circulation model is quantified using a dynamic systems approach, and the effect of incorporating a simple oceanic circulation on predictability is evaluated. The predictability and the structure of the model attractors are compared using Liapunov exponents, local divergence rates, and the correlation and Liapunov dimensions. It was found that the activation of oceanic circulation increases the average error doubling time of the atmosphere and the coupled ocean-atmosphere system by 10 percent and decreases the variance of the largest local divergence rate by 20 percent. When an oceanic circulation develops, the average predictability of annually averaged states is improved by 25 percent and the variance of the largest local divergence rate decreases by 25 percent.

Nese, Jon M.↗
NASA NTRS1993

Transition to chaos in an open unforced 2D flow

The present numerical study of unsteady, low Reynolds number flow past a 2D airfoil attempts to ascertain the bifurcation sequence leading from simple periodic to complex aperiodic flow with rising Reynolds number, as well as to characterize the degree of chaos present in the aperiodic flow and assess the role of numerics in the modification and control of the observed bifurcation scenario. The ARC2D Navier-Stokes code is used in an unsteady time-accurate mode for most of these computations. The system undergoes a period-doubling bifurcation to chaos as the Reynolds number is increased from 800 to 1600; its chaotic attractors are characterized by estimates of the fractal dimension and partial Liapunov exponent spectra.

Pulliam, Thomas H.↗
NASA NTRS1992

Neurodynamical model of collective brain

A dynamical system which mimics collective purposeful activities of a set of units of intelligence is introduced and discussed. A global control of the unit activities is replaced by the probabilistic correlations between them. These correlations are learned during a long term period of performing collective tasks, and are stored in the synaptic interconnections. The model is represented by a system of ordinary differential equations with terminal attractors and repellers, and does not contain any man-made digital devices.

Zak, Michail↗
NASA NTRS1993

The velocity field of clusters of galaxies within 100 megaparsecs. II - Northern clusters

Distances and peculiar velocities for galaxies in eight clusters and groups have been determined by means of the near-infrared Tully-Fisher relation. With the possible exception of a group halfway between us and the Hercules Cluster, we observe peculiar velocities of the same order as the measuring errors of about 400 km/s. The present sample is drawn from the northern Galactic hemisphere and delineates a quiet region in the Hubble flow. This contrasts with the large-scale flows seen in the Hydra-Centaurus and Perseus-Pisces regions. We compare the observed peculiar velocities with predictions based upon the gravity field inferred from the IRAS redshift survey. The differences between the observed and predicted peculiar motions are generally small, except near dense structures, where the observed motions exceed the predictions by significant amounts. Kinematic models of the velocity field are also compared with the data. We cannot distinguish between parameterized models with a great attractor or models with a bulk flow.

Mould, J. R.↗
NASA NTRS1993

IRAS galaxies versus POTENT mass - Density fields, biasing, and Omega

A comparison of the galaxy density field extracted from a complete redshift survey of IRAS galaxies brighter than 1.936 Jy with the mass-density field reconstructed by the POTENT procedure from the observed peculiar velocities of 493 objects is presented. A strong correlation is found between the galaxy and mass-density fields; both feature the Great Attractor, part of the Perseus-Pisces supercluster, and the large void between them. Monte Carlo noise simulations show that the data are consistent with the hypotheses that the smoothed fluctuations of galaxy and mass densities at each point are proportional to each other with the 'biasing' factor of IRAS galaxies, b(I), and that the peculiar velocity field is related to the mass-density field as expected according to the gravitational instability theory. Under these hypotheses, the two density fields can be related by specifying b(I) and the cosmological density parameter, Omega.

Dekel, Avishai↗
NASA NTRS1994

Terminal Model Of Newtonian Dynamics

Paper presents study of theory of Newtonian dynamics of terminal attractors and repellers, focusing on issues of reversibility vs. irreversibility and deterministic evolution vs. probabilistic or chaotic evolution of dynamic systems. Theory developed called "terminal dynamics" emphasizes difference between it and classical Newtonian dynamics. Also holds promise for explaining irreversibility, unpredictability, probabilistic behavior, and chaos in turbulent flows, in thermodynamic phenomena, and in other dynamic phenomena and systems.

Zak, Michail↗
NASA NTRS1993

Boundary layer flow of air over water on a flat plate

A non-similar boundary layer theory for air blowing over a water layer on a flat plate is formulated and studied as a two-fluid problem in which the position of the interface is unknown. The problem is considered at large Reynolds number (based on x), away from the leading edge. A simple non-similar analytic solution of the problem is derived for which the interface height is proportional to x(sub 1/4) and the water and air flow satisfy the Blasius boundary layer equations, with a linear profile in the water and a Blasius profile in the air. Numerical studies of the initial value problem suggests that this asymptotic, non-similar air-water boundary layer solution is a global attractor for all initial conditions.

Nelson, John↗
NASA NTRS1993

Dimensional analysis of acoustically propagated signals

Traditionally, long term measurements of atmospherically propagated sound signals have consisted of time series of multiminute averages. Only recently have continuous measurements with temporal resolution corresponding to turbulent time scales been available. With modern digital data acquisition systems we now have the capability to simultaneously record both acoustical and meteorological parameters with sufficient temporal resolution to allow us to examine in detail relationships between fluctuating sound and the meteorological variables, particularly wind and temperature, which locally determine the acoustic refractive index. The atmospheric acoustic propagation medium can be treated as a nonlinear dynamical system, a kind of signal processor whose innards depend on thermodynamic and turbulent processes in the atmosphere. The atmosphere is an inherently nonlinear dynamical system. In fact one simple model of atmospheric convection, the Lorenz system, may well be the most widely studied of all dynamical systems. In this paper we report some results of our having applied methods used to characterize nonlinear dynamical systems to study the characteristics of acoustical signals propagated through the atmosphere. For example, we investigate whether or not it is possible to parameterize signal fluctuations in terms of fractal dimensions. For time series one such parameter is the limit capacity dimension. Nicolis and Nicolis were among the first to use the kind of methods we have to study the properties of low dimension global attractors.

