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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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NASA NTRS1982

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

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

Shirts, R. B.↗
NASA NTRS1988

Numerical evidence that the motion of Pluto is chaotic

The Digital Orrery has been used to perform an integration of the motion of the outer planets for 845 million years. This integration indicates that the long-term motion of the planet Pluto is chaotic. Nearby trajectories diverge exponentially with an e-folding time of only about 20 million years.

Sussman, Gerald Jay↗
NASA NTRS1986

Determining Chaotic Instabilities in Mechanical Systems

Theoretical developments enable suppression of chaotic structual motions. Theory enables prediction, avoidance, and suppression of chaotic vibrations in structures, especially using dynamic feedback stabilization. In new formulation, motion both repeatable and predictable.

Zak, M. A.↗
NASA NTRS1984

High-resolution IUE observations of interstellar absorption lines in the Vela supernova remnant

Ultraviolet spectra of 45 stars in the vicinity of the Vela supernova remnant were recorded by the short-wavelength echelle spectrograph aboard the International Ultraviolet Explorer (IUE). Over one-third of the stars show interstellar absorption lines at large radial velocities (greater than 60 km/s). The mapping of these high-velocity components in the sky suggests the motions are chaotic, rather than from a coherent expansion of the remnant material. In accord with earlier conclusions from Copernicus data, the gas at high velocity exhibits higher than normal ionization and shows substantially less depletion of nonvolatile elements than normal interstellar material at low velocities. Relatively strong lines from neutral carbon in the two excited fine-structure states indicate that the neutral clouds within the remnant have had their pressures enhanced by the passage of the blast wave from the supernova. Also, the remnant seems to show a significant enhancement in the abundances of low-velocity Si IV, C IV, and N V over those found in the general interstellar medium.

Jenkins, E. B.↗
NASA NTRS1994

A Study of Dean Vortex Development and Structure in a Curved Rectangular Channel with Aspect Ratio of 40 at Dean Numbers up to 430

Flow in a curved channel with mild curvature, an aspect ratio of 40 to 1, and an inner to outer radius ratio of 0.979 is studied at Dean numbers De ranging from 35 to 430. For positions from the start of curvature ranging from 85 to 145 degrees, the sequence of transition events begins with curved channel Poiseuille flow at De less than 40-64. As the Dean number increases, observations show initial development of Dean vortex pairs, followed by symmetric vortex pairs which, when viewed in spanwise/radial planes, cover the entire channel height (De=90-100). At De from 40 to 125-130, the vortex pairs often develop intermittent waviness in the form of vortex undulations. Splitting and merging of vortex pairs is also observed over the same experimental conditions as well as at higher De. When Dean numbers range from 130 to 185-200, the undulating wavy mode is replaced by a twisting mode with higher amplitudes of oscillation and shorter wavelengths. The twisting wavy mode results in the development of regions where turbulence intensity is locally augmented at Dean numbers from 150 to 185-200, principally in the upwash regions between the two individual vortices which make up each vortex pair. These turbulent regions eventually increase in intensity and spatial extent as the Dean number increases further, until individual regions merge together so that the entire cross section of the channel contains chaotic turbulent motions. When Dean numbers then reach 400-435, spectra of velocity fluctuations then evidence fully turbulent flow.

Ligrani, Phillip M.↗
NASA NTRS1989

One-degree-of-freedom motion induced by modeled vortex shedding

The motion of an elastically supported cylinder forced by a nonlinear, quasi-static, aerodynamic model with the unusual feature of a motion-dependent forcing frequency was studied. Numerical solutions for the motion and the Lyapunov exponents are presented for three forcing amplitudes and two frequencies (1.0 and 1.1 times the Strouhal frequency). Initially, positive Lyapunov exponents occur and the motion can appear chaotic. After thousands of characteristic times, the motion changes to a motion (verified analytically) that is periodic and damped. This periodic, damped motion was not observed experimentally, thus raising questions concerning the modeling.

