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NASA NTRS1983

Chaotic behavior and the origin of the 3/1 Kirkwood gap

The sudden eccentricity increases discovered by Wisdom (1982) are reproduced in numerical integrations of the planar ecliptic restricted three-body problem, verifying that this phenomenon is real. Mapping derivations are qualitatively reviewed and the maximum Liapunov characteristic exponent and its importance for determining the character of a trajectory are explained. The results of a number of calculations of this exponent using the differential equations for the unaveraged three-body problem are shown and compared to equivalent calculations using a mapping. In all cases the two approaches agree whether the orbits are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 3/1 commensurability, both in the planar-ecliptic problem and in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 3/1 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.

Wisdom, J.↗

NASA NTRS1990

An investigation of chaotic Kolmogorov flows

A two dimensional flow governed by the incompressible Navier-Stokes equations with a steady spatially periodic forcing (known as the Kolmogorov flow) is numerically simulated. The behavior of the flow and its transition states as the Reynolds number (Re) varies is investigated in detail, as well as a number of the flow features. A sequence of bifurcations is shown to take place in the flow as Re varied. Two main regimes of the flow were observed: small and large scale structure regimes corresponding to different ranges of Re. Each of the regimes includes a number of quasiperiodic, chaotic, and relaminarization windows. In addition, each range contains a chaotic window with non-ergodic chaotic attractors. Spatially disordered, but temporally steady states were discovered in large scale structure regime. Features of the diverse cases are displayed in terms of the temporal power spectrum, Poincare sections and, where possible, Lyapunov exponents and Kaplan-Yorke dimension.

Platt, N.↗

NASA NTRS1990

A chaotic attractor in timing noise from the Vela pulsar?

Fourteen years of timing residual data from the Vela pulsar have been analyzed in order to determine if a chaotic dynamical process is the origin of timing noise. Using the correlation sum technique, a dimension of about 1.5 is obtained. This low dimension indicates underlying structure in the phase residuals which may be evidence for a chaotic attractor. It is therefore possible that nonlinear dynamics intrinsic to the spin-down may be the cause of the timing noise in the Vela pulsar. However, it has been found that the stimulated random walks in frequency and frequency derivative often used to model pulsar timing noise also have low fractal dimension, using the same analysis technique. Recent work suggesting that random processes with steep power spectra can mimic strange attractors seems to be confirmed in the case of these random walks. It appears that the correlation sum estimator for dimension is unable to distinguish between chaotic and random processes.

Harding, Alice K.↗

NASA NTRS1991

An investigation of chaotic Kolmogorov flows

Platt, N.↗

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 NTRS1993

Desktop chaotic systems: Intuition and visualization

This paper presents a dynamic study of the Wildwood Pendulum, a commercially available desktop system which exhibits a strange attractor. The purpose of studying this chaotic pendulum is twofold: to gain insight in the paradigmatic approach of modeling, simulating, and determining chaos in nonlinear systems; and to provide a desktop model of chaos as a visual tool. For this study, the nonlinear behavior of this chaotic pendulum is modeled, a computer simulation is performed, and an experimental performance is measured. An assessment of the pendulum in the phase plane shows the strange attractor. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined for both the model and the experimental pendulum systems. Correlation dimension results indicate that the pendulum and the model are chaotic and their fractal dimensions are similar.

Bright, Michelle M.↗

NASA NTRS1993

Modelling chaotic vibrations using NASTRAN

Due to the unavailability and, later, prohibitive cost of the computational power required, many phenomena in nonlinear dynamic systems have in the past been addressed in terms of linear systems. Linear systems respond to periodic inputs with periodic outputs, and may be characterized in the time domain or in the frequency domain as convenient. Reduction to the frequency domain is frequently desireable to reduce the amount of computation required for solution. Nonlinear systems are only soluble in the time domain, and may exhibit a time history which is extremely sensitive to initial conditions. Such systems are termed chaotic. Dynamic buckling, aeroelasticity, fatigue analysis, control systems and electromechanical actuators are among the areas where chaotic vibrations have been observed. Direct transient analysis over a long time period presents a ready means of simulating the behavior of self-excited or externally excited nonlinear systems for a range of experimental parameters, either to characterize chaotic behavior for development of load spectra, or to define its envelope and preclude its occurrence.

Sheerer, T. J.↗

DOE OSTI2025

Spectral anomalies and broken symmetries in maximally chaotic quantum maps

Spectral statistics such as the level spacing statistics and spectral form factor (SFF) are widely expected to accurately identify “ergodicity,” including the presence of underlying macroscopic symmetries, in generic quantum systems ranging from quantized chaotic maps to interacting many-body systems. By studying various quantizations of maximally chaotic maps that break a discrete classical symmetry upon quantization, we demonstrate that this approach can be misleading and fail to detect macroscopic symmetries. Notably, the same classical map can exhibit signatures of different random matrix symmetry classes in short-range spectral statistics depending on the quantization. While the long-range spectral statistics encoded in the early time ramp of the SFF are more robust and correctly identify macroscopic symmetries in several common quantizations, we also demonstrate analytically and numerically that the presence of Berry-like phases in the quantization leads to spectral anomalies, which break this correspondence. Finally, we provide numerical evidence that long-range spectral rigidity remains directly correlated with ergodicity in the quantum dynamical sense of visiting a complete orthonormal basis.

