Broadband Power Spectrum Research Articles

MEASUREMENT OF DENSITY FLUCTUATIONS OF CT-6B TOKAMAK PLASMA

The electron density fluctuations of CT-6B Tokamak plasma have been studied in detail through the observation of Hα emission intensity by multichannel optical diagnostics. Their various behavior, such as power spectrum, fluctuation level and correlation,have been measured. The experimental results show that some characteristics of density fluctuations are founded in CT-6B Tokamak plasma: the density fluctuation level is about 10%-90% in the radial range r=4-10cm; the point in the range draws, nearer to the edge of plasma the level lecomes higher. A broadband power spectrum is observed at observed points along the plasma radius. There exists a local correlation structure with short correlation time, about 0.2-0.5ms, and short correlation distance , about 0.5-1cm,in the edge of plasma.

Chaos in Biomolecular Dynamics

An analysis of molecular dynamics simulation of a model α-helix indicates that the motion of the helix system is chaotic. This system's behavior is due to an intrinsic sensitivity to initial conditions, which makes initially similar conformations of the helix evolve into completely different conformations. The chaotic properties of the system can be clearly identified in terms of nonzero Lyapunov exponents, broad-band power spectra, and strange attractors. The dominant factors resulting in the chaos are found to be (1) the nonlinear interactions inherent in the system's force field, (2) the restraints (such as the restraints on temperature, dihedral angles, and bond lengths), and (3) the stochastic forces generated by the solvent. These results suggest a limitation for the predictive capabilities of the molecular dynamics simulation method in studies of biomolecules and, especially, have important implications to the studies of the protein-folding problem.

Absolute and convective instabilities and noise-sustained structures in the Couette-Taylor system with an axial flow.

A detailed study of the Couette-Taylor system with axial flow in the range of Reynolds number Re up to 4.5, which is characterized by the propagating Taylor-vortices (PTV's) state, is presented. Two methods to measure the convective instability line are described. Comparative studies of the PTV's in the absolutely and convectively unstable regions are given. It was found that at Re1 the PTV's appear first at the outlet at the absolute instability transition. At Re>1 the PTV's are also sustained in the convectively unstable region, but the properties of the PTV's in the absolutely and convectively unstable regions differ distinctively. In both regions the PTV's are characterized by the existence of an interface separating the pattern state from the Couette-Poiseuille flow. The interface is stationary in the absolutely unstable region and fluctuates in the convectively unstable region. The distance from the inlet to the interface changes as both control parameters \ensuremath{\epsilon}\ifmmode\bar\else\textasciimacron\fi{} and Re are varied, where \ensuremath{\epsilon}\ifmmode\bar\else\textasciimacron\fi{} is the distance from the convective line. This dependence is, however, different in both regions. In the absolutely unstable region the healing length is scaled with the PTV's group velocity at all values of \ensuremath{\epsilon}\ifmmode\bar\else\textasciimacron\fi{} and Re, and diverges at the absolute instability transition line. In the convectively unstable region the healing length does not obey the general scaling but is about inversely proportional to \ensuremath{\epsilon}\ifmmode\bar\else\textasciimacron\fi{}. The most distinctive difference in the PTV's behavior in the two regions is a different sensitivity to noise.A time-dependent spatial profile of the PTV's leads to a broadband power spectrum of the velocity in the convectively unstable region near the outlet. The PTV's velocity power spectrum in the absolutely unstable region is, on the other hand, noise-free. The different sensitivity to noise was used as an experimental criterion to locate the absolute instability line for Re>1. The wave-number selection is also found to be different in both regions. As a result, we concluded that the PTV's in the convectively unstable region are noise-sustained structures (NSS's) which were recently considered theoretically and observed in numerical simulations. The selective spatial amplification of an external noise and the characteristic dependence of the healing length on the control parameters and on the aspect ratio confirm our suggestion that the mechanism of NSS generation is a selective spatial amplification of a permanent noise at the inlet. An interaction of the noise with the NSS's leads to a noise modulation of the PTV's velocity.

Capillary wave turbulence

The transition from a discrete spectrum of higher harmonics to a broadband power spectrum has been observed with a wire probe (as regards the frequency spectrum) and imaging of the surface (as regards the wave‐number spectrum). By use of laser scattering techniques we hope to obtain an absolute calibration of the surface motion in this very far‐off equilibrium state. [Work supported by the U.S. DOE Division of Engineering and Geophysics and by NASA.]

