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NASA NTRS1985

A perturbative treatment of motion near the 3/1 commensurability

A semianalytic perturbation theory for motion near the 3/1 commensurability in the planar elliptic restricted three-body problem is presented. The predictions of the theory are in good agreement with the features found on numerically generated surfaces of section; a global understanding of the phase space is achieved. The principal cause of the large chaotic zone near the 3/1 commensurability is identified, and a new criterion for the existence of large-scale chaotic behavior is presented.

Wisdom, J.↗

DOE OSTI2020

The Great Inequality and the Dynamical Disintegration of the Outer Solar System

Using an ensemble of N-body simulations, this paper considers the fate of the outer gas giants (Jupiter, Saturn, Uranus, and Neptune) after the Sun leaves the main sequence and completes its stellar evolution. Due to solar mass loss—which is expected to remove roughly half of the star’s mass—the orbits of the giant planets expand. This adiabatic process maintains the orbital period ratios, but the mutual interactions between planets and the width of mean-motion resonances (MMR) increase, leading to the capture of Jupiter and Saturn into a stable 5:2 resonant configuration. The expanded orbits, coupled with the large-amplitude librations of the critical MMR angle, make the system more susceptible to perturbations from stellar flyby interactions. Accordingly, within about 30 Gyr, stellar encounters perturb the planets onto the chaotic subdomain of the 5:2 resonance, triggering a large-scale instability, which culminates in the ejections of all but one planet over the subsequent ∼10 Gyr. After an additional ∼50 Gyr, a close stellar encounter (with a perihelion distance less than ∼200 au) liberates the final planet. Through this sequence of events, the characteristic timescale over which the solar system will be completely dissolved is roughly 100 Gyr. Our analysis thus indicates that the expected dynamical lifetime of the solar system is much longer than the current age of the universe, but is significantly shorter than previous estimates.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Isochronous and period-doubling diagrams for symplectic maps of the plane

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more thorough understanding of the underlying qualitative aspects. This paper aims to address this gap by revisiting the foundational concepts of reversibility and associated symmetries, first explored in the early works of G.D. Birkhoff. We extend the original framework proposed by Hénon by adding a period-doubling diagram to his isochronous diagram, which allows to represents the system’s bifurcations and the groups of symmetric periodic orbits that emerge in typical bifurcations of the fixed point. A qualitative and quantitative explanation of the main features of the region of parameters with bounded motion is provided, along with the application of this technique to other symplectic mappings, including cases of multiple reversibility. Modern chaos indicators, such as the Reversibility Error Method (REM) and the Generalized Alignment Index (GALI), are employed to distinguish between various dynamical regimes in the mixed space of variables and parameters. These tools prove effective in differentiating regular and chaotic dynamics, as well as in identifying twistless orbits and their associated bifurcations. Additionally, we discuss the application of these methods to real-world problems, such as visualizing dynamic aperture in accelerator physics, where our findings have direct relevance.

43 PARTICLE ACCELERATORS↗

NASA NTRS1986

Orbital resonances, unusual configurations and exotic rotation states among planetary satellites

The origin of orbital resonances is shown in the demonstration of the evolution of a pair of planetary satellites through a commensurability of the mean motions by a sequence of diagrams of constant energy curves in a two-dimensional phase space; the closed curve corresponding to the motion in each successive diagram is identified by its adiabatically conserved area. It is found that two-body resonances serve as a basis in the solution of the problem of the origin and evolution of the three-body Laplace resonance among the Galilean satellites of Jupiter. The unusual rotation state of Saturn's satellite Hyperion which is expected to tumble chaotically for an indefinite amount of time is discussed.

Peale, S. J.↗

NASA NTRS2017

The Role of Magnetic Helicity in Structuring the Solar Corona

Two of the most widely observed and striking features of the Suns magnetic field are coronal loops, which are smooth and laminar, and prominences or filaments, which are strongly sheared. Loops are puzzling because they show little evidence of tangling or braiding, at least on the quiet Sun, despite the chaotic nature of the solar surface convection. Prominences are mysterious because the origin of their underlying magnetic structure filament channels is poorly understood at best. These two types of features would seem to be quite unrelated and wholly distinct. We argue that, on the contrary, they are inextricably linked and result from a single process: the injection of magnetic helicity into the corona by photospheric motions and the subsequent evolution of this helicity by coronal reconnection. In this paper, we present numerical simulations of the response of a Parker (1972) corona to photospheric driving motions that have varying degrees of helicity preference. We obtain four main conclusions: (1) in agreement with the helicity condensation model of Antiochos (2013), the inverse cascade of helicity by magnetic reconnection in the corona results in the formation of filament channels localized about polarity inversion lines; (2) this same process removes most complex fine structure from the rest of the corona, resulting in smooth and laminar coronal loops; (3) the amount of remnant tangling in coronal loops is inversely dependent on the net helicity injected by the driving motions; and (4) the structure of the solar corona depends only on the helicity preference of the driving motions and not on their detailed time dependence. We discuss the implications of our results for high-resolution observations of the corona.