Yates, L. A.↗
NASA NTRS1988

Tidal evolution of the Uranian satellites. I - Passage of Ariel and Umbriel through the 5:3 mean-motion commensurability

A significant chaotic zone is associated with the 5:3 mean-motion commensurability resonance through which Ariel and Umbriel would have passed, in the event that Uranus' specific dissipation function were less than 100,000; the standard theory for passage through orbital resonances would in this case be inapplicable, in view of significant changes in the probability and the mechanism of capture into the resonance. It is suggested that tidal evolution within the chaotic zone may have pushed eccentrocities to comparatively high values at which the probability of escape from resonance remained. Eccentricities sufficiently great to affect Ariel's thermal history have not been found.

Tittemore, William C.↗
NASA NTRS1992

Variations of the Milankovitch frequencies in time

The sensitivity of the amplitudes and frequencies in the development of the Earth's orbital and rotational elements involved in the astronomical theory of paleoclimates (eccentricity, obliquity, and climate precession), to the Earth-Moon distance and consequently to the length of the day and to the dynamical ellipticity of the Earth has been discussed for the last billions of years. The shortening of the Earth-Moon distance and of the length of the day, as well as the lengthening of the dynamical ellipticity of the Earth back in time induce a shortening of the fundamental astronomical periods for precession and obliquity. At the same time, the amplitudes of the different terms in the development of the obliquity are undergoing a relative enlargement of about 50 percent at 2 x 10(exp 9) yr BP but the independent term is increasing very weakly (less than 0.1 percent). In other words, the value of the obliquity, which lies within a range of 21.7 to 24.9 deg over the Quarternary was restricted to a range of 22.5 to 24.1 deg at 2 x 10(exp 9) yr BP. On the other hand, the amplitudes in the development of the climatic precession do not change. Moreover, these changes in the frequencies and amplitudes for both obliquity and climatic precession are larger for longer period terms. Finally, the periods in the eccentricity development are not influenced by the variation of the lunar distance. But the motion of the solar system, especially of the inner planets, was shown to be chaotic. It means that it is impossible to compute the exact motion of the planets over more than about 100 Myr, and the fundamental frequencies of the systems are not fixed quantities, but are slowly varying with time. As long as we consider the most important terms, the maximum deviation from the present-day value of the 19-kyr precessional period due to the chaotic motion of the solar system only does not reach more than a few tens of years around 80 Myr BP. Therefore the shortening of the obliquity and climatic precession periods is mostly driven by the change in the lunar distance and the consequent variations in the dynamical ellipticity of the Earth's angular speed. At first sight, the deviation in the period for the eccentricity can be neglected, as the chaotic behavior of the solar system implies a relative change of the main periods by less than 0.2 percent, 1.4 percent, and 1.9 percent respectively, this maximum change being achieved around 80 Myr BP. This implies, in particular, that the eccentricity periods for Quarternary climate studies may be considered more or less constant for pre-Quaternay times and equal to their Quaternary values.

Loutre, Marie-France↗
NASA NTRS1994

Chaotic dynamics of corotating magnetospheric convection

The corotating plasma convection system of the Jovian magnetosphere is analyzed. The macroscopic (mhd) model introduced by Summers and Mu, (1992) that incorporates the effects of microdiffusion is extended by including previously neglected density effects. We reduce the governing partial differential equations to a third-order ordinary differential system by the Galerkin technique of mode truncation. We carry out such a severe truncation partly in the interests of tractability, and leave open the question of the efficacy of adding additional modes. Exhaustive numerical integrations are carried out to calculate the long-term solutions, and we discover that a rich array of plasma motions is possible, dependent on the value of the height-integrated ionospheric Pederson conductivity Sigma. If Sigma is less than a certain critical value Sigma(sub c), then plasma motion can be expected to be chaotic (or periodic), while if Sigma is greater than Sigma(sub c), then steady state convection is expected. In the former case, whether the plasma motion is chaotic or periodic (and, if periodic, the magnitude of the period) can be very sensitive to the value of Sigma. The value of Sigma(sub c), which is a function of a parameter q that occurs in the assumed form of the stationary radial profile (varies as L(exp -q) of the plasma mass per unit magnetic flux, lies well within the accepted range of values of Sigma for Jupiter, i.e. Sigma greater than or equal to 0.1 mho and less than or equal to 10 mho.

