The Predictability of Tropical Pacific Decadal Variability: Insights from Attractor Reconstruction
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Nandini Ramesh↗
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Is the great attractor really a great wall
Some of the cosmological consequences are discussed of a late time phase transition which produces light domain walls. The observed peculiar velocity field of the Universe and the observed isotropy of the microwave background radiation severely constrain the wall surface density in such a scenario. The most interesting consequence of such a phase transition is the possibility that the local, coherent streaming motion reported by the Seven Samurai could be explained by the repulsive effect of a relic domain wall with the Hubble volume (the Great Wall).
Stebbins, Albert↗
Is the great attractor really a great wall?
Some of the cosmological consequences of a late-time phase transition which produces light domain walls are discussed. The observed peculiar velocity field of the universe and the observed isotropy of the microwave backgroud radiation severely constrain the wall surface density in such as scenario: G(omega) less than about 0.0001 H0 (H0 is the present value of the Hubble parameter). The most interesting consequence of such a phase transition is the possibility that the local, coherent streaming motion of about 600 km/s reported by Dressler et al (1987) could be explained by the repulsive effect of a relic domain wall within the Hubble volume provided that G(omega)/H0 = 0.0001.
Stebbins, Albert↗
Calculating climate attractor dimension from delta O-18 records by the Grassberger-Procaccia algorithm
The Grassberger-Procaccia method of calculating dimension from a time series is applied to 14 late Pleistocene delta O-18 records. A step-by-step sequence leading from data to the Grassberger-Procaccia dimension is outlined, and the problems encountered when dealing with observed (as opposed to theoretical) data are discussed; for the climatic proxy data these problems include situations where the time series is not very long, is noisy and/or smoothed, and is not sampled at a constant time interval. The delta O-18 records to be used are described, and the results are presented and compared with previously published dimension calculations. New dimension interpretations are assessed, and an example using a synthetic time series that illustrates the possible error due to inconsistencies in the time scale is analyzed.
Maasch, Kirk A.↗
A revised catalog of CfA1 galaxy groups in the Virgo/Great Attractor flow field
A new identification of groups and clusters in the CfA1 Catalog of Huchra et al. is presented, using a percolation algorithm to identify density enhancements. It is shown that in the resulting catalog, contamination by interlopers is significantly reduced. The Schechter luminosity function is redetermined, including the Malmquist bias.
Nolthenius, Richard↗
The Origin of Monsoon Onset: Rotational ITCZ Attractors - Part 2
Through various specially designed numerical experiments with an aqua-planet general circulation model and theoretical arguments. Chao showed the existence of multiple quasi-equilibria of the intertropical convergence zone (ITCZ). He also showed that monsoon onset could be interpreted as an abrupt transition between the quasi-equilibria of the ITCZ. He further showed that the origin of these quasi-equilibria is related to two different types of attraction pulling the ITCZ in opposite directions. One type of attraction on the ITCZ is due to earth's rotation, which pulls the ITCZ toward the equator or two equatorial latitudes symmetric with respect to the equator depending on the choice of convection scheme, and the other due to the peak of the sea surface temperature (SST, which is given in the experiments a Gaussian profile in latitude and is uniform in longitude), which pulls the ITCZ toward a latitude just poleward of the SST peak. The strength of the attraction due to the earth's rotation has a highly nonlinear dependence on the latitude and that due to the SST peak has a linear (at least in a relative sense) dependence on the latitude.
Chao, Winston C.↗
Autonomy Teaming & TRAjectories for Complex Trusted Operational Reliability: ATTRACTOR
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Autonomous Systems↗
Coherent excitations and circular attractors in cooled ion bunches
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43 PARTICLE ACCELERATORS↗
The route to chaos for the Kuramoto-Sivashinsky equation
The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609... . Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.
Papageorgiou, Demetrios T.↗
The route to chaos for the Kuramoto-Sivashinsky equation
The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609 .... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.
Papageorgiou, Demetrios T.↗
On the dynamics of approximating schemes for dissipative nonlinear equations
Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.
Jones, Donald A.↗
On the estimation of the correlation dimension and its application to radar reflector discrimination
Recently, system theorists have recognized that low order systems of nonlinear differential equations can give rise to solutions which are neither periodic, constant, nor predictable in steady state, but which are nonetheless bounded and deterministic. This behavior, which was first described in the study of weather systems, has been termed 'chaotic.' Much study of chaotic systems has concentrated on analysis of the systems' phase space attractors. It has been recognized that invariant measures of the attractor possess inherent information about the system. One such measure is the dimension of the attractors. The dimension of a chaotic attractor has been shown to be noninteger, leading to the term 'strange attractor;' the attractor is said to have a fractal structure. The correlation dimension has become one of the most popular measures of dimension. However, many problems have been identified in correlation dimension estimation from time sequences. The most common methods for obtaining the correlation dimension have been least squares curves fitting to find the slope of the correlation integral and the Takens Estimator. However, these estimates show unacceptable sensitivity to the upper limit on the distance chosen. Here, a new method is proposed which is shown to be rather insensitive to the upper limit and to perform in a very stable manner, at least in the absence of noise. The correlation dimension is also shown to be an effective discriminant in distinguishing between radar returns resulting from weather and those from the ground. The weather returns are shown to have a correlation dimension generally between 2.0 and 3.0, while ground returns have a correlation dimension exceeding 3.0.