Knizhnik, K. J.↗

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.↗

DOE OSTI2021

Quantum dynamics of a nucleon in the Fermi accelerator

Highlights: • In nuclear physics, the Fermi accelerator is a coherent open quantum system. • There is not a true equilibrium for a nucleon‘s mean-energy in the Fermi accelerator. • Occupation of higher energy eigen-states remains even after the mean-energy plateaus. • This work paves the way for work on quantum tunnelling in the Fermi accelerator. The quantum dynamics of a particle in a one-dimensional box with an oscillating wall (the Fermi accelerator) is investigated. The model is applied to the motion of a single nucleon in the mean-field potential of a heavy atomic nucleus whose surface vibrates. By directly solving the time-dependent Schrödinger equation, both the state of the particle and its mean-energy are studied. The effects of the frequency of the wall oscillation on the nucleon’s energy are addressed. Its energy oscillates in phase with the moving wall for all frequencies, showing no chaotic behaviour. There is a large initial peak of the nucleon’s energy as the particle adjusts to the sudden change in the size of the box and a varying relaxation time as it plateaus towards lower energy and a partial equilibrium. Small oscillations in energy continue, since there cannot be a true equilibrium while the wall is moving. The quantum coherence between the different parts of the nucleon’s wave-function in real space is very much preserved. This research lays the foundation for future investigations into quantum tunnelling in the Fermi accelerator.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1989

Dynamics of Pluto

The present study of the Pluto orbit yielded by the LONGSTOP 1B, 100-Myr numerical integration of the outer planets has given attention to the 3:2 resonance in mean motion with Neptune up to degree 2 in eccentricities and inclinations. Confirmations are obtained for both the 19,900-year period longitudinal libration and the 3.78-Myr period libration of the Pluto pericenter argument. Because chaos originates with small divisors, possible resonances have been searched for among the frequencies associated with the critical arguments. The macroscopic stability of the orbit of Pluto is explainable in terms of the association of high-order resonances with small chaotic regions.

Milani, A.↗

NASA NTRS1991

Particle motion in the tail current sheet

Theory of particle motion in current sheets is reviewed. For small, approximately constant normal magnetic field, Bz, particles oscillate about the current sheet and 'live' within the sheet for one-half gyroperiod based on Bz. This lifetime replaces the mean collision time in the Lorentzian conductivity and thus gives rise to the concept of an inertial (or gyro-) conductivity. A substorm model by Coroniti utilizes this conductivity to allow reconnection to proceed without anomalous processes, due to wave-particle interactions. Chaotic particle orbits may at times be important to the dynamics, depending on parameters such as particle energy, current sheet thickness, and field line curvature. A current sheet model with neutral line predicts a ridge structure and asymmetries in the distribution function. Ion distributions near the plasma sheet boundary layer, during the CDAW 6 interval, are consistent with the model predictions. In recent studies by Mitchell et al. and Williams et al., the major current carriers during the growth phase of a substorm were found to be adiabatic electrons not more than 1 keV, but just before a current disruption event, the tail current was mainly carried by energetic ions undergoing current sheet oscillation.

Speiser, T. W.↗

NASA NTRS1992

Symplectic maps for the n-body problem - Stability analysis

The stability of new symplectic n-body maps is examined from the point of view of nonlinear dynamics. The resonances responsible for the principal artifacts are identified. These are resonances between the stepsize and the difference of mean motions between pairs of planets. For larger stepsizes resonant perturbations are evident in the variation of the energy of the system corresponding to these stepsize resonances. It is shown that the principal instability of the method can be predicted and corresponds to the overlap of the stepsize resonances. It is noted that the analysis suggests that other artifacts will occur. For example, the overlap of a stepsize resonance with a resonance of the actual system may also give a region of chaotic behavior that is an artifact. It is pointed out that the fact that the principal artifacts corresponds to a particular set of stepsize resonances suggests that it may be possible to perturbatively remove the effect when the stepsize resonances are nonoverlapping.