Shou, Laura [Univ. of Maryland, College Park, MD (↗

NASA NTRS1973

Mars: Fretted and chaotic terrains

Fretted Martian terrain is characterized by smooth, flat, lowland areas separated from a cratered upland by abrupt escarpments of complex planimetric configuration and a maximum estimated height approaching 1 to 2 km. It is the product of some unusual erosive or abstractive process that has created steep escarpments. Chaotic terrain differs from fretted terrain in having a rough floor topography featuring a haphazard jumble of large angular blocks, and by arc-shaped slump blocks on its bounding escarpments. Its existence has now been confirmed by Mariner 9 pictures, and the characteristics, location, and areal extent of chaotic terrain have been more accurately and completely defined.

Sharp, R. P.↗

NASA NTRS1973

Chaotic terrain of Mars - A tectonic interpretation from Mariner 6 imagery

Sharp et al. (1971) define chaotic terrain as an irregular jumble of topographic forms covering a certain area within Pyrrhae Regio and adjacent regions centered at about 10 deg S., 35 deg W. This area is covered by Mariner 6 television imagery. An analysis of fracture patterns in the Martian surface from high-resolution Mariner 6 imagery suggests that the lineaments observed in both the chaotic terrain and the cratered plateau areas in Pyrrhae Regio are tectonic fractures resulting from stresses within the Martian crust.

Wilson, R. C.↗

NASA NTRS1985

Relative Time-scale for Channeling Events Within Chaotic Terrains, Margaritifer Sinus, Mars

A relative time scale for ordering channel and chaos forming events was constructed for areas within the Margaritifer Sinus region of Mars. Transection and superposition relationships of channels, chaotic terrain, and the surfaces surrounding them were used to create the relative time scale; crater density studies were not used. Channels and chaos in contact with one another were treated as systems. These systems were in turn treated both separately (in order to understand internal relationships) and as members of the suite of Martian erosional forms (in order to produce a combined, master time scale). Channeling events associated with chaotic terrain development occurred over an extended geomorphic period. The channels can be divided into three convenient groups: those that pre-date intercrater plains development post-plains, pre-chasma systems; and those associated with the development of the Vallis Marineris chasmata. No correlations with cyclic climatic changes, major geologic events in other regions on Mars, or triggering phenomena (for example, specific impact events) were found.

Janke, D.↗

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 NTRS1986

Is Navier-Stokes turbulence chaotic?

Whether turbulent solutions of the Navier-Stokes equations are chaotic is considered. 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 separate exponentially with time, having a positive Liapunov characteristic exponent. Thus the turbulence is characterized as chaotic.

Deissler, R. G.↗

NASA NTRS1987

Stability analysis of two coupled Lorenz lasers and the coupling-induced periodic to chaotic transition

A linear stability analysis of two Lorenz lasers coupled by their electric fields has been performed, and it is shown that the bad cavity condition becomes a function of coupling and that a good cavity instability may occur if the injected fields are inverted before injection. In addition, it is shown that the symmetrically coupled Lorenz system is isomorphic to the original Lorenz system with new parameters. The stability analysis also predicts a lowering of the second laser threshold with coupling for both the chaotic and self-pulsing regimes. Numerical integration of the equations is in agreement with these predictions and has revealed a coupling induced transition from self-pulsing to chaotic behavior. The classification of the behavior of the coupled system in the parameter space of the coupling constants has been investigated and shows that the results of symmetric coupling allow enough of a margin for an experimental test of the theory. This would allow experimentalists to observe the actual Lorenz instability at excitations as low as 4-5 times above threshold.

Lawandy, N. M.↗

NASA NTRS1988

Thermal attractor in chaotic convection with high-Prandtl-number fluids

The onset of chaotic convection in large-Prandtl-number fluids has been found to occur in the form of elongated cells with aperiodic boundary-layer instabilities. The dynamics of this phenomenon has been monitored in the physical domain, and also in the spectral and phase spaces. In order to describe more precisely the underlying mechanisms of chaotic thermal convection, the notion of 'thermal attractor' is introduced in the physical domain and a systematic study is conducted of the evolution of the spectral energy surfaces. A relationship between this thermal attractor and the presence of a strange attractor in the phase space is proposed.

Vincent, Alain P.↗

NASA NTRS1989

Chaotic Motion Of A Two-Link Planar Mechanism

Report discusses global instability in orbital motion of two-link planar mechanism. Principal objective, contributes to understanding of chaotic motions in robot manipulators and other deterministic mechanical systems. Discussion begins with brief review of previous studies of chaotic motion and introduces notion of orbital instability in nonlinear systems. Introduces geometric approach useful in representation of orbital instability.

Lokshin, Anatoly↗

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 NTRS1990

Period and phase of the 88-year solar cycle and the Maunder minimum - Evidence for a chaotic sun

The problem of whether the solar dynamo is quasi-periodic or chaotic is addressed by examining 1500 years of sunspot, geomagnetic, and auroral activity cycles. Sub-harmonics were found of the fundamental solar cycle period during the years preceding the Maunder minimum and loss of phase of the subharmonic on emergence from it. These phenomena are indicative of chaos. They indicate that the solar dynamo is chaotic and is operating in a region close to the transition between period doubling and chaos. Since Maunder-type minima reoccur irregularly for millennia, it appears that the sun remains close to this transition to and from chaos. This is postulated to be a universal characteristic of solar type stars caused by feedback in the dynamo number.

Feynman, J.↗