Bifurcations of the respiratory pattern associated with reduced lung volume in the rat

Rats with intact vagal reflexes exhibit patterns of breathing that contain greater degrees of freedom than those seen after vagotomy. To determine how alterations in end-expiratory volume modify the respiratory pattern, continuous positive (CPAP) and negative (CNAP) airway pressure was applied to tracheal openings of nine urethan-anesthetized vagi-intact rats (+3 to -9 cmH2O). Phase portraits (e.g., volume vs. flow curves), power spectra, correlation integral curves, and inspiratory-to-expiratory duration (TI/TE) ratios are used to interpret the vagal-dependent responses to changes in mean tracheal pressure (Ptr). With CPAP, respiratory oscillation was highly periodic and one dimensional, with TI/TE near 1.0. As Ptr was reduced in a stepwise manner, transient bursts of inspiratory airflow developed local to the expiratory-inspiratory transition, with amplitude increasing proportionally with the level of CNAP. These oscillatory "expiratory interrupts" (doubling TI/TE in five of nine cases) produced highly variable and asymmetric respiratory patterns. Progressive increases in correlation dimension (maximum = 1.8-3.0) and tendencies toward broadband power spectra were seen as Ptr was lowered. The irregular phase-switching dynamics seen with CNAP (which disappeared after vagotomy) are consistent with onset of low-dimensional chaos, probably correlated with activation of feedback mechanisms responsive to lung deflation.

BISPECTRAL AND TRISPECTRAL CHARACTERIZATION OF TRANSITION TO CHAOS IN THE DUFFING OSCILLATOR

Higher-order spectral (bispectral and trispectral) analyses of numerical solutions of the Duffing equation with a cubic stiffness are used to isolate the coupling between the triads and quartets, respectively, of nonlinearly interacting Fourier components of the system. The Duffing oscillator follows a period-doubling intermittency catastrophic route to chaos. For period-doubled limit cycles, higher-order spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. However, when the Duffing oscillator becomes chaotic, global behavior of the cubic nonlinearity becomes dominant and quadratic nonlinear interactions are weak, while cubic interactions remain strong. As the nonlinearity of the system is increased, the number of excited Fourier components increases, eventually leading to broad-band power spectra for chaos. The corresponding higher-order spectra indicate that although some individual nonlinear interactions weaken as nonlinearity increases, the number of nonlinearly interacting Fourier modes increases. Trispectra indicate that the cubic interactions gradually evolve from encompassing a few quartets of Fourier components for period-1 motion to encompassing many quartets for chaos. For chaos, all the components within the energetic part of the power spectrum are cubically (but not quadratically) coupled to each other.

Method to distinguish possible chaos from colored noise and to determine embedding parameters.

We present a computational method to determine if an observed time series possesses structure statistically distinguishable from high-dimensional linearly correlated noise, possibly with a nonwhite spectrum. This method should be useful in identifying deterministic chaos in natural signals with broadband power spectra, and is capable of distinguishing between chaos and a random process that has the same power spectrum. The method compares nonlinear predictability of the given data to an ensemble of random control data sets. A nonparametric statistic is explored that permits a hypothesis testing approach. The algorithm can detect underlying deterministic chaos in a time series contaminated by additive random noise with identical power spectrum at signal to noise ratios as low as 3 dB. With less noise, this method can also be used to get good estimates of the parameters (the embedding dimension and the time delay) needed to perform the standard phase-space reconstruction of a chaotic time series.

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.

Frequency entrainment of a phase coherent attractor

In this paper the frequency entrainment of the Rössler system to a periodic pulse is discussed. It was found that a periodic orbit will entrain to weak periodic pulses with frequency near the natural frequency of oscillation, but rather more surprisingly, the phase coherent strange attractor can also be entrained, on average, to small pulses near the peak frequency in the broad band power spectra of the attractor.

Direct conversion of ordinary mode into upper hybrid wave by density irregularities in the ionosphere

We investigate direct conversion of ordinary mode into upper hybrid (UH) waves by pre‐existing electron density irregularities during ionospheric heating. The electric field amplitude and the average power absorbed by the excited UH waves are calculated when the irregularity distribution is (i) coherent, and (ii) incoherent with a broad band power spectrum modelled by a drift wave type scaling. For the typical parameters of the experiments, we find that the direct conversion process can be quite efficient.

Chaos in a chain of six particles with fixed ends (a 4+2 chain).

