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NASA NTRS1989

Experimental study of the dynamics of a ruby laser pumped by a CW argon-ion laser

A study of the dynamics of a ruby laser pumped by a CW argon-ion laser is presented. The ruby laser is predominantly stable but has two accessible unstable states. One state exhibits chaotic output, while the other results in regular self-pulsing. The conditions needed for instability are discussed and homodyne spectra and temporal maps of the phase-space attractors are obtained. In addition, a numerical simulation of nonlinear beam propagation in ruby is presented that shows that strong deviations from plane-wave behavior exist, and that transverse effects must be incorporated into theoretical models of the instability.

Afzal, R. S.↗

NASA NTRS1992

Nonlinear vibration and radiation from a panel with transition to chaos

The dynamic response of an aircraft panel forced at resonance and off-resonance by plane acoustic waves at normal incidence is investigated experimentally and numerically. Linear, nonlinear (period doubling), and chaotic responses are obtained by increasing the sound pressure level of the excitation. The response time history is sensitive to the input level and to the frequency of excitation. The change in response behavior is due to a change in input conditions, triggered either naturally or by modulation of the bandwidth of the incident waves. Off-resonance bifurcation is diffused and difficult to maintain; thus the panel response drifts into a linear behavior. The acoustic pressure emanated by the panel is either linear or nonlinear as is the vibration response. The nonlinear effects accumulate during the propagation with distance. Results are also obtained on the control of the panel response using damping tape on an aluminum panel and a graphite epoxy panel having the same size and weight. Good agreement is obtained betwen the experimental and numerical results.

Maestrello, Lucio↗

NASA NTRS1992

Nonlinear vibration and radiation from a panel with transition to chaos induced by acoustic waves

The dynamic response of an aircraft panel forced at resonance and off-resonance by plane acoustic waves at normal incidence is investigated experimentally and numerically. Linear, nonlinear (period doubling) and chaotic responses are obtained by increasing the sound pressure level of the excitation. The response time history is sensitive to the input level and to the frequency of excitation. The change in response behavior is due to a change in input conditions, triggered either naturally or by modulation of the bandwidth of the incident waves. Off-resonance, bifurcation is diffused and difficult to maintain, thus the panel response drifts into a linear behavior. The acoustic pressure emanated by the panel is either linear or nonlinear as is the vibration response. The nonlinear effects accumulate during the propagation with distance. Results are also obtained on the control of the panel response using damping tape on aluminum panel and using a graphite epoxy panel having the same size and weight. Good agreement is obtained between the experimental and numerical results.

Maestrello, Lucio↗

NASA NTRS1987

Rotational dynamics of irregularly shaped natural satellites

The rotational histories of irregularly shaped satellites are studied and are found to differ from the standard picture of tidal evolution of satellite rotations. Prior to capture into the synchronous rotation resonance, a narrow attitude-unstable chaotic zone is entered and the satellite begins to tumble chaotically. It is noted that enhanced dissipation of energy during the chaotic-tumbling phase may effect the orbital evolution. The theory suggests that, eventually, the rotation stays close to one of the accessible attitude-stable islands long enough for the weak tidal torque to remove the satellite from the chaotic zone.

Wisdom, Jack↗

NASA NTRS2002

Nonlinear Dynamic Models in Advanced Life Support

To facilitate analysis, ALS systems are often assumed to be linear and time invariant, but they usually have important nonlinear and dynamic aspects. Nonlinear dynamic behavior can be caused by time varying inputs, changes in system parameters, nonlinear system functions, closed loop feedback delays, and limits on buffer storage or processing rates. Dynamic models are usually cataloged according to the number of state variables. The simplest dynamic models are linear, using only integration, multiplication, addition, and subtraction of the state variables. A general linear model with only two state variables can produce all the possible dynamic behavior of linear systems with many state variables, including stability, oscillation, or exponential growth and decay. Linear systems can be described using mathematical analysis. Nonlinear dynamics can be fully explored only by computer simulations of models. Unexpected behavior is produced by simple models having only two or three state variables with simple mathematical relations between them. Closed loop feedback delays are a major source of system instability. Exceeding limits on buffer storage or processing rates forces systems to change operating mode. Different equilibrium points may be reached from different initial conditions. Instead of one stable equilibrium point, the system may have several equilibrium points, oscillate at different frequencies, or even behave chaotically, depending on the system inputs and initial conditions. The frequency spectrum of an output oscillation may contain harmonics and the sums and differences of input frequencies, but it may also contain a stable limit cycle oscillation not related to input frequencies. We must investigate the nonlinear dynamic aspects of advanced life support systems to understand and counter undesirable behavior.

