arXiv subjects

Henry S. Greenside

Publications and source records attributed to Henry S. Greenside.

14 recordsLinked to original sources

arXiv1999

An Explicit Space-time Adaptive Method for Simulating Complex Cardiac Dynamics

For plane-wave and many-spiral states of the experimentally based Luo-Rudy 1 model of heart tissue in large (8 cm square) domains, we show that an explicit space-time-adaptive time-integration algorithm can achieve an order of magnitude reduction in computational effort and memory - but without a reduction in accuracy - when compared to an algorithm using a uniform space-time mesh at the finest resolution. Our results indicate that such an explicit algorithm can be extended straightforwardly to simulate quantitatively large-scale three-dimensional electrical dynamics over the whole human heart.

physics.comp-ph↗

arXiv1999

Stationarity and Redundancy of Multichannel EEG Data Recorded During Generalized Tonic-Clonic Seizures

A prerequisite for applying some signal analysis methods to electroencephalographic (EEG) data is that the data be statistically stationary. We have investigated the stationarity of 21-electrode multivariate EEG data recorded from ten patients during generalized tonic-clonic (GTC) seizures elicited by electroconvulsive therapy (ECT). Stationarity was examined by calculating probability density functions (pdfs) and power spectra over small equal-length non-overlapping time windows and then by studying visually and quantitatively the evolution of these quantities over the duration of the seizures. Our analysis shows that most of the seizures had time intervals of at least a few seconds that were statistically stationary by several criteria and simultaneously for different electrodes, and that some leads were delayed in manifesting the statistical changes associated with seizure onset evident in other leads. The stationarity across electrodes was further examined by studying redundancy of the EEG leads and how that redundancy evolved over the course of the GTC seizures. Using several different measures, we found a substantial redundancy which suggests that fewer than 21 electrodes will likely suffice for extracting dynamical and clinical insights. The redundancy analysis also demonstrates for the first time posterior-to-anterior time delays in the mid-ictal region of GTC seizures, which suggests the existence of propagating waves. The implications of these results are discussed for understanding GTC seizures and ECT treatment.

chao-dyn↗

arXiv1998

Extensive Scaling and Nonuniformity of the Karhunen-Loève Decomposition for the Spiral-Defect Chaos State

By analyzing large-aspect-ratio spiral-defect-chaos (SDC) convection images, we show that the Karhunen-Loève decomposition (KLD) scales extensively for subsystem-sizes larger than 4d (d is the fluid depth), which strongly suggests that SDC is extensively chaotic. From this extensive scaling, the intensive length ξ_KLD is computed and found to have a different dependence on the Rayleigh number than the two-point correlation length ξ_2. Local computations of ξ_KLD reveal a substantial spatial nonuniformity of SDC that extends over radii 18d< r < 45d in a Γ=109 aspect-ratio cell.

chao-dyn↗

arXiv1998

Size-Dependent Transition to High-Dimensional Chaotic Dynamics in a Two-Dimensional Excitable Medium

The spatiotemporal dynamics of an excitable medium with multiple spiral defects is shown to vary smoothly with system size from short-lived transients for small systems to extensive chaos for large systems. A comparison of the Lyapunov dimension density with the average spiral defect density suggests an average dimension per spiral defect varying between three and seven. We discuss some implications of these results for experimental studies of excitable media.

chao-dyn↗

arXiv1997

Comment on "Optimal Periodic Orbits of Chaotic Systems"

In a recent Letter, Hunt and Ott argued that SHORT-period unstable periodic orbits (UPOs) would be the invariant sets associated with a chaotic attractor that are most likely to optimize the time average of some smooth scalar performance function. In this Comment, we show that their conclusion does not hold generally and that optimal time averages may specifically require long-period UPOs. This situation can arise when long-period UPOs are able to spend substantial amounts of time in a region of phase space that is close to large values of the performance function.

chao-dyn↗

arXiv1997

Spatially Localized Unstable Periodic Orbits

Using an innovative damped-Newton method, we report the first calculation of many distinct unstable periodic orbits (UPOs) of a large high-dimensional extensively chaotic partial differential equation. A majority of the UPOs turn out to be spatially localized in that time dependence occurs only on portions of the spatial domain. With a particular weighting of 127 UPOs, the Lyapunov fractal dimension D=8.8 can be estimated with a relative error of 2%. We discuss the implications of these spatially localized UPOs for understanding and controlling spatiotemporal chaos.

chao-dyn↗

arXiv1996

Fractal Basins of Attraction Associated with a Damped Newton's Method

An intriguing and unexpected result for students learning numerical analysis is that Newton's method, applied to the simple polynomial z^3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newton's method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting.