Summers, Danny↗
NASA NTRS1987

Urey Prize Lecture - Chaotic dynamics in the solar system

Attention is given to solar system cases in which chaotic solutions of Newton's equations are important, as in chaotic rotation and orbital evolution. Hyperion is noted to be tumbling chaotically; chaotic orbital evolution is suggested to be of fundamental importance to an accounting for the Kirkwood gaps in asteroid distribution and for the phase space boundary of the chaotic zone at the 3/1 mean-motion commensurability with Jupiter. In addition, chaotic trajectories in the 2/1 chaotic zone reach very high eccentricities by a route that carries them to high inclinations temporarily.

Wisdom, Jack↗
NASA NTRS1992

Periodic and chaotic spatiotemporal states in a phase-conjugate resonator using a photorefractive BaTiO3 phase-conjugate mirror

The dynamics of an externally pumped phase-conjugate resonator that uses four-wave mixing in BaTi03 is investigated experimentally. The emergence of spatiotemporal instabilities as the degree of transverse confinement is varied by changing the Fresnel number is described in detail. Local intensity time series show that relaxing the transverse confinement leads the system from a stationary state to periodic, then to quasi-periodic, motions and finally to chaotic behavior. In some regions of parameter space two frequencies are identified in the power spectra of the time series, indicating a route to chaos following the Ruelle-Takens-Newhouse scheme. Wave-front topological defects are identified by interferometry. As the system's confinement is varied, the phase-defect density and the spatial correlation index are found to follow similar trends, indicating that the observed spatiotemporal dynamics may indeed be an example of defect-mediated turbulence.

Liu, Siuying R.↗
NASA NTRS1992

Chaotic evolution of the solar system

The evolution of the entire planetary system has been numerically integrated for a time span of nearly 100 million years. This calculation confirms that the evolution of the solar system as a whole is chaotic, with a time scale of exponential divergence of about 4 million years. Additional numerical experiments indicate that the Jovian planet subsystem is chaotic, although some small variations in the model can yield quasi-periodic motion. The motion of Pluto is independently and robustly chaotic.

Sussman, Gerald J.↗
NASA NTRS1992

Fuzzy operators and cyclic behavior in formal neuronal networks

Formal neuronal networks (FNN), which are comprised of threshold gates, make use of the unit step function. It is regarded as a degenerated distribution function (DDF) and will be referred to here as a non-fuzzy threshold operator (nFTO). Special networks of this kind generating long cycles of states are modified by introduction of fuzzy threshold operators (FTO), i.e., non-degenerated distribution functions (nDDF). The cyclic behavior of the new nets is compared with the original ones. The interconnection matrix and threshold values are not modified. It is concluded that the original long cycles change the fixed points and short cycles, and as the computer simulations demonstrate, the aperiodic motion that is associated with chaotic behavior appears. The emergence of the above changes depend on the steepness of the threshold operators.

Labos, E.↗
NASA NTRS1991

Evidence of chaotic pattern in solar flux through a reproducible sequence of period-doubling-type bifurcations

Presented here is a preliminary study of the limits to solar flux intensity prediction, and of whether the general lack of predictability in the solar flux arises from the nonlinear chaotic nature of the Sun's physical activity. Statistical analysis of a chaotic signal can extract only its most gross features, and detailed physical models fail, since even the simplest equations of motion for a nonlinear system can exhibit chaotic behavior. A recent theory by Feigenbaum suggests that nonlinear systems that can be led into chaotic behavior through a sequence of period-doubling bifurcations will exhibit a universal behavior. As the control parameter is increased, the bifurcation points occur in such a way that a proper ratio of these will approach the universal Feigenbaum number. Experimental evidence supporting the applicability of the Feigenbaum scenario to solar flux data is sparse. However, given the hypothesis that the Sun's convection zones are similar to a Rayleigh-Bernard mechanism, we can learn a great deal from the remarkable agreement observed between the prediction by theory (period doubling - a universal route to chaos) and the amplitude decrease of the signal's regular subharmonics. The authors show that period-doubling-type bifurcation is a possible route to a chaotic pattern of solar flux that is distinguishable from the logarithm of its power spectral density. This conclusion is the first positive step toward a reformulation of solar flux by a nonlinear chaotic approach. The ultimate goal of this research is to be able to predict an estimate of the upper and lower bounds for solar flux within its predictable zones. Naturally, it is an important task to identify the time horizons beyond which predictability becomes incompatible with computability.