Jones, Harry↗

NASA NTRS1998

Geoid Anomalies and Dynamic Topography from Time Dependent, Spherical Axisymmetric Mantle Convection

Geoid anomalies and dynamic topography are two important diagnostics of mantle convection. We present geoid and topography results for several time-dependent convection models in spherical axisymmetric geometry for Rayleigh numbers between 10(exp 6) and 10(exp 7) with depth-dependent viscosity and mixtures of bottom and internal heating. The models are strongly chaotic, with boundary layer instabilities erupting out of both thermal boundary layers. In some instances, instabilities from one boundary layer influence the development of instabilities in the other boundary layer. Such coupling between events at the top and bottom of the mantle has been suggested to play a role in a mid-Cretaceous episode of enhanced volcanism in the Pacific. These boundary layer instabilities produce large temporal variations in the geoid anomalies and dynamic nd to the topography associated with the convection. The amplitudes of these fluctuations depend on the detailed model parameter,.% it of this but fluctuations of 30-50% relative to the time-averaged geoid and topography are common. The convective planform is strongly sensitive to the specific initial conditions. Convection cells with larger aspect ratio tend to have larger fractional fluctuations in their geoid and topography amplitudes, because boundary layer instabilities have more time to develop in long cells. In some instances, we observe low-amplitude topographic highs adjacent to the topographic lows produced by cold downwellings. We discuss applications of these results to several situations, including the temporal variability of m basis. hotspots such as Hawaii, the topography of subduction zone outer rises, and the topography of coronae on Venus.

Kiefer, Walter S.↗

NASA NTRS1993

Basic studies of baroclinic flows

A fully nonlinear 3-dimensional numerical model (GEOSIM), previously developed and validated for several cases of geophysical fluid flow, has been used to investigate the dynamical behavior of laboratory experiments of fluid flows similar to those of the Earth's atmosphere. The phenomena investigated are amplitude vacillation, and the response of the fluid system to uneven heating and cooling. The previous year's work included hysteresis in the transition between axisymmetric and wave flow. Investigation is also continuing of the flows in the Geophysical Fluid Flow Cell (GFFC), a low-gravity Spacelab experiment. Much of the effort in the past year has been spent in validation of the model under a wide range of external parameters including nonlinear flow regimes. With the implementation of a 3-dimensional upwind differencing scheme, higher spectral resolution, and a shorter time step, the model has been found capable of predicting the majority of flow regimes observed in one complete series of baroclinic annulus experiments of Pfeffer and co-workers. Detailed analysis of amplitude vacillation has revealed that the phase splitting described in the laboratory experiments occurs in some but not all cases. Through the use of animation of the models output, a vivid 3-dimensional view of the phase splitting was shown to the audience of the Southeastern Geophysical Fluid Dynamics Conference in March of this year. A study on interannual variability was made using GEOSIM with periodic variations in the thermal forcing. Thus far, the model has not predicted a chaotic behavior as observed in the experiments, although there is a sensitivity in the wavenumber selection to the initial conditions. Work on this subject, and on annulus experiments with non-axisymmetric thermal heating, will continue. The comparison of GEOSIM's predictions will result from the Spacelab 3 GFFC experiments continued over the past year, on a 'back-burner' basis. At this point, the study (in the form of a draft of a journal article) is nearly completed. The results from GEOSIM compared very well with the experiments, and the use of the model allows the demonstration of flow mechanics that were not possible with the experimental data. For example, animation of the model output shows that the forking of the spiral bands is a transient phenomenon, due to the differential east-west propagation of convection bands from different latitudes.

