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At least 73 records · Page 4

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DOE OSTI2020

Chaos, percolation and the coronavirus spread: a two-step model

We discuss a two-step model for the rise and decay of a new coronavirus (Severe Acute Respiratory Syndrome-CoV-2) first reported in December 2019, COVID-19. The first stage is well described by the same equation for turbulent flows, population growth and chaotic maps: a small number of infected, d 0 , grows exponentially to a saturation value, d ∞ . The typical growth time (aggressive spreading of the virus) is given by τ = 1/λ where λ is the Lyapunov exponent. After a time t crit determined by social distancing and/or other measures, the spread decreases exponentially analogous to nuclear decays and non-chaotic maps. Some countries, like China, S. Korea and Italy, are in this second stage while others including the USA are near the end of the growth stage. The model predicted 15,000 (±2250) casualties for the Lombardy region (Italy) at the end of the spreading around May 10, 2020. Without the quarantine, the casualties would have been more than 50,000, one hundred days after the start of the pandemic. Here, the data from the 50 US states are of very poor quality because of an extremely late and confused response to the pandemic, resulting unfortunately in a large number of casualties, more than 70,000 on May 6, 2020, and more than 170,000 on August 21, 2020. S. Korea, notwithstanding the high population density (511/km 2 ) and the closeness to China, responded best to the pandemic with 255 deceased as of May 6, 2020, and 301 on August 21, 2020.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

The multi-faceted inverted harmonic oscillator: Chaos and complexity

The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Fast Dynamic Aperture Optimization with Reversal Integration

A fast method for dynamic aperture (DA) optimization of storage rings has been developed through the use of reversal integration. Even if dynamical systems have an exact reversal symme- try, a numerical forward integration di ers from its reversal. For chaotic trajectories, cumulative round-o errors are scaled, which results in an exponential growth in the di erence. The exponential e ect, intrinsically associated with the Lyapunov exponent, is a generic indicator of chaos because it represents the sensitivity of chaotic motion to an initial condition. A chaos indicator of the charged particle motion is then obtained by comparing the forward integrations of particle trajectories with corresponding reversals, a.k.a. \backward integrations." The indicator was con rmed to be ob- servable through short-term particle tracking simulations. Therefore, adopting it as an objective function could speed up optimization. The DA of the National Synchrotron Light Source II storage ring, and another test di raction-limited light source ring, were optimized using this method for the purpose of demonstration.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems

We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Computational chaos in massively parallel neural networks

A fundamental issue which directly impacts the scalability of current theoretical neural network models to massively parallel embodiments, in both software as well as hardware, is the inherent and unavoidable concurrent asynchronicity of emerging fine-grained computational ensembles and the possible emergence of chaotic manifestations. Previous analyses attributed dynamical instability to the topology of the interconnection matrix, to parasitic components or to propagation delays. However, researchers have observed the existence of emergent computational chaos in a concurrently asynchronous framework, independent of the network topology. Researcher present a methodology enabling the effective asynchronous operation of large-scale neural networks. Necessary and sufficient conditions guaranteeing concurrent asynchronous convergence are established in terms of contracting operators. Lyapunov exponents are computed formally to characterize the underlying nonlinear dynamics. Simulation results are presented to illustrate network convergence to the correct results, even in the presence of large delays.

Barhen, Jacob↗

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 NTRS1991

Chaos in the sunspot cycle - Analysis and prediction

The variability of solar activity over long time scales, given semiquantitatively by measurements of sunspot numbers, is examined as a nonlinear dynamical system. First, a discussion of the data set used and the techniques utilized to reduce the noise and capture the long-term dynamics inherent in the data is presented. Subsequently, an attractor is reconstructed from the data set using the method of time delays. The reconstructed attractor is then used to determine both the dimension of the underlying system and also the largest Lyapunov exponent, which together indicate that the sunspot cycle is indeed chaotic and also low dimensional. In addition, recent techniques of exploiting chaotic dynamics to provide accurate, short-term predictions are utilized in order to improve upon current forecasting methods and also to place theoretical limits on predictability extent. The results are compared to chaotic solar-dynamo models as a possible physically motivated source of this chaotic behavior.