Ashrafi, S.↗
NASA NTRS1991

Evidence of chaotic pattern in solar flux through a reproducible sequence of period-doubling-type bifurcations

A preliminary study of the limits to solar flux intensity prediction, and of whether the general lack of predictability in the solar flux arises from the nonlinear chaotic nature of the Sun's physical activity is presented. Statistical analysis of a chaotic signal can extract only its most gross features, and detailed physical models fail, since even the simplest equations of motion for a nonlinear system can exhibit chaotic behavior. A recent theory by Feigenbaum suggests that nonlinear systems that can be led into chaotic behavior through a sequence of period-doubling bifurcations will exhibit a universal behavior. As the control parameter is increased, the bifurcation points occur in such a way that a proper ratio of these will approach the universal Feigenbaum number. Experimental evidence supporting the applicability of the Feigenbaum scenario to solar flux data is sparse. However, given the hypothesis that the Sun's convection zones are similar to a Rayleigh-Bernard mechanism, we can learn a great deal from the remarkable agreement observed between the prediction by theory (period doubling - a universal route to chaos) and the amplitude decrease of the signal's regular subharmonics. It is shown that period-doubling-type bifurcation is a possible route to a chaotic pattern of solar flux that is distinguishable from the logarithm of its power spectral density. This conclusion is the first positive step toward a reformulation of solar flux by a nonlinear chaotic approach. The ultimate goal of this research is to be able to predict an estimate of the upper and lower bounds for solar flux within its predictable zones. Naturally, it is an important task to identify the time horizons beyond which predictability becomes incompatible with computability.

Ashrafi, S.↗
NASA NTRS1992

Chaotic scattering of pitch angles in the current sheet of the magnetotail

The modified Harris field model is used to investigate the process of pitch angle scattering by a current sheet. The relationship between the incoming asymptotic pitch angle alpha(in) and the outgoing asymptotic pitch angle alpha(out) is studied from first principles by numerically integrating the equation of motion. Evidence that charged particles undergo chaotic scattering by the current sheet is found. For fixed alpha(in), it is shown that alpha(out) exhibits sensitive dependence on the energy parameter in certain energy ranges. For a fixed energy parameter value in the same energy ranges, alpha(out) sensitively depends on alpha(in). For other energy values, alpha(out) does not show sensitive dependence on alpha(in) for most phase angles. A distribution of alpha(in) is mapped from the asymptotic region to the midplane, and it is found that the resulting particle distribution should have beam structures with well-collimated pitch angles near each resonance energy value. Implications for the particle distribution functions in the earth's magnetotail are discussed.

Burkhart, G. R.↗
NASA NTRS1998

On the Origin of Chaos in the Asteroid Belt

We consider the effect of gravitational perturbations from Jupiter on the dynamics of asteroids, when Jupiter is itself perturbed by Saturn. The presence of Saturn introduces a number of additional frequencies into Jupiters orbit. These frequencies in turn produce chaos in narrow regions on either side of the chaotic zones associated with the mean motion resonances between the asteroids and Jupiter. The resonant arguments of these three-body resonances contain the longitudes of Jupiter and the asteroid together with either the secular frequency 9-6, or the longitude of Saturn. Resonances involving the longitude of Saturn are analogs of the Laplace resonance in the Jovian satellite system. We show that many three-body resonances involving the longitude of Saturn are chaotic. We give simple expressions for the width of the chaotic region and the associated Lyapunov time. In some cases the chaos can produce a diffusive growth in the 4 eccentricity of the asteroid that leads to ejection of the asteroid on times shorter than the age of the solar system. We give simple estimates for the diffusion time. Finally, we present the results of numerical integrations testing the theory.

Murray, N.↗