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A practically important regularization of the Navier-Stokes equations was analyzed. As a continuation of the previous work, the structure of the attractors characterizing the solutins was studied. Local as well as global invariant manifolds were found. Regularity properties of these manifolds are analyzed.
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. The solutions, 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 is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixed-point. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, is both time-dependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to small-scale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence self-interaction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.
The paper presents an artificial neural network concept (the Synchronizable Oscillator Networks) where the instants of individual firings in the form of point processes constitute the only form of information transmitted between joining neurons. In the model, neurons fire spontaneously and regularly in the absence of perturbation. When interaction is present, the scheduled firings are advanced or delayed by the firing of neighboring neurons. Networks of such neurons become global oscillators which exhibit multiple synchronizing attractors. From arbitrary initial states, energy minimization learning procedures can make the network converge to oscillatory modes that satisfy multi-dimensional constraints. Such networks can directly represent routing and scheduling problems that consist of ordering sequences of events.
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.
An efficient, adaptive neural learning paradigm for addressing the inverse kinematics of redundant manipulators is presented. The proposed methodology exploits the infinite local stability of terminal attractors - a new class of mathematical constructs which provide unique information processing capabilities to artificial neural systems. For robotic applications, synaptic elements of such networks can rapidly acquire the kinematic invariances embedded within the presented samples. Subsequently, joint-space configurations, required to follow arbitrary end-effector trajectories, can readily be computed. In a significant departure from prior neuromorphic learning algorithms, this methodology provides mechanisms for incorporating an in-training skew to handle kinematics and environmental constraints.
The history of the study of the Galaxy's motion with respect to the Local Supercluster is reviewed. Direct measurements of the infall pattern suggest an infall component at the Local Group of about 250 km/s. This value is consistent with that expected if the average mass-to-light ratio of galaxies is similar to that dynamically measured in galaxy clusters, a value that gives an Omega of only about 0.3. The infall pattern is also consistent with galaxy density maps produced by IRAS surveys, although it may be only marginally consistent with the 'Great Attractor'. On smaller scales, there is strong evidence that most of the spiral galaxies near the cluster core are still falling in. Using the best distance estimates to the Virgo cluster, the local value of the Hubble constant is about 85 km/s/Mpc.
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.
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.
Discrete-event processes are modeled by state-machines in the Ramadge-Wonham framework with control by a feedback event disablement mechanism. In this paper, concepts of stabilization of discrete-event processes are defined and investigated. The possibility of driving a process (under control) from arbitrary initial states to a prescribed subset of the state set and then keeping it there indefinitely is examined. This stabilization property is studied also with respect to 'open-loop' processes and their asymptotic behavior is characterized. Polynomial time algorithms are presented for verifying various types of attraction and for the synthesis of attractors.
A new technique to simulate Taylor vortices in a spherical gap between a rotating inner sphere and a stationary outer one has been developed and tested. Paths leading to zero-, one-, and two-vortex flows are designed heuristically. Fictitious symmetric boundaries near the equator are imposed, and the choice of the location of the fictitious boundaries is determined by either one- or two-vortex flow being stimulated. The imposition of one or two fictitious boundaries during the initial calculation generates the state suitable for one-or two-vortex flow to exist. After removing the fictitious boundaries, the flow settles down into its own attractor. Using this method, the three steady flow modes can be simulated by using a half domain. The technique can converge to desired flows very fast, and its results show excellent agreement with experimental ones.
Synaptic connections adjusted one at a time in small increments. Simplified gradient-descent learning scheme for electronic neural-network processor less efficient than better-known back-propagation scheme, but offers two advantages: easily implemented in circuitry because data-access circuitry separated from learning circuitry; and independence of data-access circuitry makes possible to implement feedforward as well as feedback networks, including those of multiple-attractor type. Important in such applications as recognition of patterns.
Researchers are continuing their studies of the nonlinear dynamics of global weather systems. Sensitivity analyses of large-scale dynamical models of the atmosphere (i.e., general circulation models i.e., GCM's) were performed to establish the role of satellite-signatures of soil moisture, sea surface temperature, snow cover, and sea ice as crucial boundary conditions determining global weather variability. To complete their study of the bimodality of the planetary wave states, they are using the dynamical systems approach to construct a low-order theoretical explanation of this phenomenon. This work should have important implications for extended range forecasting of low-frequency oscillations, elucidating the mechanisms for the transitions between the two wave modes. Researchers are using the methods of jump analysis and attractor dimension analysis to examine the long-term satellite records of significant variables (e.g., long wave radiation, and cloud amount), to explore the nature of mode transitions in the atmosphere, and to determine the minimum number of equations needed to describe the main weather variations with a low-order dynamical system. Where feasible they will continue to explore the applicability of the methods of complex dynamical systems analysis to the study of the global earth-system from an integrative viewpoint involving the roles of geochemical cycling and the interactive behavior of the atmosphere, hydrosphere, and biosphere.
