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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS2003

Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem

Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.

dynamical systems theory↗
DOE OSTI2024

Lyapunov exponent as a signature of dissipative many-body quantum chaos

A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large- q limit of a q -body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q ≥ 4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024

García-García, Antonio M.↗
DOE OSTI2024

Sparsity-Independent Lyapunov Exponent in the Sachdev-Ye-Kitaev Model

The saturation of a recently proposed universal bound on the Lyapunov exponent has been conjectured to signal the existence of a gravity dual. This saturation occurs in the low-temperature limit of the dense Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions with q body ( q > 2 ) infinite-range interactions. We calculate certain out-of-time-order correlators (OTOCs) for N ≤ 64 fermions for a highly sparse SYK model and find no significant dependence of the Lyapunov exponent on sparsity up to near the percolation limit where the Hamiltonian breaks up into blocks. This provides strong support to the saturation of the Lyapunov exponent in the low-temperature limit of the sparse SYK. A key ingredient to reaching N = 64 is the development of a novel quantum spin model simulation library that implements highly optimized matrix-free Krylov subspace methods on graphical processing units. This leads to a significantly lower simulation time as well as vastly reduced memory usage over previous approaches, while using modest computational resources. Strong sparsity-driven statistical fluctuations require both the use of a much larger number of disorder realizations with respect to the dense limit and a careful finite size scaling analysis. The saturation of the bound in the sparse SYK points to the existence of a gravity analog that would enlarge substantially the number of field theories with this feature. Published by the American Physical Society 2024

Physics↗
NASA NTRS2003

Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem

Dynamical systems theory has recently been employed for several missions to design trajectories within the three-body problem. This research applied a stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time.

local Lyapunov exponent design navigation three-bo↗
NASA NTRS2015

Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies

Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation L alpha m(sup p) between themean Lyapunov exponent L of stellar orbits chaotically scattered by a supermassive black hole (BH) in the centre of a galaxy and the mass parameter m, i.e. ratio of the mass of the BH over the mass of the galaxy. The exponent p is found numerically to obtain values in the range p approximately equals 0.3-0.5. We propose a theoretical interpretation of these exponents, based on estimates of local 'stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the BH's sphere of influence. We thus predict p = 2/3 − q with q approximately equaling 0.1-0.2. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power-law scaling of L with m holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the BH affects mainly populations of thick tube-like orbits surrounding some low-order branches of the x(sub 1) family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the inner Lindblad resonance and the 4/1 resonance. Implications of the correlations between L and m to determining the rate of secular evolution of galaxies are discussed.

chaos aEuro" galaxies:↗
DOE OSTI2023

Out-of-time-order correlators and Lyapunov exponents in sparse SYK

We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order of N, the standard result for the q-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter, k. We present an algorithm to sum the diagrams at any given order of 1/(kq) n . We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finite N becomes more significant for larger values of N.

2D gravity↗
NASA NTRS1995

Lyapunov exponents from CHUA's circuit time series using artificial neural networks

In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks.

Gonzalez, J. Jesus↗
NASA NTRS1990

Short-time Lyapunov exponent analysis

A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.

Vastano, J. A.↗
NASA NTRS1987

Lyapunov exponents for infinite dimensional dynamical systems

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

Mhuiris, Nessan Mac Giolla↗
NASA NTRS1991

Lyapunov exponent of magnetospheric activity from AL time series

A correlation dimension analysis of the AE index indicates that the magnetosphere behaves as a low-dimensional chaotic system with a dimension close to 4. Similar techniques are used to determine if the system's behavior is due to an intrinsic sensitivity to initial conditions and thus is truly chaotic. The quantity used to measure the sensitivity to initial conditions is the Liapunov exponent. Its calculation for AL shows that it is nonzero (0.11 + or - 0.05/min). This gives the exponential rate at which initially similar configurations of the magnetosphere evolve into completely different states. Also, predictions of deterministic nonlinear models are expected to deviate from the observed behavior at the same rate.

Vassiliadis, D.↗
NASA NTRS1991

Short-time Lyapunov exponent analysis and the transition to chaos in Taylor-Couette flow

The physical mechanism driving the weakly chaotic Taylor-Couette flow is investigated using the short-time Liapunov exponent analysis. In this procedure, the transition from quasi-periodicity to chaos is studied using direct numerical 3D simulations of axially periodic Taylor-Couette flow, and a partial Liapunov exponent spectrum for the flow is computed by simultaneously advancing the full solution and a set of perturbations. It is shown that the short-time Liapunov exponent analysis yields more information on the exponents and dimension than that obtained from the common Liapunov exponent calculations. Results show that the chaotic state studied here is caused by a Kelvin-Helmholtz-type instability of the outflow boundary jet of Taylor vortices.

