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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Modeling aerodynamic discontinuities and the onset of chaos in flight dynamical systems

Various representations of the aerodynamic contribution to the aircraft's equation of motion are shown to be compatible within the common assumption of their Frechet differentiability. Three forms of invalidating Frechet differentiality are identified, and the mathematical model is amended to accommodate their occurrence. Some of the ways in which chaotic behavior may emerge are discussed, first at the level of the aerodynamic contribution to the equation of motion, and then at the level of the equations of motion themselves.

Tobak, M.↗
NASA NTRS1987

Modeling aerodynamic discontinuities and onset of chaos in flight dynamical systems

Various representations of the aerodynamic contribution to the aircraft's equation of motion are shown to be compatible within the common assumption of their Frechet differentiability. Three forms of invalidating Frechet differentiability are identified, and the mathematical model is amended to accommodate their occurrence. Some of the ways in which chaotic behavior may emerge are discussed, first at the level of the aerodynamic contribution to the equations of motion, and then at the level of the equations of motion themselves.

Tobak, M.↗
NASA NTRS1990

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.↗
NASA NTRS1989

Dynamics of Pluto and Charon

The dynamics of the Pluto-Charon system are reviewed from a historical perspective. Although Pluto's orbit crosses Neptune's, an intricate system of nested resonances keeps these planets apart. Pluto's orbit is apparently chaotic as well. Pluto always keeps the same face turned toward Charon, and vice versa. Tides also damp Charon's orbital eccentricity and inclination. Precession of Pluto's orbital plane causes Pluto's obliquity to vary periodically from formally prograde to retrograde. Pluto is probably an original member of the Solar system, but not an escaped satellite of Neptune.

Dobrovolskis, Anthony R.↗
NASA NTRS1989

Evanescent radiation, quantum mechanics and the Casimir effect

An attempt to bridge the gap between classical and quantum mechanics and to explain the Casimir effect is presented. The general nature of chaotic motion is discussed from two points of view: the first uses catastrophe theory and strange attractors to describe the deterministic view of this motion; the underlying framework for chaos in these classical dynamic systems is their extreme sensitivity to initial conditions. The second interpretation refers to randomness associated with probabilistic dynamics, as for Brownian motion. The present approach to understanding evanescent radiation and its relation to the Casimir effect corresponds to the first interpretation, whereas stochastic electrodynamics corresponds to the second viewpoint. The nonlinear behavior of the electromagnetic field is also studied. This well-understood behavior is utilized to examine the motions of two orbiting charges and shows a closeness between the classical behavior and the quantum uncertainty principle. The evanescent radiation is used to help explain the Casimir effect.

Schatten, Kenneth H.↗
NASA NTRS1997

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↗
NASA NTRS1991

Predicting chaos for infinite dimensional dynamical systems: The Kuramoto-Sivashinsky equation, a case study

The results of extensive computations are presented in order to accurately characterize transitions to chaos for the Kuramoto-Sivashinsky equation. In particular, the oscillatory dynamics in a window that supports a complete sequence of period doubling bifurcations preceding chaos is followed. As many as thirteen period doublings are followed and used to compute the Feigenbaum number for the cascade and so enable, for the first time, an accurate numerical evaluation of the theory of universal behavior of nonlinear systems, for an infinite dimensional dynamical system. Furthermore, the dynamics at the threshold of chaos exhibit a fractal behavior which is demonstrated and used to compute a universal scaling factor that enables the self-similar continuation of the solution into a chaotic regime.

Smyrlis, Yiorgos S.↗
NASA NTRS1991

Future mission studies: Forecasting solar flux directly from its chaotic time series

The mathematical structure of the programs written to construct a nonlinear predictive model to forecast solar flux directly from its time series without reference to any underlying solar physics is presented. This method and the programs are written so that one could apply the same technique to forecast other chaotic time series, such as geomagnetic data, attitude and orbit data, and even financial indexes and stock market data. Perhaps the most important application of this technique to flight dynamics is to model Goddard Trajectory Determination System (GTDS) output of residues between observed position of spacecraft and calculated position with no drag (drag flag = off). This would result in a new model of drag working directly from observed data.

Ashrafi, S.↗
NASA NTRS1991

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↗
NASA NTRS1989

An introduction to chaotic and random time series analysis

The origin of chaotic behavior and the relation of chaos to randomness are explained. Two mathematical results are described: (1) a representation theorem guarantees the existence of a specific time-domain model for chaos and addresses the relation between chaotic, random, and strictly deterministic processes; (2) a theorem assures that information on the behavior of a physical system in its complete state space can be extracted from time-series data on a single observable. Focus is placed on an important connection between the dynamical state space and an observable time series. These two results lead to a practical deconvolution technique combining standard random process modeling methods with new embedded techniques.

Scargle, Jeffrey D.↗
NASA NTRS2004

Flow field topology of transient mixing driven by buoyancy

Transient mixing driven by buoyancy occurs through the birth of a symmetric Rayleigh-Taylor morphology (RTM) structure for large length scales. Beyond its critical bifurcation the RTM structure exhibits self-similarity and occurs on smaller and smaller length scales. The dynamics of the RTM structure, its nonlinear growth and internal collision, show that its genesis occurs from an explosive bifurcation which leads to the overlap of resonance regions in phase space. This event shows the coexistence of regular and chaotic regions in phase space which is corroborated with the existence of horseshoe maps. A measure of local chaos given by the topological entropy indicates that as the system evolves there is growth of uncertainty. Breakdown of the dissipative RTM structure occurs during the transition from explosive to catastrophic bifurcation; this event gives rise to annihilation of the separatrices which drives overlap of resonance regions. The global bifurcation of explosive and catastrophic events in phase space for the large length scale of the RTM structure serves as a template for which mixing occurs on smaller and smaller length scales. Copyright 2004 American Institute of Physics.