The equations of motion for a chain of six particles with fixed ends (a 4+2 chain), interacting via nearest-neighbor linear and cubic forces, have been numerically integrated for the case of odd-mode excitations. Positive Lyapunov exponents are seen for sufficiently large anharmonic forces. However, no corresponding stochastic Poincar\'e surface of section intersections or broadband power spectra are observed. These latter chaotic phenomena may in this case be at a level below the resolution of the numerical data.

Structural stability in two-dimensional model flows: Lagrangian and Eulerian turbulence

A two-dimensional approximation of the Navier–Stokes equations illustrates a specific instance of transition from Lagrangian to Eulerian turbulence. As the Reynolds number (Re) increases, the system describing the dynamics of the velocity field undergoes a transition from steady state to a limit cycle. At this point the flow displays chaotic advection—i.e., manifolds intersect transversely and Poincaré maps show the typical chaotic structure—but the velocity field itself is time-periodic and the power spectrum presents a fundamental frequency and its harmonics. As Re increases still further, the limit cycle bifurcates into a strange attractor producing a broadband power spectrum. The model suggests that Lagrangian turbulence (complex particle trajectories) might serve as a springboard for Eulerian turbulence (complex signal at a fixed point) and indicates a possible link between a kinematical view of flows and mixing and other viewpoints of turbulence based on strange attractors. The ideas can be generalized to three-dimensional flows; however, due to their simplicity, the flows generated are unable to mimic some key features of turbulence such as spatial uncorrelation of Eulerian signals.

Quantum ergodicity of a coherence-preserving SU(2) Hamiltonian system with quasiperiodic kicking.

In this paper we study the time evolution of a quasiperiodically kicked spin system whose Hamiltonian is linear in the SU(2) generators. Since this Hamiltonian preserves SU(2) coherent states under time evolution, there is a close correspondence between the quantum evolution and the classical evolution of the system on the Bloch sphere. Such a system has previously been studied by Milonni, Ackerhalt, and Goggin [Phys. Rev. A 35, 1714 (1987)] and, with two incommensurate driving frequencies, the quantum evolution was characterized by (a) decay of the autocorrelation function of the state vector, (b) broadband power spectrum of observables, (c) ergodicity on the Bloch sphere. We expand upon their work to study the corresponding behavior of the motion on a symplectic phase space and the decay of the autocorrelation function of the state vector. When the forcing strength is such that the motion is localized in phase space, the autocorrelation function shows revivals. When the motion is ergodic the autocorrelation function rapidly decays without revivals. The classical motion however is not chaotic, there is no sensitive dependence on initial conditions. For three incommensurate frequencies, the motion is more delocalized than for two frequencies for a fixed forcing strength.

Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra.

We consider the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space that captures the attractor (usually strange) on which the system evolves and give a procedure for constructing parametrized maps that evolve points in the phase space into the future. The predictor of future points in the phase space is a combination of operation on past points by the map and its iterates. Thus the map is regarded as a dynamical system and not just a fit to the data. The invariants of the dynamical system, the Lyapunov exponents and optimum moments of the invariant density on the attractor, are used as constraints on the choice of mapping parameters. The parameter values are chosen through a constrained least-squares optimization procedure, constrained by the values of these invariants. We give a detailed discussion of methods to extract the Lyapunov exponents and optimum moments from data and show how to equate them to the values for the parametric map in the constrained optimization. We also discuss the motivation and methods we utilize for choosing the form of our parametric maps. Their form has a strong similarity to the work in statistics on kernel density estimation, but the goals and techniques differ in detail. Our methodology is applied to "data" from the H\'enon map and the Lorenz system of differential equations and shown to be feasible. We find that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps that do reproduce the values of the invariants are not optimum in the least-squares sense, yet still are excellent predictors. We discuss several technical and general problems associated with prediction and system identification on strange attractors. In particular, we consider the matter of the evolution of points that are off the attractor (where few or no data are available), onto the attractor where long-term motion takes place. We find that we are able to realize maps that give a least-squares approximation to the data with rms variation over the attractor of 0.5% or less and still reproduce the dynamical invariants to 5% or better. The dynamical invariants are the classifiers of the dynamical system producing the broadband time series in the first place, so this quality of the maps is essential in representing the correct dynamics.