chao-dyn↗

arXiv1996

Spatiotemporal Chaos in Large Systems: The Scaling of Complexity with Size

The dynamics of a nonequilibrium system can become complex because the system has many components (e.g., a human brain), because the system is strongly driven from equilibrium (e.g., large Reynolds-number flows), or because the system becomes large compared to certain intrinsic length scales. Recent experimental and theoretical work is reviewed that addresses this last route to complexity. In the idealized case of a sufficiently large, nontransient, homogeneous, and chaotic system, the fractal dimension D becomes proportional to the system's volume V which defines the regime of extensive chaos. The extensivity of the fractal dimension suggests a new way to characterize correlations in high-dimensional systems in terms of an intensive dimension correlation length $ξ_δ$. Recent calculations at Duke University show that $ξ_δ$ is a length scale smaller than and independent of some commonly used measures of disorder such as the two-point and mutual-information correlation lengths. Identifying the basic length and time scales of extensive chaos remains a central problem whose solution will aid the theoretical and experimental understanding of large nonequilibrium systems.

chao-dyn↗

arXiv1996

Karhunen-Lo`eve Decomposition of Extensive Chaos

We show that the number of KLD (Karhunen-Lo`eve decomposition) modes D_KLD(f) needed to capture a fraction f of the total variance of an extensively chaotic state scales extensively with subsystem volume V. This allows a correlation length xi_KLD(f) to be defined that is easily calculated from spatially localized data. We show that xi_KLD(f) has a parametric dependence similar to that of the dimension correlation length and demonstrate that this length can be used to characterize high-dimensional inhomogeneous spatiotemporal chaos.

chao-dyn↗

arXiv1995

Lyapunov spectral analysis of a nonequilibrium Ising-like transition

By simulating a nonequilibrium coupled map lattice that undergoes an Ising-like phase transition, we show that the Lyapunov spectrum and related dynamical quantities such as the dimension correlation length~$ξ_δ$ are insensitive to the onset of long-range ferromagnetic order. As a function of lattice coupling constant~$g$ and for certain lattice maps, the Lyapunov dimension density and other dynamical order parameters go through a minimum. The occurrence of this minimum as a function of~$g$ depends on the number of nearest neighbors of a lattice point but not on the lattice symmetry, on the lattice dimensionality or on the position of the Ising-like transition. In one-space dimension, the spatial correlation length associated with magnitude fluctuations and the length~$ξ_δ$ are approximately equal, with both varying linearly with the radius of the lattice coupling.

chao-dyn↗

arXiv1994

Characterization of the transition from defect- to phase-turbulence

For the complex Ginzburg-Landau equation on a large periodic interval, we show that the transition from defect- to phase-turbulence is more accurately described as a smooth crossover rather than as a sharp continuous transition. We obtain this conclusion by using a powerful parallel computer to calculate various order parameters, especially the density of space-time defects, the Lyapunov dimension density, and the correlation lengths of the field phase and amplitude. Remarkably, the correlation length of the field amplitude is, within a constant factor, equal to the length scale defined by the dimension density. This suggests that a correlation measurement may suffice to estimate the fractal dimension of some large homogeneous chaotic systems.

chao-dyn↗

arXiv1993

Dependence of extensive chaos on the spatial correlation length (substantial revision)

We consider spatiotemporal chaotic systems for which spatial correlation functions decay substantially over a length scale xi (the spatial correlation length) that is small compared to the system size L. Numerical simulations suggest that such systems generally will be extensive, with the fractal dimension D growing in proportion to the system volume for sufficiently large systems (L >> xi). Intuitively, extensive chaos arises because of spatial disorder. Subsystems that are sufficiently separated in space should be uncorrelated and so contribute to the fractal dimension in proportion to their number. We report here the first numerical calculation that examines quantitatively how one important characterization of extensive chaos---the Lyapunov dimension density---depends on spatial disorder, as measured by the spatial correlation length xi. Surprisingly, we find that a representative extensively chaotic system does not act dynamically as many weakly interacting regions of size xi.

chao-dyn↗

arXiv1993

Spatial Variation of Correlation Times for 1D Phase Turbulence

For one-dimensional phase-turbulent solutions of the Kuramoto-Sivashinsky equation with rigid boundary conditions, we show that there is a substantial variation of the correlation time~$τ_c(x)$ with spatial position $x$ in moderately large systems of size $L$. These results suggest that some time-averaged properties of spatiotemporal chaos do not become homogeneous away from boundaries for large systems and for long times.

chao-dyn↗

arXiv1993

Stochastic to deterministic crossover of fractal dimension for a Langevin equation

Using algorithms of Higuchi and of Grassberger and Procaccia, we study numerically how fractal dimensions cross over from finite-dimensional Brownian noise at short time scales to finite values of deterministic chaos at longer time scales for data generated from a Langevin equation that has a strange attractor in the limit of zero noise. Our results suggest that the crossover occurs at such short time scales that there is little chance of finite-dimensional Brownian noise being incorrectly identified as deterministic chaos.

chao-dyn↗