Wisdom, Jack↗

NASA NTRS2002

Atmospheric Dynamics of the Outer Planets

The giant planets-Jupiter, Saturn, Uranus, and Neptune-are fluid objects. The winds are powered by absorbed sunlight, as on earth, and by internal heat left over from planetary formation. The main constituents of the atmospheres are hydrogen and helium. The clouds are made of ammonia, hydrogen sulphide, and water. All four giant planets are banded, with multiple zonal jet streams. Even Uranus, whose spin axis is tipped by 98deg relative to the orbit axis, shows latitudinal banding and zonal jets. Equator-to-pole temperature differences are close to zero. Wind speeds are larger than on earth and do not decrease with distance from the sun. Although the power/area at Neptune is only 1/20 that at Jupiter, the winds at Neptune are three times stronger. Stable vortices like the Great Red Spot of Jupiter and similar spots on Neptune come in all size ranges and exhibit a variety of behaviours including merging, orbiting, filament ejection, and oscillating in both shape and position. At least at cloud-top levels, 90% of the long-lived vortices are anticyclonic and sit in anticyclonic shear zones. Features in the cyclonic zones tend to be chaotic, with lifetimes of several days or less. These mesoscale eddies tend to have lightning in them, which suggests that they get their energy from moist convection. The rate of conversion of eddy kinetic energy into kinetic energy of the zonal jets is more than 10% of the power/area radiated by Jupiter. This fraction is more than an order of magnitude larger than on earth. Several lines of evidence now indicate that the winds at cloud-top levels are the surface manifestation of deep-rooted motions that extend into the interior and are presumably driven by internal heat.

Ingersoll, Andrew P.↗

NASA NTRS2018

The Chaotic Long-Term X-Ray Variability of 4U 1705-44

The low-mass X-ray binary 4U1705-44 exhibits dramatic long-term X-ray time variability with a time-scale of several hundred days. The All-SkyMonitor (ASM) aboard the Rossi X-ray Timing Explorer (RXTE) and the Japanese Monitor of All-sky X-ray Image (MAXI) aboard the International Space Station together have continuously observed the source from 1995 December through 2014 May. The combined ASM-MAXI data provide a continuous time series over 50 times the length of the time-scale of interest. Topological analysis can help us identify 'fingerprints' in the phase space of a system unique to its equations of motion. The Birman-Williams theorem postulates that if such fingerprints are the same between two systems, then their equations of motion must be closely related. The phase-space embedding of the source light curve shows a strong resemblance to the double-welled non-linear Duffing oscillator. We explore a range of parameters for which the Duffing oscillator closely mirrors the time evolution of 4U1705-44. We extract low period, unstable periodic orbits from the 4U1705-44 and Duffing time series and compare their topological information. The Duffing and 4U1705-44 topological properties are identical, providing strong evidence that they share the same underlying template. This suggests that we can look to the Duffing equation to help guide the development of a physical model to describe the long-term X-ray variability of this and other similarly behaved X-ray binary systems.

Rossi X-ray Timing Explorer (RXTE)↗

NASA NTRS1988

A physical model for earthquakes. I - Fluctuations and interactions. II - Application to southern California

The idea that earthquakes represent a fluctuation about the long-term motion of plates is expressed mathematically through the fluctuation hypothesis, under which all physical quantities which pertain to the occurance of earthquakes are required to depend on the difference between the present state of slip on the fault and its long-term average. It is shown that under certain circumstances the model fault dynamics undergo a sudden transition from a spatially ordered, temporally disordered state to a spatially disordered, temporally ordered state, and that the latter stages are stable for long intervals of time. For long enough faults, the dynamics are evidently chaotic. The methods developed are then used to construct a detailed model for earthquake dynamics in southern California. The result is a set of slip-time histories for all the major faults, which are similar to data obtained by geological trenching studies. Although there is an element of periodicity to the events, the patterns shift, change and evolve with time. Time scales for pattern evolution seem to be of the order of a thousand years for average recurring intervals of about a hundred years.

Rundle, John B.↗

NASA NTRS1991

Randomization of particle motions and the observed morphology of cometary heads

A great diversity is known to exist in the coma morphology of comets. In particular, some comets show much structural detail in their heads (such as jets, halos, fans, plumes, streamers), while others have completely structureless comas. Obvious questions arise as to why is this so and what does the presence or absence of features tell us about the emission processes on cometary nuclei. In an effort to investigate these problems, a computer code that generates synthetic images of dust comets was modified by introducing random perturbations into motions of ejected particles. It is noted that the introduction of perturbations has made the computer generated images simulate the appearance of comets quite faithfully and that by increasing the perturbations beyond a certain limit it has been possible to erase the coma morphology diagnostic of the details of the ejection process. It is proposed that the degree of collimation of an ejecta flow from discrete active sources on the nucleus surface and possible emissions of dust coma. Molecules of comet gases that radiate in the spectral region employed (and whose velocity distribution is much more chaotic than that of dust particles) and limited atmospheric seeing likewise contribute to blurring structural detail in ground-based imaging observation of comets. The absence of discrete features in the coma does no means imply the absence of localized sources of activity on the nucleus.