Vastano, John A.↗
DOE OSTI2022

Holographic Kolmogorov-Sinai entropy and the quantum Lyapunov spectrum

In classical chaotic systems the entropy, averaged over initial phase space distributions, follows a universal behavior. While approaching thermal equilibrium it passes through a stage where it grows linearly, while the growth rate, the Kolmogorov-Sinai entropy (rate), is given by the sum over all positive Lyapunov exponents. A natural question is whether a similar relation is valid for quantum systems. We argue that the Maldacena-Shenker-Stanford bound on quantum Lyapunov exponents implies that the upper bound on the growth rate of the entropy, averaged over states in Hilbert space that evolve towards a thermal state with temperature T, should be given by πT times the thermal state’s von Neumann entropy. Strongly coupled, large N theories with black hole duals should saturate the bound. To test this we study a large number of isotropization processes of random, spatially homogeneous, far from equilibrium initial states in large N, $\mathcal{N}$ = 4 Super Yang Mills theory at strong coupling and compute the ensemble averaged growth rate of the dual black hole’s apparent horizon area. We find both an analogous behavior as in classical chaotic systems and numerical evidence that the conjectured bound on averaged entropy growth is saturated granted that the Lyapunov exponents are degenerate and given by λ i = ±2πT. This fits to the behavior of classical systems with plus/minus symmetric Lyapunov spectra, a symmetry which implies the validity of Liouville’s theorem.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Many-body quantum dynamics slows down at low density

We study quantum many-body systems with a global U(1) conservation law, focusing on a theory of N interacting fermions with charge conservation, or N interacting spins with one conserved component of total spin. We define an effective operator size at finite chemical potential through suitably regularized out-of-time-ordered correlation functions. The growth rate of this density-dependent operator size vanishes algebraically with charge density; hence we obtain new bounds on Lyapunov exponents and butterfly velocities in charged systems at a given density, which are parametrically stronger than any Lieb-Robinson bound. We argue that the density dependence of our bound on the Lyapunov exponent is saturated in the charged Sachdev-Ye-Kitaev model. We also study random automaton quantum circuits and Brownian Sachdev-Ye-Kitaev models, each of which exhibit a different density dependence for the Lyapunov exponent, and explain the discrepancy. We propose that our results are a cartoon for understanding Planckian-limited energy-conserving dynamics at finite temperature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Scrambling versus relaxation in Fermi and non-Fermi liquids

In this work, we compute the Lyapunov exponent characterizing quantum scrambling in a family of generalized Sachdev- Ye-Kitaev models, which can be tuned between different low temperature states from Fermi liquids, through non-Fermi liquids, to fast scramblers. The analytic calculation, controlled by a small coupling constant and large $\textit{N}$, allows us to clarify the relations between the quasiparticle relaxation rate $\textit{1/τ}$ and the Lyapunov exponent $λ_L$ characterizing scrambling. In the Fermi liquid states we find that the quasiparticle relaxation rate dictates the Lyapunov exponent. In non-Fermi liquids, where $1/τ \gg T$, we find that $λ_L$ is always $\textit{T}$ linear with a prefactor that is independent of the coupling constant in the limit of weak coupling. Instead it is determined by a scaling exponent that characterizes the relaxation rate. $λ_L$ approaches the general upper bound 2$\textit{πT}$ at the transition to a fast scrambling state. Finally, in a marginal Fermi liquid state the exponent is linear in temperature with a prefactor that vanishes as a nonanalytic function ~$g\text{ln}(1/g)$ of the coupling constant $\textit{g}$.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1989

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.↗
DOE OSTI2021

The divergence of nearby trajectories in soft-sphere DEM

The n-body instability is investigated with the soft-sphere discrete element method. The divergence of nearby trajectories is quantified by the dynamical memory time. Using the inverse proportionality between the dynamical memory time and the largest Lyapunov exponent, the soft-sphere discrete element method results are compared to previous hard-sphere molecular dynamics data for the first time. Good agreement is observed at low concentrations and the degree of instability is shown to increase asymptotically with increasing spring stiffness. At particle concentrations above 30%, the soft-sphere Lyapunov exponents increase faster than the corresponding hard-sphere data. Finally, this paper concludes with a demonstration of how this case study may be used in conjunction with regression testing and code verification activities.

42 ENGINEERING↗
DOE OSTI2021

The quantum p -spin glass model: a user manual for holographers

We study a large-N bosonic quantum mechanical sigma-model with a spherical target space subject to disordered interactions, more colloquially known as the p-spin spherical model. Replica symmetry is broken at low temperatures and for sufficiently weak quantum fluctuations, which drives the system into a spin glass (SG) phase. The first half of this paper is dedicated to a discussion of this model's thermodynamics, with particular emphasis on the marginally stable SG. This phase exhibits an emergent conformal symmetry in the strong coupling regime, which dictates its thermodynamic properties. It is associated with an extensive number of nearby states in the free energy landscape. We discuss in detail an elegant approximate solution to the SG equations, which interpolates between the conformal regime and an ultraviolet-complete short distance solution. In the second half of this paper we explore the real-time dynamics of the model and uncover quantum chaos as measured by out-of-time-order four-point functions, both numerically and analytically. We find exponential Lyapunov growth, which intricately depends on the model's couplings and becomes strongest in the quantum critical regime. We emphasize that the SG phase also exhibits quantum chaos, albeit with parametrically smaller Lyapunov exponent than in the replica symmetric phase. An analytical calculation in the marginal SG phase suggests that this Lyapunov exponent vanishes in a particular infinite coupling limit. Finally, we comment on the potential meaning of these observations from the perspective of holography.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