Miller, Timothy L.↗

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 NTRS2019

Long and Short Term Geomagnetic Prediction (Chapter 19)

Prediction of geomagnetic variability depends on the accuracy of geomagnetic field modeling, dynamical modeling of source regions that contribute to geomagnetic signals, and advanced assimilation algorithms that combine effectively the results of geomagnetic field and dynamic models to make accurate estimates of the dynamic states of the sources and, therefore, accurate forecast of geomagnetic variations. In this chapter we provide an overview of recent research efforts in these three research areas, focusing primarily on geomagnetic variations from the dynamic outer core and from solar and lunar tidal effects, but also including a review of relevant research results and developments. Prediction of weak but periodic tidal phenomena, and of strong but chaotic secular variation (SV) showcases two very important new developments which will lead to new opportunities in geomagnetic research and application.

Weijia Kuang↗

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

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 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 NTRS1992

Turbulence 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, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low Reynolds number fully developed turbulence are compared. Several flows are noted: fully chaotic, complex periodic, weakly chaotic, simple periodic, and fixed-point. Of these, only fully chaotic is classified as turbulent. Besides the sustained flows, a flow which decays as it becomes turbulent is examined. For the finest grid, 128(exp 3) points, the spatial resolution appears to be quite good. As a final note, the variation of the velocity derivatives skewness of a Navier-Stokes flow as the Reynolds number goes to zero is calculated numerically. The value of the skewness is shown to become small at low Reynolds numbers, in agreement with intuitive arguments that nonlinear terms should be negligible.

Deissler, Robert G.↗

NASA NTRS1992

On chaotic conductivity in the magnetotail

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.

Holland, Daniel L.↗

NASA NTRS2002

The Statistical Mechanics of Ideal Homogeneous Turbulence

Plasmas, such as those found in the space environment or in plasma confinement devices, are often modeled as electrically conducting fluids. When fluids and plasmas are energetically stirred, regions of highly nonlinear, chaotic behavior known as turbulence arise. Understanding the fundamental nature of turbulence is a long-standing theoretical challenge. The present work describes a statistical theory concerning a certain class of nonlinear, finite dimensional, dynamical models of turbulence. These models arise when the partial differential equations describing incompressible, ideal (i.e., nondissipative) homogeneous fluid and magnetofluid (i.e., plasma) turbulence are Fourier transformed into a very large set of ordinary differential equations. These equations define a divergenceless flow in a high-dimensional phase space, which allows for the existence of a Liouville theorem, guaranteeing a distribution function based on constants of the motion (integral invariants). The novelty of these particular dynamical systems is that there are integral invariants other than the energy, and that some of these invariants behave like pseudoscalars under two of the discrete symmetry transformations of physics, parity, and charge conjugation. In this work the 'rugged invariants' of ideal homogeneous turbulence are shown to be the only significant scalar and pseudoscalar invariants. The discovery that pseudoscalar invariants cause symmetries of the original equations to be dynamically broken and induce a nonergodic structure on the associated phase space is the primary result presented here. Applicability of this result to dissipative turbulence is also discussed.

Shebalin, John V.↗

NASA NTRS1988

The theoretical and experimental study of the nonlinear and chaotic response of curved panels to intense acoustic excitation

Assuming a single-mode transverse displacement, a simple formula is derived for the transverse load-displacement relationship of a plate under in-plane compression. The formula is used to derive a simple analytical expression for the nonlinear dynamic response of postbuckled plates under sinusoidal or random excitation. The highly nonlinear motion of snap-through can be easily interpreted using the single-mode formula. Experimental results are obtained using buckled and cylindrical aluminum panels using discrete frequency and broadband excitation of mechanical and acoustic forces.

Ng, C. F.↗

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 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.↗