Prediction and system identification in chaotic nonlinear systems: Time series with broadband spectra

We consider the problem of prediction and system identification for time series having broadband power spectra which arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space which captures the attractor (usually strange) on which the system evolves, and give a procedure for constructing parameterized maps which evolve points in the phase space into the future. The predictor of future points in the phase space is a combination of operation on past points by the map and its iterates. Thus the map is regarded as a dynamical system, not just a fit to the data. The invariants of the dynamical system — the Lyapunov exponents and aspects of the invariant density on the attractor — are used as constraints on the choice of mapping parameters. The parameter values are chosen through a least-squares optimization procedure. The method is applied to “data” from the Hénon map and shown to be feasible. It is found that the parameter values which minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps which do reproduce the values of the invariants are not optimum in the least-squares sense, yet still are excellent predictors. We discuss several technical and general problems associated with prediction and system identification on strange attractors. In particular, we consider the matter of the evolution of points that are off the attractor (where little or no data is available), onto the attractor, where long-term motion takes place.

Atropine unmasks bed-rest effect: a spectral analysis of cardiac interbeat intervals.

Bed-rest deconditioning is suspected to reduce cardiac reserve, possibly by impairing autonomic function. Heart rate response in normal subjects reveals considerable variability, reflected by a relatively broadband interbeat interval power spectrum. A reduction in this autonomically modulated variability would be predicted to cause a narrowing of the spectrum. We retrospectively analyzed data from 10 aerobically conditioned men (age range 35-49 yr) who had undergone orthostatic tolerance testing with lower body negative pressure pre-bed rest and after 7-10 days of bed rest, while on placebo and after intravenous atropine. Spectra were derived by Fourier analysis of 128 interbeat interval data sets. Spectral power was estimated by computing the root-mean-square (rms) values (mean +/- SD) for the band encompassing the 2nd to 64th harmonics from subjects with a sufficient number of beats: placebo rms is 93 +/- 33 ms for pre-bed rest and 84 +/- 38 ms for bed rest (NS, n = 6); atropine rms is 63 +/- 24 ms for pre-bed rest and 40 +/- 23 ms for bed rest (P less than 0.01; n = 7). These data suggest that atropine "unmasks" a deconditioning effect of bed rest in athletic men, evidenced by a reduction in interbeat interval spectral power not apparent with placebo. Spectral analysis offers a useful means of quantitating the effects of bed-rest deconditioning and autonomic perturbations on cardiac dynamics.

Time-dependent convective instabilities in a closed vertical cylinder heated from below

The convective behaviour of xenon gas in a vertical thermally conducting cylinder (height/radius = 6) heated from below was investigated. Convectively induced temperature fluctuations in the gas were analysed with digital signal-processing techniques over a range of Rayleigh number 0 ≤ Ra [lsim ] 2300. Quiescent, steady-state, periodic and weakly turbulent convective regimes were characterized. Bistability of steady states (mode switching) was observed in the range 400 [lsim ] Ra [lsim ] 700. At Ra = 1550 a strictly periodic flow developed. With increasing Ra two additional incommensurate frequencies appeared, leading to ‘turbulence’ at Ra ≈ 2000. This turbulence, characterized by a broadband power spectrum, intermittently showed periodic flow. A periodic window with a period-doubling sequence appeared between 2100 [lsim ] Ra [lsim ] 2200. The spectral features of this sequence can be followed into the broad band noise at higher Ra. Although these experiments were conducted quasistatically, a strong hysteresis was observed with decreasing Ra. Furthermore, it was demonstrated that the sequence of convective regimes can be fundamentally altered by minor perturbations (self-heating) from the flow sensors.

Limits on photospheric Doppler signatures for solar giant cells

Mount Wilson solar-velocity data taken since improvement of the spectrograph in May 1982 are analyzed to search for photospheric traces of persistent velocity patterns that are anticipated in recent model predictions. The method involves time-averaged autocorrelations and cross correlations of the residuals that remain after least-squares fits for differential rotation, limb shift, and meridional circulation are extracted from the daily-magnetogram velocity arrays. It is argued that, owing to the supergranular motions in the photosphere, the sensitivity in applying the present method to the new Mount Wilson data is close to the ultimate sensitivity possible for detection of this phenomenon. The following limits are currently established through this analysis: (1) there is no sharply peaked power spectrum with amplitude above about 1 m/s per wavenumber, and (2) there is no broad-band power spectrum for which the total integrated power is greater than about 10 sq m/sq sec.

Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibres

The response of the membrane of mammalian ventricular myocardial fibres, described by the Beeler-Reuter differential equation system, is investigated in the case where the system is subjected to sinusoidal stimulation. For some stimulation frequencies the system exhibits periodic behaviour, frequency locked to the driving cycle. For other values of the stimulation frequency we find chaotic responses clearly distinguished by broad band power spectra. The Lyapunov exponents are computed for this case from which the fractal dimension of the corresponding strange attractor can be estimated.