Nonlinear Dynamics↗
NASA NTRS1987

Lyapunov exponents for infinite dimensional dynamical systems

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

Mhuiris, Nessan Mac Giolla↗
NASA NTRS1990

Non-linear dynamic analysis of geared systems, part 2

A good understanding of the steady state dynamic behavior of a geared system is required in order to design reliable and quiet transmissions. This study focuses on a system containing a spur gear pair with backlash and periodically time-varying mesh stiffness, and rolling element bearings with clearance type non-linearities. A dynamic finite element model of the linear time-invariant (LTI) system is developed. Effects of several system parameters, such as torsional and transverse flexibilities of the shafts and prime mover/load inertias, on free and force vibration characteristics are investigated. Several reduced order LTI models are developed and validated by comparing their eigen solution with the finite element model results. Several key system parameters such as mean load and damping ratio are identified and their effects on the non-linear frequency response are evaluated quantitatively. Other fundamental issues such as the dynamic coupling between non-linear modes, dynamic interactions between component non-linearities and time-varying mesh stiffness, and the existence of subharmonic and chaotic solutions including routes to chaos have also been examined in depth.

Singh, Rajendra↗
NASA NTRS1991

Indications of low dimensionality in magnetospheric dynamics

Using three separate but related approaches, the question of whether the dynamic response of the magnetosphere to the solar wind input may be described by a low-order system of equations is examined. First, it is determined that the dimension of the subset (the attractor) in the high-dimensional magnetospheric phase space associated with the westward auroral electrojet (AL) index for some of the data sets compiled by Bargatze et al. (1985) is 4.0 + or - 0.2, seemingly independent of activity level. Second, direct modeling of the magnetosphere, considering the bulk properties of the tail plasma, leads to a system of equations that is similar to those previously reported as a dripping faucet model; here, the focus is specifically on the prediction of a natural frequency in this model. Finally, a peak is identified with the predicted frequency in power spectra of AL computed for intervals with both low and high activity. Peaks at other frequencies also appear in the spectra, and such resonances would be expected for a chaotic nonlinear oscillator. Combining these approaches it is concluded that at least some aspects of magnetospheric dynamics may be meaningfully modeled by low-dimensional sets of equations.

Roberts, D. A.↗
NASA NTRS2017

A new concept of stability in orbit propagation, useful for quantifying numerical errors

We present the concept of topological stability in the numerical propagation of orbits, and show how it results in a useful new method for measuring the global numerical error of an orbit propagation. The concept applies to any problem in orbital dynamics. Moreover, it can be extended to any three-dimensional system of di erential equations of second order. In order to assess the topological stability of a given integration a special metric is introduced, which can be used to estimate the numerical errors robustly. The method is particularly well suited for dealing with strongly perturbed and chaotic systems. The construction is based on the constraint imposed by the Hopf map that supports the Kustaanheimo-Stiefel transformation. Generic concepts of stability are translated to KS space.

Pelaez, Jesus↗
NASA NTRS2005

Stochastic Representation of Chaos using Terminal Attractors

A nonlinear version of the Liouville equation based upon terminal attractors is proposed for describing post-instability motions of dynamical systems with exponential divergence of trajectories such as those leading to chaos and turbulence. As a result, the post-instability motions are represented by expectations, variances, and higher moments of the state variables as functions of time. The proposed approach can be applied to conservative chaos, and in particular, to n-bodies problem, as well as to dissipative systems, and in particular, to chaotic attractors and turbulence.

stochastic processes↗
NASA NTRS2009

Unsteady Full Annulus Simulations of a Transonic Axial Compressor Stage

Two recent research endeavors in turbomachinery at NASA Glenn Research Center have focused on compression system stall inception and compression system aerothermodynamic performance. Physical experiment and computational research are ongoing in support of these research objectives. TURBO, an unsteady, three-dimensional, Navier-Stokes computational fluid dynamics code commissioned and developed by NASA, has been utilized, enhanced, and validated in support of these endeavors. In the research which follows, TURBO is shown to accurately capture compression system flow range-from choke to stall inception-and also to accurately calculate fundamental aerothermodynamic performance parameters. Rigorous full-annulus calculations are performed to validate TURBO s ability to simulate the unstable, unsteady, chaotic stall inception process; as part of these efforts, full-annulus calculations are also performed at a condition approaching choke to further document TURBO s capabilities to compute aerothermodynamic performance data and support a NASA code assessment effort.

Herrick, Gregory P.↗
NASA NTRS1988

Dynamics of coorbital satellite rings

The dynamical behavior of a coorbital satellite ring is studied for N = 2-9 satellites, in terms of a simplified dynamical description, in which the motion is reduced to the separation angles between satellites. The number and stability of different kinds of stationary configurations is explored, revealing that equally spaced rings are not stable against small perturbations for N not greater than 6, while for N = 2-8 there exists another, stable compact solution. Integrations of exact equations confirm these results. Moreover, the systems are found to display chaotic characteristics for a certain range of energy. The behavior can be interpreted in terms of maximum velocity curves, defining the allowed region of the motion in the phase space.

Salo, H.↗