Mundt, Michael D.↗

NASA NTRS1991

An investigation of chaotic Kolmogorov flows

Platt, N.↗

NASA NTRS1991

The Applicability of Nonlinear Systems Dynamics Chaos Measures to Cardiovascular Physiology Variables

Three measures of nonlinear chaos (fractal dimension, Approximate Entropy (ApEn), and Lyapunov exponents) were studied as potential measures of cardiovascular condition. It is suggested that these measures have potential in the assessment of cardiovascular condition in environments of normal cardiovascular stress (normal gravity on the Earth surface), cardiovascular deconditioning (microgravity of space), and increased cardiovascular stress (lower body negative pressure (LBNP) treatments).

Hooker, John C.↗

NASA NTRS1992

Nonlinear techniques for forecasting solar activity directly from its time series

Numerical techniques for constructing nonlinear predictive models to forecast solar flux directly from its time series are presented. This approach makes it possible to extract dynamical invariants of our system without reference to any underlying solar physics. We consider the dynamical evolution of solar activity in a reconstructed phase space that captures the attractor (strange), given a procedure for constructing a predictor of future solar activity, and discuss extraction of dynamical invariants such as Lyapunov exponents and attractor dimension.

Ashrafi, S.↗

NASA NTRS1992

Detecting and disentangling nonlinear structure from solar flux time series

Interest in solar activity has grown in the past two decades for many reasons. Most importantly for flight dynamics, solar activity changes the atmospheric density, which has important implications for spacecraft trajectory and lifetime prediction. Building upon the previously developed Rayleigh-Benard nonlinear dynamic solar model, which exhibits many dynamic behaviors observed in the Sun, this work introduces new chaotic solar forecasting techniques. Our attempt to use recently developed nonlinear chaotic techniques to model and forecast solar activity has uncovered highly entangled dynamics. Numerical techniques for decoupling additive and multiplicative white noise from deterministic dynamics and examines falloff of the power spectra at high frequencies as a possible means of distinguishing deterministic chaos from noise than spectrally white or colored are presented. The power spectral techniques presented are less cumbersome than current methods for identifying deterministic chaos, which require more computationally intensive calculations, such as those involving Lyapunov exponents and attractor dimension.

Ashrafi, S.↗

NASA NTRS1993

Nonlinear techniques for forecasting solar activity directly from its time series

This paper presents numerical techniques for constructing nonlinear predictive models to forecast solar flux directly from its time series. This approach makes it possible to extract dynamical in variants of our system without reference to any underlying solar physics. We consider the dynamical evolution of solar activity in a reconstructed phase space that captures the attractor (strange), give a procedure for constructing a predictor of future solar activity, and discuss extraction of dynamical invariants such as Lyapunov exponents and attractor dimension.

Ashrafi, S.↗

NASA NTRS1994

Modal decomposition of Hamiltonian variational equations

Over any finite arc of trajectory, the variational equations of a Hamiltonian system can be separated into 'normal' modes. This transformation is canonical, and the Lyapunov exponents over the trajectory arc occur as positive/negative pairs for conjugate modes, while the modal vectors remain unit vectors. This decomposition effectively solves the variational equations for any canonical, linear-dependent system. As an example, we study the Voyager I trajectory. In an interplanetary flyby, some of the modal variables increase by very large multiplicative factors, but this means that their conjugate modal variables decrease by those same very large multiplicative vectors. Maneuver strategies for this case are explored, and the minimum delta upsilon maneuver is found.

Wiesel, William E.↗

NASA NTRS1995

Neurodynamic oscillators

Oscillation of electrical activity has been found in many nervous systems, from invertebrates to vertebrates including man. There exists experimental evidence of very simple circuits with the capability of oscillation. Neurons with intrinsic oscillation have been found and also neural circuits where oscillation is a property of the network. These two types of oscillations coexist in many instances. It is nowadays hypothesized that behind synchronization and oscillation there is a system of coupled oscillators responsible for activities that range from locomotion and feature binding in vision to control of sleep and circadian rhythms. The huge knowledge that has been acquired on oscillators from the times of Lord Rayleigh has made the simulation of neural oscillators a very active endeavor. This has been enhanced with more recent physiological findings about small neural circuits by means of intracellular and extracellular recordings as well as imaging methods. The future of this interdisciplinary field looks very promising; some researchers are going into quantum mechanics with the idea of trying to provide a quantum description of the brain. In this work we describe some simulations using neuron models by means of which we form simple neural networks that have the capability of oscillation. We analyze the oscillatory activity with root locus method, cross-correlation histograms, and phase planes. In the more complicated neural network models there is the possibility of chaotic oscillatory activity and we study that by means of Lyapunov exponents. The companion paper shows an example of that kind.