Barnett, Kevin D.↗
Causal horizons, geodesic completeness and stability in slow contraction cosmology
We show that cosmological models with a semi-infinite phase of slow contraction (ekpyrosis) possess a combination of properties that can address several fundamental problems in cosmology, otherwise faced in contracting de Sitter phases or standard big bang expansion. In particular, flat or open slow contraction admits a stable past attractor that asymptotes to Minkowski space and is past geodesically complete, as well as a stable, flat, homogeneous, and isotropic future attractor with negligible Weyl curvature (and, therefore, negligible gravitational entropy). In bouncing cosmologies, this contracting attractor is terminated by a smooth, non-singular bounce that transforms the attractor properties at the end of contraction into the initial conditions for the subsequent expanding phase. Cosmologies incorporating a slow contraction phase have no particle horizon and therefore avoid the causal horizon problem. The past Minkowski attractor also generates an initial spectrum of vacuum-like quantum fluctuations on all wavelengths. Moreover, because the averaged expansion rate along past-directed geodesics is non-positive, models incorporating a semi-infinite phase of slow contraction also evade the Borde–Guth–Vilenkin theorem. By contrast, contracting de Sitter space possesses a finite particle horizon and becomes unstable in the presence of scalar fields, matter, or radiation.
Cosmology and Nongalactic Astrophysics (astro-ph.C↗
Modal dynamics of high-frequency transverse combustion instabilities
This paper investigates the modal dynamics of self-excited high frequency transverse instabilities in a multi-nozzle, swirl-stabilized can combustor. It utilizes results from multiple pressure sensors to decompose the disturbance into clockwise (CW) and counter-clockwise (CCW) waves, enabling a reconstruction of the pressure amplitude, phase, anti-nodal line location, and spin ratio. While these types of decompositions have been demonstrated on small scale combustors, particularly annular combustors, this is the first study to analyze these results for a high power, multi-nozzle can combustor. Its particular focus is in characterizing the system evolution in phase space and the nature of the system attractors under linearly unstable conditions; results are presented in spin ratio-phase difference space, so that limit cycle oscillations appear as fixed points. Results are shown for conditions where three different attractors dominate (standing, CW, and CCW) but are taken for very long time periods (~50,000 cycles) so that the system's interaction with these attractors can be observed. For the standing mode-dominated case, the phase portrait shows spiral trajectories converging to a single fixed point. For the spinning mode dominated cases, however, multiple fixed points and saddle points were observed, and the trajectories showed a more complicated structure. It was also shown that even though the CW mode is dominant, this mode intermittently switches to standing or CCW modes for short period. These results suggest that the strength of each attractor depends on test conditions and that random noise stochastically perturbs the system so that large regions of the phase space are explored. Lastly, one interesting observation is that the angular velocity of the anti-nodal line linearly varies with the spin ratio under standing-mode dominant conditions.
42 ENGINEERING↗
Chaotic saddles and interior crises in a dissipative nontwist system
Here, we consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems; they are responsible for chaotic transients, fractal basin boundaries, and chaotic scattering, and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increases the transient times and we analyze the phenomenon of crisis induced intermittency.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Predictability of weather and climate in a coupled ocean-atmosphere model: A dynamical systems approach
A dynamical systems approach is used to quantify the instantaneous and time-averaged predictability of a low-order moist general circulation model. Specifically, the effects on predictability of incorporating an active ocean circulation, implementing annual solar forcing, and asynchronously coupling the ocean and atmosphere are evaluated. The predictability and structure of the model attractors is compared using the Lyapunov exponents, the local divergence rates, and the correlation, fractal, and Lyapunov dimensions. The Lyapunov exponents measure the average rate of growth of small perturbations on an attractor, while the local divergence rates quantify phase-spatial variations of predictability. These local rates are exploited to efficiently identify and distinguish subtle differences in predictability among attractors. In addition, the predictability of monthly averaged and yearly averaged states is investigated by using attractor reconstruction techniques.
Nese, Jon M.↗
A Distributed Simulation-to-Flight Framework to Support Investigating Trust/Trustworthiness in Multi-Agent Systems
As autonomous systems continue to grow both in use and complexity, the necessity for robust and extensible simulation-to-flight frameworks is paramount for establishing an effective architecture for autonomous systems. Hardware test flights are time-consuming and cost prohibitive during early system design and development. Simulation environments can be useful tools to accelerate algorithm development and testing. However, transitions from simulation to flight (sim-to-flight) can be challenging, unless systems are designed with this transition in mind and with the necessary capabilities built into the architecture and framework. One of the objectives of Autonomy Teaming and TRAjectories for Complex Trusted Operational Reliability (ATTRACTOR) was to design and develop a distributed mixed-reality simulation environment to begin establishing a basis for certification of autonomous systems via research into trust and trustworthiness. ATTRACTOR’s objective was to construct computational concepts of trustworthiness and justifiable trust in multi-agent autonomous teams, to inform future certification of safety-critical and time-critical autonomous systems in aviation. In this paper, we present an autonomous systems architecture and development framework paired with a persistent distributed modeling and simulation (ModSim) environment for test and evaluation of autonomous systems. They were designed under ATTRACTOR in order to measure and establish trustworthiness and trust in single-and multi-agent human-machine systems whether these machines are fixed-wing general aviation, rotary-wing Unmanned Aerial Vehicles (UAVs), ground rovers, or even spacecraft. The Autonomous Entity Operational Network (AEON) framework enables autonomous system development with an easily extensible collection of libraries and plug-n-play nodes facilitated by the Data Distribution Service (DDS) communication protocol standard. The Baseline Environment for Autonomous Modeling (BEAM) simulation environment is a distributed mixed-reality Unity™-based environment built around the same DDS communication paradigm allowing for easy integration with AEON-based autonomous applications, enabling sim-to-flight with minimal configuration changes. Using AEON and BEAM, source code that runs in simulation ports directly to hardware and has successfully flown in the National Airspace System (NAS) at NASA LaRC many times over the lifetime of ATTRACTOR.
Benjamin N Kelley↗
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.↗