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The motion of charged particles is studied numerically and analytically using a test particle approach. Acceleration near the X-type neutral line is described in terms of adiabatic invariants and separatrix crossings. The relation between regular and chaotic properties of the process is discussed for a realistic source in the plasma mantle.
The present study examines how the changes in the generalized first adiabatic invariant J derived from the separatrix crossing theory can be incorporated into the drift variable approach to generating distribution functions. A method is proposed for determining distribution functions for an ensemble of particles following interaction with the tail current sheet by treating the interaction as a scattering problem characterized by changes in the invariant. Generalized drift velocities are obtained for a 1D tail configuration by using the generalized first invariant. The invariant remained constant except for the discrete changes caused by chaotic scattering as the particles cross the separatrix.
Chaotic inflation is analyzed in the frame of scalar-tensor theories of gravity. Fluctuations in the energy density arise from quantum fluctuations of the Brans-Dicke field and of the inflation field. The spectrum of perturbations is studied for a class of models: it is non scale-invarient and, for certain values of the parameters, it has a peak. If the peak appears at astrophysically interesting scales, it may help to reconcile the Cold Dark Matter scenario for structure formation with large scale observations.
We propose a new class of galactic discrete gamma-ray sources, the chaotic, high mass-loss-rate winds from luminous early-type stars. Early-type stellar winds are highly unstable due to intrinsic line-driven instabilities, and so are permeated by numerous strong shocks. These shocks can accelerate a small fraction of thermal electrons and ions to relativistic energies via the first-order Fermi mechanism. A power-law-like photon spectrum extending from keV to above 10 MeV energies is produced by inverse Compton scattering of the extremely abundant stellar UV photons by the relativistic electrons. In addition, a typical pi(sup 0)-decay gamma-ray spectrum is generated by proton-ion interactions in the densest part of the winds.
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.
Using techniques from shadowing theory, the solution of the scalar advection-diffusion equation is studied. It is shown that, under certain circumstances, the effect of small scalar diffusivity is to smooth the zero-diffusivity solution by averaging local fine-scaled structure against a Gaussian. The method of study depends on shadowing and thus fails for nonuniformly stretching systems, its failure suggesting the ways in which the effects of asymptotically small molecular diffusion can become nonlocal in chaotic fluid flows.
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.
The concept of chaotic conductivity and the acceleration of particles due to a constant dawn dusk electric field are studied in a magnetotail-like magnetic field. A test particle simulation is used including the full nonlinear dynamics. It is found that the acceleration process can be understood without invoking chaos and that the cross tail current is determined by the particle dynamics and distributions. It is concluded that in general there is no simple relationship between the electric field and the current.
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.
Nonlinear phenomena associated with the dynamics of free drops and bubbles are investigated analytically, numerically and experimentally. Although newly developed levitation and measurement techniques have been implemented, the full experimental validation of theoretical predictions has been hindered by interfering artifacts associated with levitation in the Earth gravitational field. The low gravity environment of orbital space flight has been shown to provide a more quiescent environment which can be utilized to better match the idealized theoretical conditions. The research effort described in this paper is a closely coupled collaboration between predictive and guiding theoretical activities and a unique experimental program involving the ultrasonic and electrostatic levitation of single droplets and bubbles. The goal is to develop and to validate methods based on nonlinear dynamics for the understanding of the large amplitude oscillatory response of single drops and bubbles to both isotropic and asymmetric pressure stimuli. The first specific area on interest has been the resonant coupling between volume and shape oscillatory modes isolated gas or vapor bubbles in a liquid host. The result of multiple time-scale asymptotic treatment, combined with domain perturbation and bifurcation methods, has been the prediction of resonant and near-resonant coupling between volume and shape modes leading to stable as well as chaotic oscillations. Experimental investigations of the large amplitude shape oscillation modes of centimeter-size single bubbles trapped in water at 1 G and under reduced hydrostatic pressure, have suggested the possibility of a low gravity experiment to study the direct coupling between these low frequency shape modes and the volume pulsation, sound-radiating mode. The second subject of interest has involved numerical modeling, using the boundary integral method, of the large amplitude shape oscillations of charged and uncharged drops in the presence of a static or time-varying electric field. Theoretically predicted non linearity in the resonant frequency of the fundamental quadrupole mode has been verified by the accompanying experimental studies. Additional phenomena such as hysteresis in the frequency response of ultrasoncially levitated droplets in the presence of a time varying electric field, and mode coupling in the oscillations of ultrasonically modulated droplets, have also been uncovered. One of the results of this ground-based research has been the identification and characterization of phenomena strictly associated with the influence of the gravitational field. This has also allowed us to identify the specific requirements for potential microgravity investigations yielding new information not obtainable on Earth.
The stability of ion distributions with velocity space holes caused by chaotic ion acceleration in the vicinity of the field reversal region in the Earth's magnetotail is investigated. It is shown that such distributions are unstable to electrostatic waves with wave vector directed along the Sun-Earth line. The solution of the quasi-linear equation shows that the saturation of the instability occurs due to velocity space trapping of the low-energy background ions and results in a flattening of the reduced ion distribution function in the region of the original velocity space hole. The energy level of the excited waves in the vicinity of the Earth's midplane is estimated as a function of the initial size of the empty velocity space region.