Hansen, Scott D.↗
NASA NTRS1995

Lyapunov exponents from CHUA's circuit time series using artificial neural networks

In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks.

Gonzalez, J. Jesus↗
NASA NTRS1994

Bias properties of extragalactic distance indicators. 3: Analysis of Tully-Fisher distances for the Mathewson-Ford-Buchhorn sample of 1355 galaxies

The observational selection bias properties of the large Mathewson-Ford-Buchhorn (MFB) sample of axies are demonstrated by showing that the apparent Hubble constant incorrectly increases outward when determined using Tully-Fisher (TF) photometric distances that are uncorreted for bias. It is further shown that the value of H(sub 0) so determined is also multivlaued at a given redshift when it is calculated by the TF method using galaxies with differenct line widths. The method of removing this unphysical contradiction is developed following the model of the bias set out in Paper II. The model developed further here shows that the appropriate TF magnitude of a galaxy that is drawn from a flux-limited catalog not only is a function of line width but, even in the most idealistic cases, requires a triple-entry correction depending on line width, apparent magnitude, and catalog limit. Using the distance-limited subset of the data, it is shown that the mean intrinsic dispersion of a bias-free TF relation is high. The dispersion depends on line width, decreasing from sigma(M) = 0.7 mag for galaxies with rotational velocities less than 100 km s(exp-1) to sigma(M) = 0.4 mag for galaxies with rotational velocities greater than 250 km s(exp-1). These dispersions are so large that the random errors of the bias-free TF distances are too gross to detect any peculiar motions of individual galaxies, but taken together the data show again the offset of 500 km s(exp-1) fond both by Dressler & Faber and by MFB for galaxies in the direction of the putative Great Attractor but described now in a different way. The maximum amplitude of the bulk streaming motion at the Local Group is approximately 500 km s(exp-1) but the perturbation dies out, approaching the Machian frame defined by the CMB at a distance of approximately 80 Mpc (v is approximately 4000 km s(exp -1)). This decay to zero perturbation at v is approximately 4000 km s(exp -1) argues against existing models with a single attraction at approximately 4500 km s(exp -1) (the Great Attactor model) pulling the local region. Rather, the cause of the perturbation appears to be the well-known clumpy mass distribution within 4000 km s(exp -1) in the busy directions of Hydra, Centaurus, Antila and Dorado, as postulated earlier (Tammann & Sandage 1985).

Federspiel, Martin↗
NASA NTRS1996

Dimension Determination of Precursive Stall Events in a Single Stage High Speed Compressor

This paper presents a study of the dynamics for a single-stage, axial-flow, high speed compressor core, specifically, the NASA Lewis rotor stage 37. Due to the overall blading design for this advanced core compressor, each stage has considerable tip loading and higher speed than most compressor designs, thus, the compressor operates closer to the stall margin. The onset of rotating stall is explained as bifurcations in the dynamics of axial compressors. Data taken from the compressor during a rotating stall event is analyzed. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined during the bifurcations leading to rotating stall. The intent of this study is to examine the behavior of precursive stall events so as to predict the entrance into rotating stall. This information may provide a better means to identify, avoid or control the undesirable event of rotating stall formation in high speed compressor cores.

Bright, Michelle M.↗
NASA NTRS1996

Turbulent Fluid Motion 6: Turbulence, Nonlinear Dynamics, and Deterministic Chaos

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The turbulence is sustained by a nonrandom time-independent external force. The solutions, on the average, separate exponentially with time, having a positive Liapunov exponent. Thus, the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number (or strength of the forcing) is reduced. Several qualitatively different flows are noted. These are, respectively, fully chaotic, complex periodic, weakly chaotic, simple periodic, and fixed-point. Of these, we classify only the fully chaotic flows as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows, although having positive Liapunov exponents, have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, as generally understood, is both time-dependent and aperiodic.

Deissler, Robert G.↗
NASA NTRS2001

Single and Double ITCZ in Aqua-Planet Models with Globally Uniform Sea Surface Temperature and Solar Insolation: An Interpretation

It has been known for more than a decade that an aqua-planet model with globally uniform sea surface temperature and solar insolation angle can generate ITCZ (intertropical convergence zone). Previous studies have shown that the ITCZ under such model settings can be changed between a single ITCZ over the equator and a double ITCZ straddling the equator through one of several measures. These measures include switching to a different cumulus parameterization scheme, changes within the cumulus parameterization scheme, and changes in other aspects of the model design such as horizontal resolution. In this paper an interpretation for these findings is offered. The latitudinal location of the ITCZ is the latitude where the balance of two types of attraction on the ITCZ, both due to earth's rotation, exists. The first type is equator-ward and is directly related to the earth's rotation and thus not sensitive to model design changes. The second type is poleward and is related to the convective circulation and thus is sensitive to model design changes. Due to the shape of the attractors, the balance of the two types of attractions is reached either at the equator or more than 10 degrees away from the equator. The former case results in a single ITCZ over the equator and the latter case a double ITCZ straddling the equator.

Chao, Winston C.↗