Linear instability calculations at MSFC have suggested that the Geophysical Fluid Flow Cell (GFFC) should exhibit classic baroclinic instability at accessible parameter settings. Interest was in the mechanisms of transition to temporal chaos and the evolution of spatio-temporal chaos. In order to understand more about such transitions, high resolution numerical experiments for the physically simplest model of two layer baroclinic instability were conducted. This model has the advantage that the numerical code is exponentially convergent and can be efficiently run for very long times, enabling the study of chaotic attractors without the often devastating effects of low-order trunction found in many previous studies. Numerical algorithms for implementing an empirical orthogonal function (EOF) analysis of the high resolution numerical results were completed. Under conditions of rapid rotation and relatively low differential heating, convection in a spherical shell takes place as columnar banana cells wrapped around the annular gap, but with axes oriented along the axis of rotation; these were clearly evident in the GFFC experiments. The results of recent numerical simulations of columnar convection and future research plans are presented.
The chaos in a one-dimensional system, which would be nonlinear stationary Alfven waves in the absence of an external driver, is characterized. The evolution equations are numerically integrated for the transverse wave magnetic field amplitude and phase using the derivative nonlinear Schroedinger equation (DNLS), including resistive wave damping and a long-wavelength monochromatic, circularly polarized driver. A Poincare map analysis shows that, for the nondissipative (Hamiltonian) case, the solutions near the phase space (soliton) separatrices of this system become chaotic as the driver amplitude increases, and 'strong' chaos appears when the driver amplitude is large. The dissipative system exhibits a wealth of dynamical behavior, including quasiperiodic orbits, period-doubling bifurcations leading to chaos, sudden transitions to chaos, and several types of strange attractors.
A potential flow reconstruction algorithm has been applied to the real universe to reconstruct the three-dimensional potential, velocity, and mass density fields smoothed on large scales. The results are shown as maps of these fields, revealing the three-dimensional structure within 6000 km/s distance from the Local Group. The dominant structure is an extended deep potential well in the Hydra-Centaurus region, stretching across the Galactic plane toward Pavo, broadly confirming the Great Attractor (GA) model of Lynden-Bell et al. (1988). The Local Supercluster appears to be an extended ridge on the near flank of the GA, proceeding through the Virgo Southern Extension to the Virgo and Ursa Major clusters. The Virgo cluster and the Local Group are both falling toward the bottom of the GA potential well with peculiar velocities of 658 + or - 121 km/s and 565 + or - 125 km/s, respectively.
Using three separate but related approaches, the question of whether the dynamic response of the magnetosphere to the solar wind input may be described by a low-order system of equations is examined. First, it is determined that the dimension of the subset (the attractor) in the high-dimensional magnetospheric phase space associated with the westward auroral electrojet (AL) index for some of the data sets compiled by Bargatze et al. (1985) is 4.0 + or - 0.2, seemingly independent of activity level. Second, direct modeling of the magnetosphere, considering the bulk properties of the tail plasma, leads to a system of equations that is similar to those previously reported as a dripping faucet model; here, the focus is specifically on the prediction of a natural frequency in this model. Finally, a peak is identified with the predicted frequency in power spectra of AL computed for intervals with both low and high activity. Peaks at other frequencies also appear in the spectra, and such resonances would be expected for a chaotic nonlinear oscillator. Combining these approaches it is concluded that at least some aspects of magnetospheric dynamics may be meaningfully modeled by low-dimensional sets of equations.
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.
The decoding and error-correction of data transmitted over a noisy channel is in principle equivalent to the operation of a neural network performing as a content-addressable memory. For a successful application, however, the neural network has to be capable of storing arbitrary words, and it has to be guaranteed that the stored words represent the only stable attractors of the memory. This paper presents a novel high-order neural network architecture that has these characteristics. The analog nature of the network can be used to perform soft-decision decoding with any block code. The performance in terms of postdecoding bit error rate versus SNR is demonstrated for two exemplary block codes. The comparison with a conventional decoding algorithm for a (15,5) cyclic redundancy code shows, for example, that the bit error rate at 7dB SNR can be decreased by two orders of magnitude.