Recent calculations of the Martian obliquity suggests that it varies chaotically on timescales longer than about 10(exp 7) years and varies between about 0 and 60 deg. We examine the seasonal water behavior at obliquities between 40 and 60 deg. Up to several tens of centimeters of water may sublime from the polar caps each year, and possibly move to the equator, where it is more stable. The CO2 frost and CO2-H2O clathrate hydrate are stable in thepolar deposits below a few tens of meters depth, so that the polar cap could contain a significant CO2 reservoir. If CO2 is present, it could be left over from the early history of Mars; also, it could be released into the atmosphere during periods of high obliquity, causing occasional periods of more-clement climate.
Irregular fluctuations in intensity have long plagued the operation of a wide variety of solid-state lasers. We are exploring the possibility of exploiting rather than avoiding a laser's chaotic output. As an important step in that direction, we have applied a novel control technique to stabilize a solid state laser. By making small periodic changes in only one input parameter of the laser, we are able to stabilize complex periodic waveforms and steady state behavior in the laser output. We demonstrate the application of this approach in a diode pumped Nd:/YAG laser system.
The motion of planetary bodies is the archetypal clockwork system. Indeed, clocks and calendars were developed to keep track of the relative motions of the Earth, the Sun and the Moon. However, studies over the past few decades imply that this predictable regularity does not extend to small bodies, nor does it apply to the precise trajectories of the planets themselves over long timescale.s. Various examples of chaotic motion within our Solar System and, extrasolar planetary systems will be discussed.
The shape and spin of Neptune's outermost satellite Nereid are still unknown. Ground-based photometry indicates large brightness variations, but different observers report very different lightcurve amplitudes and periods. On the contrary, Voyager 2 images spanning 12 days show no evidence of variations greater than 0.1 mag. The latter suggest either that Nereid is nearly spherical, or that it is rotating slowly. We propose that tides have already despun Nereid's rotation to a period of a few weeks, during the time before the capture of Triton when Nereid was closer to Neptune. Since Nereid reached its present orbit, tides have further despun Nereid to a period on the order of a month. For Nereid's orbital eccentricity of 0.75, tidal evolution ceases when the spin period is still approximately 1/8 of the orbital period. Furthermore, the synchronous resonance becomes quite weak for such high eccentricities, along with other low-order spin orbit commensurabilities. In contrast, high-order resonances become very strong particularly the 6:1, 6.5:1, 7:1, 7.5:1, and 8:1 spin states. If Nereid departs by more than approximately 1% from a sphere, however, these resonances overlap, generating chaos. Our simulations show that Nereid is likely to be in chaotic rotation for any spin period longer than about 2 weeks.
Atmospheric data assimilation is the name scientists give to the techniques of blending atmospheric observations with atmospheric model results to obtain an accurate idea of what the atmosphere looks like at any given time. Because two pieces of information are used, observations and model results, the outcomes of data assimilation procedure should be better than what one would get by using one of these two pieces of information alone. There is a number of different mathematical techniques that fall under the data assimilation jargon. In theory most these techniques accomplish about the same thing. In practice, however, slight differences in the approaches amount to faster algorithms in some cases, more economical algorithms in other cases, and even give better overall results in yet some other cases because of practical uncertainties not accounted for by theory. Therefore, the key is to find the most adequate data assimilation procedure for the problem in hand. In our Data Assimilation group we have been doing extensive research to try and find just such data assimilation procedure. One promising possibility is what we call retrospective iterated analysis (RIA) scheme. This procedure has recently been implemented and studied in the context of a very large data assimilation system built to help predict and study weather and climate. Although the results from that study suggest that the RIA scheme produces quite reasonable results, a complete evaluation of the scheme is very difficult due to the complexity of that problem. The present work steps back a little bit and studies the behavior of the RIA scheme in the context of a small problem. The problem is small enough to allow full assessment of the quality of the RIA scheme, but it still has some of the complexity found in nature, namely, its chaotic-type behavior. We find that the RIA performs very well for this small but still complex problem which is a result that seconds the results of our early studies.
Time series from biological system often display fluctuations in the measured variables. Much effort has been directed at determining whether this variability reflects deterministic chaos, or whether it is merely "noise". Despite this effort, it has been difficult to establish the presence of chaos in time series from biological sytems. The output from a biological system is probably the result of both its internal dynamics, 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, i.e., a positive characteristic exponent that leads to sensitivity to initial conditions. 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.
Canonical equations governing an electron motion in electromagnetic field of the whistler mode waves propagating along the direction of an ambient magnetic field are derived. The physical processes on which the equations of motion are based .are identified. It is shown that relativistic electrons interacting with these fields demonstrate chaotic motion, which is accompanied by the particle stochastic heating and significant pitch angle diffusion. Evolution of distribution functions is described by the Fokker-Planck-Kolmogorov equations. It is shown that the whistler mode waves could provide a viable mechanism for stochastic energization of electrons with energies up to 50 MeV in the Jovian magnetosphere.