Espinosa, Ismael↗

NASA NTRS1994

Mixing by barotropic instability in a nonlinear model

A global, nonlinear, equivalent barotropic model is used to study the isentropic mixing of passive tracers by barotropic instability. Basic states are analytical zonal-mean jets representative of the zonal-mean flow in the upper stratosphere, where the observed 4-day wave is thought to be a result of barotropic, and possibly baroclinic, instability. As is known from previous studies, the phase speed and growth rate of the unstable waves is fairly sensitive to the shape of the zonal-mean jet; and the dominant wave mode at saturation is not necessarily the fastest growing mode; but the unstable modes share many features of the observed 4-day wave. Lagrangian trajectories computed from model winds are used to characterize the mixing by the flow. For profiles with both midlatitude and polar modes, mixing is stronger in midlatitude than inside the vortex; but there is little exchange of air across the vortex boundary. There is a minimum in the Lyapunov exponents of the flow and the particle dispersion at the jet maximum. For profiles with only polar unstable modes, there is weak mixing inside the vortex, no mixing outside the vortex, and no exchange of air across the vortex boundary. These results support the theoretical arguments that, whether wave disturbances are generated by local instability or propagate from other regions, the mixing properties of the total flow are determined by the locations of the wave critical lines and that strong gradients of potential vorticity are very resistant to mixing.

Bowman, Kenneth P.↗

NASA NTRS1993

Solar flux forecasting using mutual information with an optimal delay

Solar flux F(sub 10.7) directly affects the atmospheric density, thereby changing the lifetime and prediction of satellite orbits. For this reason, accurate forecasting of F(sub 10.7) is crucial for orbit determination of spacecraft. Our attempts to model and forecast F(sub 10.7) uncovered highly entangled dynamics. We concluded that the general lack of predictability in solar activity arises from its nonlinear nature. Nonlinear dynamics allow us to predict F(sub 10.7) more accurately than is possible using stochastic methods for time scales shorter than a characteristic horizon, and with about the same accuracy as using stochastic techniques when the forecasted data exceed this horizon. The forecast horizon is a function of two dynamical invariants: the attractor dimension and the Lyapunov exponent. In recent years, estimation of the attractor dimension reconstructed from a time series has become an important tool in data analysis. In calculating the invariants of the system, the first necessary step is the reconstruction of the attractor for the system from the time-delayed values of the time series. The choice of the time delay is critical for this reconstruction. For an infinite amount of noise-free data, the time delay can, in principle, be chosen almost arbitrarily. However, the quality of the phase portraits produced using the time-delay technique is determined by the value chosen for the delay time. Fraser and Swinney have shown that a good choice for this time delay is the one suggested by Shaw, which uses the first local minimum of the mutual information rather than the autocorrelation function to determine the time delay. This paper presents a refinement of this criterion and applies the refined technique to solar flux data to produce a forecast of the solar activity.

Ashrafi, S.↗

NASA NTRS1996

Regular and Chaotic Quantum Dynamics of Two-Level Atoms in a Selfconsistent Radiation Field

Dynamics of two-level atoms interacting with their own radiation field in a single-mode high-quality resonator is considered. The dynamical system consists of two second-order differential equations, one for the atomic SU(2) dynamical-group parameter and another for the field strength. With the help of the maximal Lyapunov exponent for this set, we numerically investigate transitions from regularity to deterministic quantum chaos in such a simple model. Increasing the collective coupling constant b is identical with 8(pi)N(sub 0)(d(exp 2))/hw, we observed for initially unexcited atoms a usual sharp transition to chaos at b(sub c) approx. equal to 1. If we take the dimensionless individual Rabi frequency a = Omega/2w as a control parameter, then a sequence of order-to-chaos transitions has been observed starting with the critical value a(sub c) approx. equal to 0.25 at the same initial conditions.

Konkov, L. E.↗

NASA NTRS2000

Controlling Chaos Via Knowledge of Initial Condition for a Curved Structure

Nonlinear response of a flexible curved panel exhibiting bifurcation to fully developed chaos is demonstrated along with the sensitivity to small perturbation from the initial conditions. The response is determined from the measured time series at two fixed points. The panel is forced by an external nonharmonic multifrequency and monofrequency sound field. Using a low power time-continuous feedback control, carefully tuned at each initial condition, produces large long-term effects on the dynamics toward taming chaos. Without the knowledge of the initial conditions, control may be achieved by destructive interference. In this case, the control power is proportional to the loading power. Calculation of the correlation dimension and the estimation of positive Lyapunov exponents, in practice, are the proof of chaotic response.

Maestrello, L.↗