Sekanina, Zdenek↗

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.↗

DOE OSTI2020

The microstructure of a selective laser melting (SLM)-fabricated NiTi shape memory alloy with superior tensile property and shape memory recoverability

A selective laser melting (SLM)-fabricated NiTi with superior tensile property and shape memory recoverability was obtained by using a unique stripe rotation scanning strategy. Here, the alloy characteristics, formation mechanisms and evolution in terms of twins, dislocations and precipitations of the alloy were systematically studied. Compared to the conventional smelting-followed-by-machining, the SLM fabrication process involves rapid solidification and repeated heating, which confer distinctive characteristics to the microstructures of SLM-fabricated NiTi alloys. Rapid solidification promotes the formation of a supersaturation solid solution matrix containing a high concentration vacancies, which in turn aggregate to generate a high density of dislocations. During subsequent repeated heating stages, these dislocations occur thermal motion along three directions of <001 > , <111> and <110 >, leading to the formation of thermal kinks, helical dislocations and wave morphology. Simultaneously, precipitated particles Ti 3 Ni 4 repeatedly nucleate and heterogeneous grow with the movement of dislocation. Such precipitation behavior, termed repeated precipitation, has not been previously reported in the conventional NiTi alloys, suggesting that it could be a unique characteristic of such alloy. After martensitic transformation, only two twins, {1 1¯ 1} type I twin and compound twin, are detected. The twinning lamellae of these two twins, where precipitation and dislocation pile-ups exist, often have uneven thickness and chaotic arrangement. Besides, the unique self-accommodated microstructures, such as secondary {1 1¯ 1} type I twin and compound twin with “herring-bone’’ lamellae, which often appear in the deformed or nanocrystalline NiTi, can also be observed. These unique microstructures may confer the distinctive properties to the SLM-fabricated NiTi.

36 MATERIALS SCIENCE↗

NASA NTRS2008

Fomalhaut's Debris Disk and Planet: Constraining the Mass of Formalhaut B from Disk Morphology

Following the optical imaging of exoplanet candidate Fomalhaut b (Fom b), we present a numerical model of how Fomalhaut's debris disk is gravitationally shaped by a single interior planet. The model is simple, adaptable to other debris disks, and can be extended to accommodate multiple planets. If Fom b is the dominant perturber of the belt, then to produce the observed disk morphology it must have a mass M(sub pl) < 3M(sub J), an orbital semimajor axis a(sub pl) > 101.5AU, and an orbital eccentricity e(sub pl) = 0.11 - 0.13. These conclusions are independent of Fom b's photometry. To not disrupt the disk, a greater mass for Fom b demands a smaller orbit farther removed from the disk; thus, future astrometric measurement of Fom b's orbit, combined with our model of planet-disk interaction, can be used to determine the mass more precisely. The inner edge of the debris disk at a approximately equals 133AU lies at the periphery of Fom b's chaotic zone, and the mean disk eccentricity of e approximately equals 0.11 is secularly forced by the planet, supporting predictions made prior to the discovery of Fom b. However, previous mass constraints based on disk morphology rely on several oversimplifications. We explain why our constraint is more reliable. It is based on a global model of the disk that is not restricted to the planet's chaotic zone boundary. Moreover, we screen disk parent bodies for dynamical stability over the system age of approximately 100 Myr, and model them separately from their dust grain progeny; the latter's orbits are strongly affected by radiation pressure and their lifetimes are limited to approximately 0.1 Myr by destructive grain-grain collisions. The single planet model predicts that planet and disk orbits be apsidally aligned. Fomalhaut b's nominal space velocity does not bear this out, but the astrometric uncertainties are difficult to quantify. Even if the apsidal misalignment proves real, our calculated upper mass limit of 3 M(sub J) still holds. Parent bodies are evacuated from mean-motion resonances with Fom b; these empty resonances are akin to the Kirkwood gaps opened by Jupiter. The belt contains at least 3M(sub Earth) of solids that are grinding down to dust, their velocity dispersions stirred so strongly by Fom b that collisions are destructive. Such a large mass in solids is consistent with Fom b having formed in situ.

Chiang, E.↗

DOE OSTI2021

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices

The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi et al. [Los Alamos Scientific Laboratory Report No. LA-1940, 1955 (unpublished)]. Here, we introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a box or stadium with hard walls. The stadia are arranged on a one-dimensional lattice but they individually do not have to be one dimensional, thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N . Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh Gordon theory, we do not see any evidence for the kind of nonergodicity famously seen in the work of Fermi et al. Finally, we examine the chain with particles confined to two-dimensional stadia where the individual stadium is already chaotic and find a much more chaotic phase space at small system sizes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